[    {        "title": "0909.4609v1.Electrical__magnetic__magnetodielectric_and_magnetoabsorption_studies_in_multiferroic_GaFeO3.pdf",        "content": "Electrical, magnetic, magnetodielectric and magnetoabsorption \nstudies in multiferroic GaFeO 3 \n \nV. B. Naik and R. Mahendiran1  \nDepartment of Physics and NUS Nanos cience & Nanotechnology Initiative \n(NUSNNI), Faculty of Science, National University of Singapore, \n2 Science Drive 3, Singapore -117542, Singapore \n \nAbstract \nWe report electrical, magnetic, magnetodielec tric and magnetoabsorp tion properties of a \npolycrystalline GaFeO 3. The resistivity measurement shows that the sample is highly \ninsulating below 200 K and the resistivity a bove 200 K obey the Arrhenius law with an \nactivation energy of Ea = 0.67 eV. An anomaly occurs in  the temperature dependence of \npermittivity ( ) near the ferrimagnetic transition temperature (T C = 228 K) in a zero \nmagnetic field and it is suppressed under 0H = 60 mT which indicates a possible \nmagnetoelectric coupling in GaFeO 3 with a fractional change of / = -1.8% at 60 mT \naround TC. The coercivity (H C) of the sample increases dr amatically with lowering \ntemperature below 200 K from 0.1 T at 200 K to 0.9 T at 5 K. Magnetoabsorption was \nstudied with a LC resonance technique and we found a close correlation between the shift \nin the resonance frequency due to applied ma gnetic field and the coercive field measured \nusing dc magnetization measurements. Our resu lts obtained with multiple techniques \n                                                 \n1 Corresponding author – phyrm@nus.edu.sg  \n 1suggest that GaFeO 3 is an interesting ferrimagnet with potential applications in future \nmultiferroic devices. \n PACS number(s): 75.50.Gg, 75.60.Ej, 77.22.Ch, 77.22.Ej          \n \n         \n 2I.  INTRODUCTION  \n \nThe m\nagnetoelectric (ME) multiferroic materials which show a strong coupling \nbetween ferromagnetic and ferro electric order parameters ar e promising for applications \nin new kind of multistate non volatile memories such as magnetically tunable \nferroelectric random access memories (FeRA Ms), electrically tunable magnetic random \naccess memories (MRAMs) and high frequency filters etc.1,2,3,4,5 In addition, the \nobservations of magnetic control of ferroelectric polarization in TbMnO 3,6  electric-field-\ninduced spin flop in BiFeO 3,7 magnetic-field-induced ferroelectric state in DyFeO 38 and  \nthe ME memory effect in MnWO 49\n are quite interesting from  both a fundamental and \ntechnical perspectives. However, one of th e biggest challenges facing the field of \nmultiferroics is the need for room temperature multifunctionality, since there exist a very few single-phase multiferroic materials. Because, ferromagnetism requires an odd \nnumber of d-electrons while the ferroelectricity  generally occurs only  in the absence of d-\nelectrons.\n10 This apparent incompatibility to s how multiferroicit y can indeed be \novercome in materials such as BiFeO 3, GaFeO 3 and rare earth manganites of AMnO 3 (A \n= Y, Tb, Gd, Ho)11 in which A and Mn are the sources of ferroelectricity and magnetism, \nrespectively. \n \nThe magnetic and magnetoelectric properties of GaFeO 3 was a topic of intensive \nresearch in 1960 since Remeika et al .12 reported the occurrence of piezoelectricity at \nroom temperature above the ferromagnetic transition temperature TC = 260 K in Ga 2-\nxFexO3 (x =1). Soon after, Rado et al.13 observed the magnetoelectric effect in Ga 2-xFexO3 \n 3(x ≈ 1\n) and showed this effect to be larger by an order of magnit ude than previously \nknown magnetoelectric material, Cr 2O3.14  Petrov et al .15 found that the line shape and \nthe line width of electron spin resonance spectra of the piezoelectric ferrimagnet \nGa0.85Fe1.15O3 were affected by an external electrical  field and that could not be attributed \nto change in the conductiv ity. Folen and Rado suggested16 the electric field induced \nchange in the electron spin res onance line width observed by Petrov et al.15 could have \nbeen caused by Joule heating. Kaneko et al.17 found that the TC in single crystals of Ga 2-\nxFexO3 can be continuously increased from 120 K for x = 0.08 to 370 K for x = 0.14.    \n \nGaFeO 3 crystallizes in an orthorhombic structure with space group Pc21n.18 This \ncompound has a spontaneous polarization along the b axis and has a collinear \nferrimagnetic structure19 where the ferrimagnetism is due to the unequal distribution of \nFe spins of nearly equal magnitude on the subl attices with a magnetic moment of the spin \nalong the c  axis (instead of a canted antiferroma gnetic structure as previously assumed13 \nor inferred20). Recent reports of magnetization- induced second harmonic generation,21 \noptical22 and dc23 ME effect, and the ultrafast electric  and magnetic response of GaFeO 3 \ninduced by irradiation of a femtosecond laser pulse24 makes this compound more \nattractive for potential app lications. Moreover, GaFeO 3 was reported to show the largest \nFaraday rotation.25 However, detailed magnetic and electrical studies in GaFeO 3 \npolycrystal have not yet been  reported. In this work, we report electrical, magnetic, \nmagnetodielectric and magnetoabsorption pr operties of a polycrystalline GaFeO 3 sample.  \n \nII. EXPERIMENT DETAILS  \n 4Polycrystalline sample of GaFeO 3 was synthesized using a conventional solid state \nroute. The solid solution was prepared by mixing the stoichiometric mixtures of Ga 2O3 \nand Fe 2O3 precursors, and was homogenized in an ag ate mortar before it reacted at 1000 \n°C for 12 hours. Final sintering was done at 1200 °C for 12 hours. Single phase \nidentification was performed by the powder X -ray diffraction (XRD) experiment (Philips \nX’pert Pro) using Cu K α radiation. The compound was pressed into pellet and heat \ntreated at 1250 °C for 5 hours to get a relati vely dense pellet. A parallel plate capacitor \nstructure was made for dielectric and ferroelec tric measurements by using the silver paint \nto make electrodes. The typical dimensions ( A – area and t – thickness) of the samples for \ndielectric and ferroelectric measurements were A = 10.56 mm2, t = 0.91 mm and A = 9.11 \nmm2, t = 0.38 mm, respectively. Temperature depe ndence of dielectr ic properties was \nstudied using the Agilent 4285A Precision LC R meter and an opti cal cryostat (Janis \nmodel CCS102). Two probe resistivity m easurement was carried out using an \nelectrometer (Keithley 6517A) with Janis cryo stat. A precision LC ferroelectric tester \n(Radiant Technologies) was used to m easure the ferroelectric properties ( P-E loops). \nTemperature and field dependences of ma gnetization measurements were performed \nusing a vibrating sample magnetometer (VSM) with a superc onducting magnet and a \ncommercial cryostat (PPMS, Quantum Desi gn Inc). Temperature and magnetic field \ndependent radio-frequency ( rf) electromagnetic absorption has been investigated using a \nhome built LC resonant circuit powered by an integrated chip oscillator (ICO)26 with \nPPMS.  \n \nIII.  RESULTS AND DISCUSSION  \n 5Fig. 1(a) shows the XRD pattern of the GaFeO 3 compound at room temperature, \nwhich reveals the single phase pattern quite similar to the JCPDS-ICDD 76-1005.27 The \nobserved XRD pattern (blue color) was indexed (red color) to orthorhombic structure of \nGaFeO 3 with space group Pc21n (Pna21, recommended notation used in the international \ntable) using TOPAS software  version 2.1. The prominent peaks in the XRD pattern are \nindexed by its crystallographic planes, which are represented by the miller indices ( hkl). \nThe lattice parameters of the compound determ ined by the above Reitveld refinement are \nfound to be a = 8.74042 Å, b = 9.38931 Å and  c = 5.07955 Å which are in close \nagreement with earlier report.27 There is no report on the resistivity measurement for \nGaFeO 3 compound, perhaps because the compound is highly insulating. Temperature \ndependence of the resistivity, (T) is shown in the fig. 1(b) in the temperature interval of \n360 K to 265 K. The (T) increases from 50 k m at 360 K to 60 M m at 265 K. The \ninset in the fig. 1(b) shows the Arrhenius plot  for the resistivity which reveals the linear \nrelationship between ln  and 1/ T. The activation energy deduced from an Arrhenius \nlaw, = 0 exp ( Ea/kBT) is found to be Ea = 0.675 eV.  \n \nThe fig. 2(a) shows the temperature dependences of the magnetization ( M) under \ndifferent magnetic fields, 0H = 0.2 T, 0.5 T, 1 T, 2 T and 5 T. The rapid increase of \nmagnetization in a magnetic field of 0.2 T at TC ≈ 228 K  (calculated from the maximum \nchange in the slope) signals the onset  of ferrimagnetic transition in GaFeO 3. The \nmagnetization gradually increases with lowering temperature below TC and becomes \nnearly temperature independent as it approaches to the lowest temperature T = 10 K. The \nferrimagnetic transition broadens with increasing strength of the external magnetic field.  \n 6The m\nagnitude of M at 10 K is 0.667 B/Fe atom under the applied field of 0H = 5 T. \nThe sample shows well defined hysteresi s loops below 230 K [fig. 2(b)]. The \nspontaneous magnetization ( MS) obtained from the extra polation of the high field M to H  \n= 0 decreases appreciably with incr easing temperature. The value of MS at 5 K is 0.45 \nB/Fe atom. The significant point here is that the coercive field ( HC) is tremendously \nlarge which elucidates the hard magnetic material nature of GaFeO 3, and it increases \nrapidly below 200 K from 0.1 T at 200 K to 0.9 T at 5 K [fig. 2(c)].  \n \nTemperature dependences of the dielectric constant [ (T)] and dissipation factor \n(D = tan) in zero field  for three different frequencies (100 kHz, 500 kHz and 1 MHz) are \nshown in the main panels of the fig. 3. The (T) showed an anomaly near ferrimagnetic \ntransition temperature, TC ≈ 228 K, but the D  does not show any clear anomaly close to \nTC. Interestingly, a small magnetic field of 0H = 60 mT suppresses the dielectric \nanomaly found near TC, which is shown in the inset of the fig. 3(a) for  f = 1 MHz (left \nscale). The magnetodielectric  (MD) coefficient calculated from the equation MD = \n[(H)(0)(0) is plotted as a function of temperature in the inset of the fig. 3(a) \n(right scale) and it is found to be -1.8% close to TC in GaFeO 3. The temperature \ndependence of dissipation factor in 0H = 0 and 60 mT for f = 1 MHz shown in the inset \nof the fig. 3(b) does not show up any change in  response to the applie d field as like the \n(T). In a ferroelectromagnet, the difference in the dielectric constant [ = (H)(0)] \nbelow TC is proportional to the square of the magnetization i.e., M2, where  is the \nmagnetoelectric coupling constant.28 Z H Sun et al.29 has observed a linear relationship \nbetween and M 2 as an indication of the magnetoelec tric (ME) coupli ng, and reported \n 7the m\nagnetocapacitance (MC) of -0.5 % close to TC in GaFeO 3 based on the extrapolation \nof the zero field anomaly. The appearance of dielectric anomaly around TC in zero \nmagnetic field and its suppression in a small magnetic field of 0H = 60 mT suggest an \nactive magnetoelectric coupling in our GaFeO 3 sample. A similar weak anomaly has been \nfound in (T) in the hexagonal YMnO 330 at T N = 70 K. Very recently, T. Kimura et al.31 \nhas found a pronounced anomaly in (T) at the onset of paramagnetic to spiral \nantiferromagnetic transition ( TC = 270 K) in CuO. It is also  to be noted that the observed \n-1.8% MD coupling coefficient in our sample is higher than -0.6%  MC observed in the \nferromagnetic BiMnO 328 at T C ≈ 100 K and 0H = 9 T, and lower than -8% MD effect \nfound in the E type antiferromagnetic HoMnO 332\n at T = 4.5 K and 0H = 7 T. \n \nThe P-E loops measured at f = 1 kHz (frequency of the hysteresis cycle or \nmeasurement frequency) at selected temperatures, T = 150 K, 200 K and 225 K are \nshown in the fig. 4(a). The hysteresis loop observed at T = 225 K is unsaturated and \nrounded at the highest field, which reveal s the leakage curren t contribution that \novershadows the true polarization due to the or ientation of the electr ic dipoles. Because, \nfor an insulating ferroelectric material, the switched charge Q due to applied electric field \nE depends only on the remanent polarization Pr through the relation Q = 2PrA where A is \nthe surface area of the capacitor, wherea s for a lossy dielectric material, extra \ncontribution comes from the conductivity  through the relation Q = 2PrA + EAt, where \nt is the time for hysteresis measurement.33 The P-E loops at T  = 200 K and 150 K shown \nin fig. 4(a) are unsaturated ev en at the higher limit of elect ric field of 5 kV/cm in our \nexperiment, but it elucidates the considerab le suppression of leaky behavior of the \n 8com\npound below T = 225 K as supported by our resi stivity measurement, where the \nresistivity increases exponentially as temper ature decreases, or in other words, the \nconductivity ( ) contribution to the polarization is less at lower temperatures. The highly \nfrequency dependent P-E  loops at T  = 150 K [fig. 4(b)] measured  in a narrow range of \nmeasurement frequency ( f = 1 kHz – 50 Hz) suggest th at our polycrystalline GaFeO 3 \nsample does not appear to be ferroelectric because the P-E loops opens up as the \nmeasurement frequency decreases which is an artifact due to the leakage current \ncontribution.33 However, electrical polarization study at higher electric field is necessary \nto confirm the ferroelectricity in our bulk sample. Next we will see the behavior of magnetic absorption studied by the LC resonance technique.  \n  \nIn the main panel of fig. 5, we show the temperature dependences of the \nresonance frequency (f\nr) (right scale) and the current ( I) through an ICO (l eft scale) for a \nfew dc magnetic fields, 0H = 0 T, 60 mT and 0.2 T applied along the coil axis. \nTemperature dependence of fr and I  for the empty coil is also shown. The fr of an ICO in a \nzero magnetic field ( 0H = 0 T) decreases gradually from T = 300 K, shows a minimum \nclose to the ferrimagnetic transition ( TC = 228 K), and then increases gradually with \nlowering temperature. A small magnetic field of 0H = 60 mT appreciably suppresses the \nminimum in fr and broadens it, but it hardly affects the fr in the temperature range T < 190 \nK and T > 250 K. The peak in fr becomes more broader in a magnetic field of 0H = 0.2 \nT and the behavior of fr is similar to an empty coil. On the other hand, the current I in a \nzero magnetic field showed a gradual decrease  with lowering temperature as like an \nempty coil, except a weak anom aly showed up very close to T C in the temperature \n 9interval of 200 K to 250 K which can be clearly s een in the inset. This is in contrast to the \nlarge decrease in I found at T C for La 0.67Ba0.33MnO 3.26 The weak anomaly present in the \nzero field I is completely suppressed in the magnetic fields of 0H = 60 mT and 0.2 T.  \nThe fig. 6(a) shows the magnetic field dependence of fr at selected temperatures T \n= 10 K – 200 K. These data were taken by monitoring the f r while sweeping the magnetic \nfield from 0H = 0 T → +3 T → -3 T → +3 T. We have not shown the initial field sweep \n(0 T → +3 T) here, since it closely matches with  the down field sweep (+3 T to 0 T). We \nhave also not shown here the field dependence of I, since there has been no considerable \nchange in  I with the magnetic field sweep. The fr at T = 10 K shows a butterfly curve \nwithin the field interval of 0H = +1.5 T – -1.5 T and shows a peak on either side of the \norigin at 0H = ±0.75 T (i.e., HP, the field at which peak occurs) with a large hysteresis, \nbut saturates at the highest field 0H = ±3 T. The field dependent fr at temperatures above \nT = 10 K showed the similar behavior as for T = 10 K, except for T = 200 K where the \npeak is completely vanished with absolutely no hysteresis. The significant point here is \nthat the peak position shifts toward s the origin and the hysteresis in fr becomes narrower \nat higher temperatures ( T  > 10 K) as like the hysteresis in M-H  loops [fig. 2(b)]. This can \nbe clearly concluded from the figures (b) – (e). The magnetic field dependences of M and \nfr at T = 10 K are shown in the figu res (b) and (c), respectively. A large hysteresis is seen \nin both the cases. As descri bed above for fig. 6(a), the fr in the figure (c) showed a peak \nin the negative side at HP = -0.75 T while the field is swept from 0H = +3 T →  -3 T. \nThis position of the peak ( HP) is closely matching with the coercive field ( HC), but it is \nslightly less in value. Interestingly,  the temperature dependences of both HC and H P show \n 10a sim\nilar behavior i.e., the rapid in crease with lowering temperature below T = 200 K and \nare shown in the figures (d ) and (e), respectively.  \n \nThe resonance frequency of an ICO,  fr = 1/(2πLC) where L – the inductance of \nthe empty coil and C – the capacitance in the circuit, changes due to the  change in the \nreal part of rf magnetic permeability ( ′) of the sample. This is unlike the case of a \nmetallic La 0.67Ba0.33MnO 3 which showed an extra contribution arising from the \nincomplete penetration of ac magnetic field due to the induced eddy current.26 Thus, the \nincrease in ′ upon transition from paramagnetic to ferrimagnetic while cooling leads to a \ndecrease in the fr around T C. An external applied magne tic field suppresses the spin \nfluctuations in the sample and leads to a considerable decrease in ′ which in turn \ndecreases the fr at T C. On the other hand, the dynamical magnetic and electrical losses in \nthe sample lead to rf power absorption in the sample th at changes the complex impedance \n(Z) of the tank circuit, which in turn lead s to a change in the current through the ICO \ncircuit. The expression for the impedance of  the inductance coil can be modified for a \nhighly insulating materials (in which electro magnetic field completely penetrates the \nsample) as Z = ( R0+L0)+jL0′, where R 0 is the resistance, L0 is the inductance of \nthe empty coil, and ′ and  are the real and imaginary parts of the permeability of the \nsample, respectively. Therefore, the effective resistance of the coil changes mainly due to \nthe magnetic loss characterized by the  which reflects the electromagnetic power \nabsorption in the sample [P = ½ Hac2Re(Z)],26 and hence the current in the circuit shows \nan anomaly near the ferrimagnetic transiti on. The suppression of this peak in the \n 11m\nagnetic field of 0H = 60 mT and 0.2 T is due to the suppression of ′ and  by the \nmagnetic field.  \n \nThere is a close correlation between the temperature dependence of HC and HP \n[fig. 6 (d) and (e)]. The slight difference in the values of HC and HP is possibly because \nthe change in fr (which is related with th e dynamical magnetization i.e. M/H) is \nmeasured at fr ~ 1 MHz in our ICO experiment and the low frequency ( f <100 kHz) \nmeasurement might lead to a closer agreement between the HC and HP values. It is worth \nmentioning that the coercivity of GaFeO 3 is high, H C = 0.9 T at T  = 5 K. Since \n where MS – saturation magnetization, it im plies a rapid increase in the \nanisotropic constant ( K1) with lowering temper ature. Indeed, Pinto et al .12/ CHK M S\n34 reported \nunusually large anisotropic magnetizati on in single crystal of orthorhombic \nGa0.92Fe1.08O3. The coercivity is expected to increas e if a Ga is replac ed by a rare earth \nion R. For example, H C = 1.5 T, 1.65 T and 1.8 T for R = Dy, Sm and Y, respectively in \nthe RFeO 3 series.35      \n   \nIV.  CONCLUSIONS  \nWe have studied electrical, magnetic, magnetodielectric and magnetoabsorption \nproperties of GaFeO 3. The dielectric permittivity exhibited a weak anomaly at TC which \nis suppressed in 0H = 60 mT suggesting a ME coupling in GaFeO 3. The sample showed \nwell defined magnetic hysteresis loops belo w 230 K with a rapid increase in the \ncoercivity below 200 K from 0.1 T at 200 K to  0.9 T at 5 K. The resistivity of this \ncompound above 200 K obey the Arrhenius law with an activation energy of E a = 0.67 \n 12eV. The P-E loops suggested that the leakage cu rrent is drastically reduced below T = \n200 K. The m\nagnetoabsorption study showed an anomaly in both fr and I  at T C and we \nfound that there is a close correlation between the temperature dependence of HC and HP, \nthe peak found in the fr versus magnetic field. Very  recently, Tokunaga et al.36 showed \nthat GdFeO 3, one of the most orthodox perovskite  oxides, possesses a ferroelectric \nground state in which ferroelectric polarization is generated by exchange interaction \nbetween Gd3+ and Fe3+ ions. They have also demonstrated  the electrical field control of \nmagnetization and the magnetic control of ferr oelectric polarization below 2.5 K. In view \nof these results, it will be interesting to investigate the magnetic and magnetoelectric \neffects in Ga 1-xGdxFeO 3. \n ACKNOWLEDGMENTS  \nR. M. acknowledges the National Research  Foundation (Singapore)  for supporting this \nwork through the grant NRF-CRP-G-2007.   \n \n        \n 13 14                                                References:  \n \n1 S. W. 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Arima and Y. Tokura, Nat. \nMater. 8, 558 (2009). \n \n \n          17                                                                                                                                                 \nFigures : \n \n \n Fig. 1 (color online) (a) Observed (blue colo r) and Reitveld refinement (red color) of \nthe XRD pattern for the GaFeO\n3 compound with space group Pc21n at room \ntemperature, (b) temperature dependence of the resistivity ( ) in a narrow \ntemperature interval (265 K – 360 K) and the inset shows the linear behavior of \nArrhenius plot for the resistivity.       18                                                                                                                                                 \n \n \n \n Fig. 2 (color online) (a) Temperatu re dependences of magnetization (M ) at different \nmagnetic fields (0.2 T – 5 T), (b) fi eld dependences of magnetization ( M) at different \ntemperatures (5 K – 300 K), (c) temp erature dependence of coercive field ( H\nC). \n  19                                                                                                                                                 \n \n \nFig. 3 (color online) Temperature dependen ces of (a) the dielectric constant ( ) and \n(b) the dissipation factor ( D = tan) for f = 100 kHz, 500 kHz and 1 MHz. The insets \nshow the temperature dependences of  and D (left scale) in 0H = 0 and 60 mT for f  \n= 1 MHz. The inset in the figure (a) show s the magnetodielectric (MD) coefficient \n(right scale) as a function of temperature for f = 1 MHz.  \n  20                                                                                                                                                 \n \nFig. 4 (color online) (a) P-E loops at selected temper atures (frequency of the \nhysteresis cycle f = 1 kHz) (b) P-E loops at T  = 150 K with different frequencies of \nthe hysteresis cycle ( f = 1 kHz – 50 Hz). \n  21                                                                                                                                                 \n \n \nFig. 5 (color online) Temperature dependences of the resonance frequency ( fr) (right \nscale) and current ( I) (left scale) through the circu it for different strengths of dc \nmagnetic fields  0H = 0 T, 60 mT and 0.2 T. The da ta for the empty coil are also \nshown. The inset shows the temperature dependence of I in a narrow temperature \ninterval (190 K – 270 K).     22                                                                                                                                                 \n \nFig. 6 (color online) The magnetic field dependences of the (a) resonance frequency \n(fr) at selected temperatures T = 10 K – 200 K, (b) magnetization ( M) at T = 10 K, (c) \nresonance frequency ( fr) at T = 10 K. Temperature dependences of the (d) coercive \nfield ( HC) and (e) position of the peak ( HP) which is observed in the fr versus field.   \n "    },    {        "title": "0906.3201v1.Magnetostructural_Effect_in_the_Multiferroic_BiFeO3_BiMnO3_Checkerboard_from_First_Principles.pdf",        "content": "arXiv:0906.3201v1  [cond-mat.mtrl-sci]  17 Jun 2009Magnetostructural Effect in the Multiferroic BiFeO 3-BiMnO 3Checkerboard from\nFirst Principles\nL. P´ alov´ a, P. Chandra and K. M. Rabe\nDepartment of Physics and Astronomy, Rutgers University, P iscataway, NJ 08854\n(Dated: June 10, 2021)\nUsing first principles calculations, we present a magnetost ructural effect in the BiFeO 3-BiMnO 3\nnanocheckerboard that is not found in either bulk parent com pound or in BiFeO 3-BiMnO 3super-\nlattices. We also demonstrate that the atomic-scale checke rboard has a multiferroic ground state\nwith the desired properties of each constituent material: p olar and ferrimagnetic due to BiFeO 3and\nBiMnO 3respectively.\nPACS numbers: 75.80.+q,77.80.-e,75.75.+a\nThere is currently tremendous interest in finding new\nmultiferroic (ferroelectric and ferromagnetic) materials\nwith large magnetoelectric coupling. Advances in the\nsynthesis of artificially structured materials have stimu-\nlated efforts to design new multiferroic heterostructures,\nwith first principles methods being an essential tool for\nthe identification and investigation of promising systems.\nIn this Letter, we report the first-principles identifica-\ntion and characterization of an unusual heterostructure,\na multiferroic atomic-scale 2D nanocheckerboard1,2,3,4of\nBiFeO 3-BiMnO 3, with properties that critically depend\non the geometry and are not present in either bulk or\nlayered structures of the constituent materials. In par-\nticular, the 2D checkerboard geometry leads to magnetic\nfrustration and to quasi-degenerate magnetic states that\ncan be tuned by an external perturbation that changes\nthe crystal structure, such as an electric field. This re-\nsults in a novel magnetostructural effect, adding to pre-\nvious examples of magnetostructural coupling such as\nbulk5and layered6manganites, epitaxial EuTiO 37and\nEuSe/PbSe 1−xTexmultilayers.8\nOur first principles calculations are performed using\ndensity functional theory within the local spin-density\napproximation (LSDA)+U method as implemented in\nthe Vienna Ab-initio Simulation Package VASP-4.6.349.\nWe test the robustness of our results with two different\nimplementations of the rotationally invariant LSDA+U\nversion, the first as introduced by Liechtenstein10with\nUFe=UMn= 5eV,JFe=JMn= 1eV, and the sec-\nond due to Dudarev11, withUeff\nMn= 5.2eV,Ueff\nFe= 4eV,\nwhereUeff=U−J. It has been shown that these U\nandJvalues match experimental data in bulk BiFeO 312;\nthe value Ueff= 5.2eVhas been used for previous bulk\nBiMnO 3ground state calculations13. We use projector-\naugmented wave potentials (PAW)14,15and treat ex-\nplicitly 15 valence electrons for Bi (5 d106s26p3), 14 for\nFe (3p63d64s2), 13 for Mn (3 p63d54s2), and 6 for O\n(2s22p4). The cutoff energies for the plane wave ba-\nsis set are 550 eVand 800eVin the ionic relaxations and\nfor subsequent self-consistent energy calculations respec-\ntively. Gaussian broadening of the partial occupancies\nfor each wavefunction is 0 .05eV. A Monkhorst-Pack\nk-point grid16is generated with density 4 ×4×4 forFigure 1: (i) BiFeO 3-BiMnO 3superlattice with alternation\nof Fe/Mn planes. (ii) (left) BiFeO 3-BiMnO 3checkerboard.\nCheckerboard ordering of Fe/Mn atoms in the ( xy) plane,\npillars of the same composition form along the z-direction.\n(right) Ideal perovskite unit cell. Perovskite cells with F e/Mn\natoms on the B-site repeat according to the checkerboard pat -\ntern (ii), or layered geometry (i).\n(√\n2×√\n2×1)a0double perovskite and 4 ×4×2 for\n(√\n2×√\n2×2)a0four perovskitecells. Ions arerelaxed to-\nwardsequilibrium positions until the Hellmann-Feynman\nforces are less than 10−3eV/˚A. The spontaneous polar-\nization is calculated by the Berry phase method17with\nk-point mesh twice as dense as in the energycalculations.\nWe consider four formula units (perovskite cells), two\neach with Fe and Mn atoms on the B-site, which we re-\npeat periodically in space. For the planar checkerboard,\nwe alternate iron (Fe) and manganese (Mn) atoms at the\natomic level to form pillars of the same composition as\nin Fig. 1(ii). For the layered superlattice, we alternate\nsingle unit cell layers along z, as in Fig. 1(i). In both\ncases, the supercell is√\n2a0×√\n2a0×2a0.\nWe study various collinear spin orderings of the mag-\nnetic Fe and Mn atoms, shown for the checkerboard in\nFig. 2. FeFM and FeAFM refer to ferromagnetic and\nantiferromagnetic ordering respectively for the Fe mo-\nments in the relevant structural component (pillar for\nthe checkerboard, layer for the superlattice); similarly\nMnFM and MnAFM describe the spin ordering of the2\nFigure 2: From top left to bottom right: (i) G-AFM: rocksalt\ntype antiferromagnetic (AFM) order, (ii) C-FIM: AFM order\nin horizontal planes, ferromagnetic (FM) order along Fe/Mn\npillars, (iii) FM order, (iv) FeAFMMnFM: AFM order along\nFe pillars, FM order along Mn pillars, (v) FeFMMnAFM:\nFM order along Fe pillars, AFM order along Mn pillars, (vi)\nFeAFMMnAFM: AFM order along Fe/Mn pillars, but FM\norder in horizontal planes.\nMn moments. In the checkerboard, this notation fully\nspecifies the states considered. For the superlattice, the\nremaining ambiguity is resolved as follows: FeAFMM-\nnAFM magnetic order designates AFM ordered Fe and\nMn planes with FM order along the mixed Fe-Mn chains\nin thezdirection, while G-AFM designates the case with\nAFM order along the mixed chains; similarly, FeFMM-\nnFMdesignatesFMorderedFeandMnplaneswithAFM\norder along the mixed Fe-Mn chains in the zdirection,\nwhile FM designates the case with FM order along the\nmixed chains.\nIn searching for the ground state crystal structure for\neach spin ordering, we consider structures generated by\nthree typically unstable modes of the cubic perovskite\nstructure:18the zone center polar mode Γ−\n4, the M+\n3oxy-\ngenoctahedronrotations(allrotationsaboutagivenaxis\nin phase) and R+\n4rotations (sense of rotations alternates\nalong the rotation axis). We freeze in selected modes\nand combinations of modes and optimize atomic dis-\nplacements and lattice parameters in the resulting space\ngroups.\nFirst, toinvestigatetheeffectof B-sitecationgeometry\non the magnetic ordering, we present results in Table I\nfor the magnetic ordering energies when the structures\nare held fixed to the ideal perovskite reference structure.\nIn the layered superlattice and both bulk parent sys-\ntems, the difference in energy between magnetic ground\nstate (FeAFMMnFM in the superlattice, G-AFM in bulk\nBiFeO 3and FM in bulk BiMnO 3) and the first alter-\nnative state is 0 .11−0.12eV/f.u.; this difference corre-\nsponds to a relatively large energy and we do not ex-\npect a transition to a different magnetic state. The high-\nest energy magnetic states are more than 0 .26eV/f.u.\napart. On the other hand in the checkerboard, all mag--1.2-1-0.8-0.6-0.4-0.2 0\nRΓ(d) R4+(d) Γ4-(y) R4+(y) M3+(z) Γ4-(z) Pm-3mEnergy difference per f.u. [eV]\nFigure 3: Structural energetics of bulk BiFeO 3. Energy differ-\nence per perovskite cell (f.u.) for different magnetic order ings\n(see Fig. 2) and structural distortions: (1) Pm¯3m: no dis-\ntortion - ideal perovskite, (2) Γ−\n4(z): polar distortion along\nzaxis, (3) M+\n3(z): + oxygen octahedra tilts about zaxis,\n(4)R+\n4(y):−oxygen octahedra tilts about yaxis, (5) R+\n4(y)\nand Γ−\n4(y) (RΓ(y)): linear combination of (4) and (2) along\nyaxis, relaxes back to polar Γ−\n4(y) with zero tilting angle,\n(6)R+\n4([111]) (R+\n4(d)):−oxygen octahedra tilts about [111]\naxis, (7) R+\n4([111]) and Γ−\n4([111]) (RΓ(d)): linear combina-\ntion of (6) and (2) along [111] ( d), where drefers to the cube\ndiagonal direction.\nnetic states are found quasi-degenerate and are confined\nwithin the energetical window of 0 .12eV/f.u., that is, all\nare lower than the lowest states in the layered super-\nlattice and bulk parent compounds. Indeed, the closest\nmagnetic state to the FeAFMMnFM ground state is now\nonly 0.022eV/f.u.higher, making it much more plausible\nfor a magnetic transition to occur.\nNext we study the energetics of the structural distor-\ntion and its effect on the spin order. Before discussing\nresults for the BiFeO 3-BiMnO 3checkerboard, we look at\nthe structural energetics of the two bulk constituent ma-\nterials,BiFeO 3andBiMnO 3. Weplotenergiesforvarious\nmagnetic orderings in seven types of structural distor-\ntions of bulk BiFeO 3in Fig. 3. Our calculation verifies\nthe R3c ground state of BiFeO 3: counter-rotations of the\noxygen octahedra ( R+\n4) and polar ionic distortions (Γ−\n4)\nalong the [111] axis are most energetically favorable.12,19\nThe ground state structure has G-type AFM order and\nspontaneouspolarization90 µC/cm2along[111]axis. For\nall structural distortions considered, the lowest energy\nmagnetic ordering is G-AFM.\nWe study the structural energetics of bulk BiMnO 3\nin a similar way; the plotted energies for various struc-\ntural distortions and magnetic orderings are presented in\nFig.4. WefindthelowestenergystructurewithR3csym-\nmetry, the same as the ground state of BiFeO 3. The low-\nest energy structure has a half-metallic character and is3\nTable I: Calculated magnetic energies in an ideal perovskit e setting with lattice constant a0= 3.893˚Afor various magnetic\nstates in the checkerboard, layered superlattice, and bulk BiFeO 3and BiMnO 3. Value of U= 5eVandJ= 1eVis used.\ncheckerboard layered superlattice BiFeO 3 BiMnO 3\nmagnetic state ∆ E[eV/f.u.]magnetic state ∆ E[eV/f.u.]mag. state ∆ E[eV/f.u.]mag. state ∆ E[eV/f.u.]\nFeAFMMnFM 0.000 FeAFMMnFM 0.000 - -\nFM 0.022 FM 0.111 FM 0.360 FM 0.000\nC-FIM 0.076 FeFMMnFM 0.136 C-AFM 0.115 C-AFM 0.293\nFeAFMMnAFM 0.081 FeAFMMnAFM 0.135 A-AFM 0.223 A-AFM 0.116\nG-AFM 0.114 G-AFM 0.181 G-AFM 0.000 G-AFM 0.494\nFeFMMnAFM 0.119 FeFMMnAFM 0.260 - -\n-1.2-1-0.8-0.6-0.4-0.2 0\nRΓ(d) RΓ(y) R4+(y) MΓ(z) M3+(z) Γ4-(z) P4/mmmPm-3mEnergy difference per f.u. [eV]\nFigure 4: Structural energetics of bulk BiMnO 3. Energy dif-\nference per perovskite cell (f.u.) for various structural d is-\ntortions (see Fig. 3) and magnetic orderings (see Fig. 2);\nP4/mmmcorresponds to a tetragonally distorted perovskite\ncell with ideally positioned atoms and MΓ(z) is a linear com-\nbination of rotational M+\n3(z) and polar Γ−\n4(z) modes.\nferromagnetic. This structure is not the monoclinic cen-\ntrosymmetric ground state C2/cof bulk BiMnO 3which\nhas a larger unit cell than that considered here.20How-\never our calculation shows that it lies close to the ground\nstate (only 43 meV/f.u. above the GS). For all struc-\ntural distortions considered, the lowest energy magnetic\nordering is FM.\nIn the layered BiFeO 3-BiMnO 3superlattice, we cal-\nculate magnetic energies for the rocksalt type G-AFM\nand FeAFMMnFM layered magnetic states in two struc-\ntural distortions. For R+\n4(y)&Γ−\n4(y), we find ∆ E=\n−0.504eV/f.u.for G-AFM and ∆ E=−0.553eV/f.u.for\nFeAFMMnFM with respect to the FeAFMMnFM mag-\nnetic state in the ideal perovskite cell (see Table I). For\nR+\n4([111])&Γ−\n4([111]), we find ∆ E=−0.752eV/f.u.for\nG-AFM and ∆ E=−0.761eV/f.u. for FeAFMMnFM.\nFor both structural distortions considered, the lowest en-\nergy magnetic ordering is FeAFMMnFM.-1-0.8-0.6-0.4-0.2 0\nR4+,Γ4−(d) R4+,Γ4-(y) Γ4-(z) P4/mmmEnergy difference per f.u. [eV]-0.92-0.9-0.88-0.86-0.84-0.82-0.8\nFigure 5: Structural energetics of BiFeO 3-BiMnO 3checker-\nboard. Energydifferenceperperovskitecell (f.u.) for diffe rent\nmagnetic orderings (see Fig. 2) and structural distortions : (1)\nP4/mmm, (2)Γ−\n4(z), (3)R+\n4(y)andΓ−\n4(y), (4)R+\n4([111]) and\nΓ−\n4([111]) (R+\n4,Γ−\n4(d)). Inset: zoomed view of the magnetic\nenergies of c-R3c (4) distortion.\nLet us now look at the results for the structural en-\nergetics of the BiFeO 3-BiMnO 3checkerboard. In Fig. 5,\nwe present the energies for four different types of struc-\ntural distortions. These types of distortions show the\nlowest energies among a larger set of structures that we\nexplored.21Notice that the variation ofthe structuralen-\nergy is much larger than that of the magnetic energy of\nthe checkerboard.\nNot surprisingly, the R+\n4([111]) and Γ−\n4([111]) (R3c)\ntype of structural distortion is energetically the most fa-\nvorable; it is the BiFeO 3ground state and the BiMnO 3\nlowest energy structural distortion. The R3c symmetry\nis now broken due to pillar cation ordering and the space\ngroup of the checkerboard ground state becomes P1;\nwe use the notation c-R3c, where c designates “checker-\nboard”, as a reminder of the origin of the distortions.\nAs shown in the inset of Fig. 5, the two lowest magnetic\nstates G-AFM and FeAFMMnFM, are only 2 meV/f.u.4\napart. The ground state of the checkerboard has the\nFeAFMMnFM magnetic order, where Fe spins are or-\ndered antiferromagnetically (AFM) along the Fe pillars,\nMn spins are ordered ferromagnetically (FM) along the\nMn pillars, reflecting “AFM” and “FM” nature of the\nparent BiFeO 3and BiMnO 3compounds respectively. A\ntotal magnetic moment 3 .7µBper Fe -Mn pair results\nfrommanganesechains. TheFeAFMMnFM groundstate\nis insulating with energy gap 0 .88eV, and we calculate\na value of the polarization 62 µC/cm2pointing in the\n[0.85,0.85,1] direction. The ground state of the checker-\nboard is multiferroic, being ferroelectric and ferrimag-\nnetic.\nIn particular we want to relate and contrast the prop-\nerties ofthe BiFeO 3-BiMnO 3checkerboardto those ofits\ntwo bulk constituent materials; we recall that BiFeO 3is\npolar and antiferromagnetic, while BiMnO 3is non-polar\nand ferromagnetic, and we have found that the checker-\nboard assumes the desired ferromagnetic-ferroelectric\nproperties of each leading to a multiferroic ground-state\nwhose magnetic behavior is structurally sensitive. We\nattribute this behavior to the development of a quasi-\ndegenerate manifold of magnetic states in the checker-\nboard, in contrast to the gapped states in the layered\nsuperlattice and in the bulk; this can be understood in\ntermsoffrustrationofthecationsinherentinthechecker-\nboard geometry. Since bulk BiFeO 3is known to be G-\nAFM, and bulk BiMnO 3FM, the Fe-Fe and Mn-Mn\ninteractions tend to be AFM and FM-like respectively.\nIn the layered superlattice, each Fe(Mn) atom has four\nFe(Mn) and only two Mn(Fe) nearest neighbours, so that\nthe Fe/Mn layers prefer to be AFM/FM, leading to mi-\nnor frustration between the minority of mixed Fe-Mn\nbonds. The FeAFMMnFM layered ground state is much\nmore preferable and lower in energy than any other mag-\nneticstate. Inthecheckerboardtherearemorefrustrated\nbonds per eachcation, and thereforemoreweightis given\nto the mixed Fe-Mn bonds and various magnetic states\ncompete energetically.We study the sensitivity of the closely spaced magnetic\nlevels in the checkerboard to a structural distortion. As\nwe tune the system from the checkerboard c-R3c ground\nstate to c-I4cm state with R+\n4(y)&Γ−\n4(y) distortions, ei-\nther the FeAFMMnFM (filled diamond) or the G-AFM\n(open circle) lowest state is favored. Switching between\nthese two magnetic states occurs as we tune the sys-\ntem to other structural distortions (see Fig. 5). It is the\ncompetition between these two magnetic types that al-\nlows switching between nonzero and zero magnetization;\nthe magnetostructural effect leads to the possibility of\na structurally-driven magnetic transition in the checker-\nboard. This could be realized, for example, by imposing\nexpitaxial strain constraints.\nIn summary, we present a magnetostructural effect in\nthe atomic-scale checkerboard BiFeO 3-BiMnO 3, which\nis not present in either bulk or in layered structures of\nthese two materials. Furthermore, unlike its parent com-\npounds, the checkerboardhas a multiferroic ground state\nwith a nonzero magnetization and polarization; this is a\nnew example of a nanocomposite with properties that\ncan be designed. We note that this behavior is due to\nthe magnetic frustration in this system inherent to the\ncheckerboard geometry; as a result the magnetic states\narequasi-degenerateandcanbetunedbysmallperturba-\ntionsincludingstrain. Weremarkthatourfirstprinciples\ncalculations do not include spin-orbit coupling which is\nknown to lead to weak ferromagnetism in bulk BiFeO 3.22\nWe expect that such corrections will not change our re-\nsults fundamentally, but this is certainly worth pursuing\nin future work. We would also plan to investigate simi-\nlar checkerboards on longer length-scales to make more\ndirect contact with the possibility of future experiments.\nWe thank V. R. Cooper, M. Dawber, C.-J. Eklund,\nC. Fennie, A. Malashevich, M. Marsman and D. Vander-\nbilt for helpful discussions. This work was supported in\npart by NSF MRSEC DMR-0820404, NSF NIRT-ECS-\n0608842 and by the US Army Research Office through\nMURI-DAAD 19-01-1-0517.\n1H. Zheng et al., Science303, 661 (2004).\n2S. Yeo et al., Appl. Phys. Lett. 89, 233120 (2006); C. L.\nZhang et al., Appl. Phys. Lett. 90, 133123 (2007); C. L.\nZhang et al., Appl. Phys. Lett. 91, 233110 (2007).\n3B. S. Guiton and P. K. Davies, Nature Mater. 6, 586\n(2007).\n4J. L. MacManus-Driscoll et al., Nature Mater. 7, 314\n(2008).\n5D. P. Kozlenko et al., J. Magn. Mag. Mat. 258-259 , 290\n(2003)\n6T. Murata et al., J. Magn. Mag. Mat. 310, 1555 (2007).\n7C. J. Fennie and K. M. Rabe, Phys. Rev. Lett. 97, 267602\n(2006).\n8R. T. Lechner et al., Phys. Rev. Lett. 94, 157201 (2005).\n9G. Kresse and J. Hafner, Phys. Rev. B 47, R558 (1993); G.\nKresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).\n10A. I. Liechtenstein et al., Phys. Rev. B 52, R5467 (1995).11S. L. Dudarev et al., Phys. Rev. B 57, 1505 (1998).\n12J. B. Neaton et al., Phys. Rev. B 71, 014113 (2005).\n13P. Baettig et al., J. Am. Chem. Soc. 129, 9854 (2007).\n14P. E. Blochl, Phys. Rev. B 50, 17953 (1994).\n15G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n16H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188\n(1976).\n17R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,\n1651 (1993).\n18H. T. Stokes et al., Acta Cryst. B B58, 934 (2002).\n19C. Michel et al., Solid State Commun. 7, 701 (1969).\n20A. A. Belik et al., J. Am. Chem. Soc. 128, 706 (2006); A.\nA. Belik et al., J. Am. Chem. Soc. 129, 971 (2007).\n21L. P´ alov´ a, P. Chandra and K.M. Rabe, in preparation.\n22C. Ederer and N. A. Spaldin, Phys. Rev. B 71, 060401(R)\n(2005)."    },    {        "title": "1803.00235v1.Calculating_the_Magnetic_Anisotropy_of_Rare_Earth_Transition_Metal_Ferrimagnets.pdf",        "content": "Calculating the Magnetic Anisotropy of\nRare-Earth|Transition-Metal Ferrimagnets\nChristopher E. Patrick,1,\u0003Santosh Kumar,1Geetha Balakrishnan,1Rachel\nS. Edwards,1Martin R. Lees,1Leon Petit,2and Julie B. Staunton1\n1Department of Physics, University of Warwick,\nCoventry CV4 7AL, United Kingdom\n2Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom\n(Dated: March 2, 2018)\nAbstract\nMagnetocrystalline anisotropy, the microscopic origin of permanent magnetism, is often ex-\nplained in terms of ferromagnets. However, the best performing permanent magnets based on rare\nearths and transition metals (RE-TM) are in fact ferrimagnets, consisting of a number of mag-\nnetic sublattices. Here we show how a naive calculation of the magnetocrystalline anisotropy of\nthe classic RE-TM ferrimagnet GdCo 5gives numbers which are too large at 0 K and exhibit the\nwrong temperature dependence. We solve this problem by introducing a \frst-principles approach\nto calculate temperature-dependent magnetization vs. \feld (FPMVB) curves, mirroring the exper-\niments actually used to determine the anisotropy. We pair our calculations with measurements on\na recently-grown single crystal of GdCo 5, and \fnd excellent agreement. The FPMVB approach\ndemonstrates a new level of sophistication in the use of \frst-principles calculations to understand\nRE-TM magnets.\n1arXiv:1803.00235v1  [cond-mat.mtrl-sci]  1 Mar 2018High-performance permanent magnets, as found in generators, sensors and actuators, are\ncharacterized by a large volume magnetization and a high coercivity [1]. The coercivity |\nwhich measures the resistance to demagnetization by external \felds | is upper-bounded\nby the material's magnetic anisotropy [2], which in qualitative terms describes a preference\nfor magnetization in particular directions. Magnetic anisotropy may be partitioned into\ntwo contributions: the shape anisotropy, determined by the macroscopic dimensions of the\nsample, and the magnetocrystalline anisotropy (MCA), which depends only on the mate-\nrial's crystal structure and chemical composition. Horseshoe magnets provide a practical\ndemonstration of shape anisotropy, but the MCA is less intuitive, arising from the relativistic\nquantum mechanical coupling of spin and orbital degrees of freedom [3].\nPermanent magnet technology was revolutionized with the discovery of the rare-earth/transition-\nmetal (RE-TM) magnet class, beginning with Sm-Co magnets in 1967 [4] (whose high-\ntemperature performance is still unmatched [5]), followed by the world-leading workhorse\nmagnets based on Nd-Fe-B [6, 7]. With the TM providing the large volume magnetization,\ncareful choice of RE yields MCA values which massively exceed the shape anisotropy con-\ntribution [8]. RE-TM magnets are now indispensable to everyday life, but their signi\fcant\neconomic and environmental cost has inspired a global research e\u000bort aimed at replacing\nthe critical materials required in their manufacture [9].\nIn order to perform a targeted search for new materials it is necessary to fully understand\nthe huge MCA of existing RE-TM magnets. An impressive body of theoretical work based\non crystal \feld theory has been built up over decades [10], where model parameters are\ndetermined from experiment (e.g. Ref. [11]) or electronic structure calculations [12{14]. An\nalternative and increasingly more common approach is to use these electronic structure\ncalculations, usually based on density-functional theory (DFT), to calculate the material's\nmagnetic properties directly without recourse to the crystal \feld picture [15{19].\nCalculating the MCA of RE-TM magnets presents a number of challenges to electronic\nstructure theory. The interaction of localized RE-4 felectrons with their itinerant TM\ncounterparts is poorly described within the most widely-used \frst-principles methodology,\nthe local spin-density approximation (LSDA) [12]. Indeed, the MCA is inextricably linked to\norbital magnetism whose contribution to the exchange-correlation energy is missing in spin-\nonly DFT [20, 21]. MCA energies are generally a few meV per formula unit, necessitating\na very high degree of numerical convergence [22]. Finally, the MCA depends strongly on\n2temperature, so a practical theory of RE-TM magnets must go beyond zero-temperature\nDFT and include thermal disorder [23].\nEven when these signi\fcant challenges have been overcome, there is a more fundamental\nproblem. Experiments access the MCA indirectly, measuring the change in magnetization of\na material when an external \feld is applied in di\u000berent directions. By contrast, calculations\nusually access the MCA directly by evaluating the change in energy when the material is\nmagnetized in di\u000berent directions, with no reference to an external \feld. These experimental\nand computational approaches arrive at the same MCA energy provided one is studying a\nferro magnet. However, the majority of RE-TM magnets (and many other technologically-\nimportant magnetic materials) are ferrimagnets, i.e. they are composed of sublattices with\nmagnetic moments of distinct magnitudes and orientations. Crucially the application of\nan external \feld may introduce canting between these sublattices, a\u000becting the measured\nmagnetization. Thus the standard theoretical approach of ignoring the external \feld is hard\nto reconcile with real experiments on ferrimagnets.\nIn this Letter, through a combination of calculations and experiments, we provide the\nhitherto missing link between electronic structure theory and practical measurements of the\nMCA. Speci\fcally, we show how to directly simulate experiments by calculating, from \frst\nprinciples (FP), how the measured magnetization ( M) varies as a function of \feld ( B) applied\nalong di\u000berent directions and at di\u000berent temperatures. We apply our \\FPMVB\" approach\nto the RE-TM ferro and ferrimagnets YCo 5and GdCo 5, which are isostructural to the\ntechnologically-important SmCo 5[24] and, in the case of GdCo 5, a source of controversy in\nthe literature [25{35]. Pairing FPMVB with new measurements of the MCA of GdCo 5allows\nus to resolve this controversy. More generally, FPMVB enables a new level of collaboration\nbetween theory and experiment in understanding the magnetic anisotropy of ferrimagnetic\nmaterials.\nThe electronic structure theory behind FPMVB treats magnetic disorder at a \fnite tem-\nperatureTwithin the disordered local moment (DLM) picture [36, 37]. The methodology\nallows the calculation of the magnetization of each sublattice i,M i(T) =Mi(T)^M i, and the\ntorque quantity @F(T)=@^M i, whereFis an approximation to the temperature dependent\nfree energy. @F(T)=@^M iaccounts for the anisotropy arising from the spin-orbit interaction,\nwhile the contribution from the classical magnetic dipole interaction is computed numer-\nically [38]. Many of the technical details of the DFT-DLM calculations [36, 39{43] were\n3FIG. 1. Data points and \fts of dF=d\u0012 calculated for GdCo 5(blue, empty symbols; Gd and Co\nmoments held antiparallel) and YCo 5(green, \flled symbols), at 0 and 300 K.\ndescribed in our recent study of the magnetization of the same compounds [44]; the exten-\nsions to calculate the torques are described in Ref. [37]. The Gd-4 felectrons are treated\nwith the local self-interaction correction [43], and we have also implemented the orbital\npolarization correction [20] following Refs. [45, 46] using reported Racah parameters [47].\nDetails are given as Supplemental Material (SM) [48].\nYCo 5and GdCo 5crystallize in the CaCu 5structure, consisting of alternating hexagonal\nRCo 2c/Co 3glayers [24]. Y is nonmagnetic, while in GdCo 5the large spin moment of Gd\n(originating mainly from its half-\flled 4 fshell) aligns antiferromagnetically with the Co\nmoments. We now consider a \\standard\" calculation of the MCA based on a rigid rotation\nof the magnetization. If the Gd and Co moments are held antiparallel, GdCo 5is e\u000bectively\na ferromagnet with reduced moment MCo\u0000MGd. Then, from the hexagonal symmetry we\nexpect the angular dependence of the free energy to follow \u00141sin2\u0012+\u00142sin4\u0012+O(sin6\u0012),\nwhere\u0012is the polar angle between the crystallographic caxis and the magnetization direc-\ntion. The constants \u00141;\u00142determine the change in free energy \u0001 F, calculated e.g. from the\nforce theorem [49] or the torque dF=d\u0012 [50].\nIn Fig. 1 we show dF=d\u0012 calculated for ferromagnetic YCo 5and GdCo 5at 0 and 300 K.\nFitting the data to the derivative of the textbook expression, sin 2 \u0012(\u00141+ 2\u00142sin2\u0012), \fnds\u00141\nand\u00142to be positive (easy caxis) with\u00141an order of magnitude larger than \u00142. Considering\nexperimentally measured anisotropy constants in the literature, for YCo 5our\u00141value of\n3.67 meV (all energies are per formula unit, f.u.) at 0 K compares favorably to the values of\n3.6 and 3.9 meV reported in Refs. [28] and [51]. At 300 K, our value of 2.19 meV exhibits a\nslightly faster decay with temperature compared to experiment (2.6 and 3.0 meV), which we\nattribute to our use of a classical spin hamiltonian in the DLM picture [36, 44]. However, for\nGdCo 5our calculated values of \u00141show very poor agreement with experiments [26, 29]. First,\n4at 0 K we \fnd \u00141to be larger than YCo 5(4.26 meV), while experimentally the anisotropy\nconstant is much smaller (1.5, 2.1 meV). Second, we \fnd \u00141decreases with temperature\n(2.39 meV at 300 K) while experimentally the anisotropy constant increases (2.7, 2.8 meV).\nTo understand these discrepancies we must ask how the anisotropy energies were actually\nmeasured. Torque magnetometry provides an accurate method of accessing the MCA [52],\nbut is technically challenging in RE-TM magnets, which require very high \felds to reach\nsaturation [53]. Singular point detection [54] and ferromagnetic resonance [55] has also\nbeen used to investigate the MCA of polycrystalline and thin-\flm samples. However, the\nmost commonly-used method for RE-TM magnets, employed in Refs. [26, 29], is based on\nthe seminal 1954 work by Sucksmith and Thompson [56] on the anisotropy of hexagonal\nferromagnets. This work provides a relation between the measured magnetization Mab\nand \feldBapplied in the hard plane in terms of \u00141,\u00142and the easy axis magnetization\nM0[48, 56]:\n(BM 0=2)=(Mab=M0)\u0011\u0011=\u00141+ 2\u00142(Mab=M0)2: (1)\nFurther introducing m= (Mab=M0), equation 1 shows that a plot of \u0011againstm2should\nyield a straight line with \u00141as the intercept. Even though this \\Sucksmith-Thompson\nmethod\" was derived for ferromagnets, the technical procedure of plotting \u0011againstm2can\nbe performed also for ferrimagnets like GdCo 5[26, 29]. In this case, the quantity extracted\nfrom the intercept is an e\u000bective anisotropy constant Ke\u000bso, unlike YCo 5, the anisotropy\nconstants reported in Refs. [26, 29] are distinct from the \u00141values extracted from Fig. 1. As\nrecognized at the time of the original experiments [27{30], the reduced value of Ke\u000bwith\nrespect to\u00141of YCo 5is a \fngerprint of canting between the Gd and Co sublattices.\nMaking contact with previous experiments thus requires we obtain Ke\u000b. To this end\nwe have developed a scheme of calculating \frst-principles hard-plane magnetization vs. \feld\n(FPMVB) curves, on which we perform the Sucksmith-Thompson analysis to directly mirror\nthe experiments. The central concept of FPMVB is that at equilibrium, the torques from\nthe exchange, spin-orbit and dipole interactions must balance those arising from the external\n\feld. Then,\nB=@F(T)\n@\u0012i1\nMicos\u0012i+P\njsin\u0012j@M j\n@\u0012i: (2)\nThe magnetization at a given B;T is determined by the angle set f\u0012Gd;\u0012Co1;\u0012Co2;:::gwhich\nsatis\fes equation 2 for every magnetic sublattice. The spin-orbit interaction breaks the\n5FIG. 2. Magnetization of GdCo 5vs. applied magnetic \feld shown on a standard plot (left panel)\nor after the Sucksmith-Thompson analysis (eq. 1, right panel). Crosses/circles are calculated with\nmethods (i)/(ii) discussed in the text, and the area between them shaded as a guide to the eye.\nNote the two methods are e\u000bectively indistinguishable in the left panel. The dashed/solid lines are\ncalculated from the model free energies F1andF2. The right panel also shows the geometry of the\nmagnetization and \feld with respect to the crystallographic c-axis (thick gray arrow).\nsymmetry of the Co 3gatoms such that altogether there are four independent angles to vary\nfor GdCo 5. The second term in the denominator of equation 2 re\rects that the magnetic\nmoments themselves might depend on \u0012i(magnetization anisotropy). We have tested (i)\nneglecting this contribution and (ii) modeling the dependence as Mi(\u0012i) =M0i(1\u0000pisin2\u0012i),\nwhereM0iandpiare parameterized from our calculations.\nFigure 2 shows FPMVB curves of GdCo 5calculated using equation 2 with methods (i)\nand (ii), (crosses and circles) which yield virtually identical values of Ke\u000b. TheMvs.B\ncurves in the left panel resemble those of a ferromagnet where, as the temperature increases,\nit becomes easier to rotate the moments away from the easy axis so that a given B\feld\ninduces a larger magnetization. However, plotting \u0011againstm2in the right panel tells a\nmore interesting story. The e\u000bective anisotropy constant Ke\u000b(y-axis intercept) at 0 K is\n1.53 meV, much smaller than \u00141of YCo 5. Furthermore Ke\u000bincreases with temperature,\nto 1.74 meV at 300 K. Therefore, in contrast to the standard calculations of Fig. 1, the\nFPMVB approach reproduces the experimental behavior of Refs. [26, 29].\n6Our FPMVB calculations provide a microscopic insight into the magnetization process.\nFor instance at 0 K and 9 T, we calculate that the cobalt moments rotate away from the\neasy axis by 6.1\u000e. By contrast the Gd moments have rotated by only 3.9\u000e, i.e. the ideal\n180\u000eGd-Co alignment has reduced by 2.2\u000e(the geometry is shown in Fig. 2). We also \fnd\ncanting between the di\u000berent Co sublattices, but not by more than 0.1\u000eat both 0 and 300 K\n(the calculated angles as a function of \feld are shown in the SM [48]). This Co-Co canting\nis small thanks to the Co-Co ferromagnetic exchange interaction, which remains strong over\na wide temperature range [44]. The temperature dependence of Ke\u000bcan be traced to the\nfact that the easy axis magnetization M0of GdCo 5initially increases with temperature [44].\nEven ifMabincreases with temperature at a given \feld, a faster increase in M0can lead to\nan overall hardening in Ke\u000b(equation 1).\nWe assign the canting in GdCo 5to a delicate competition between the exchange interac-\ntion favoring antiparallel Co/Gd moments, uniaxial anisotropy favoring c-axis (anti)alignment,\nand the external \feld trying to rotate all moments into the hard plane. We can quantify\nthese interactions by looking for a model parameterization of the free energy F. Crucially\nwe can train the model with an arbitrarily large set of \frst-principles calculations exploring\nsublattice orientations not accessible experimentally, and test its performance against the\ntorque calculations of equation 2. Neglecting the 0.1\u000ecanting within the cobalt sublattices\ngives two free angles, \u0012Gdand\u0012Co. Including Gd-Co exchange A, uniaxial Co anisotropy\nK1;Coand a dipolar contribution S(\u0012Gd;\u0012Co) [31, 48] leads naturally to a two-sublattice\nmodel [30],\nF1(\u0012Gd;\u0012Co) =\u0000Acos(\u0012Gd\u0000\u0012Co) +K1;Cosin2\u0012Co\n+S(\u0012Gd;\u0012Co): (3)\nThe training calculations showed additional angular dependences not captured by F1, so we\nalso investigated:\nF2(\u0012Gd;\u0012Co) =F1(\u0012Gd;\u0012Co) +K2;Cosin4\u0012Co\n+K1;Gdsin2\u0012Gd: (4)\nAs discussed below the training calculations showed no strong evidence of Gd-Co exchange\nanisotropy [31{34].\nThe dashed (solid) lines in Fig. 2 are the calculated Mvs.Bcurves obtained by mini-\nmizingF1(2)\u0000P\niM i\u0001B. The second term includes magnetization anisotropy on the cobalt\n7FIG. 3. Anisotropy constants Ke\u000bvs. temperature of YCo 5(green) and GdCo 5(blue). The left\npanel shows calculations using equation 2 at 0 and 300 K (stars), or using parameterized model\nexpressions F1(diamonds) and F2(circles), and from Ref. [57] (YCo 5, squares). For GdCo 5we\nalso show in red \u00141extracted from \\standard\" calculations where the Gd and Co moments were\nheld rigidly antiparallel (cf. Fig. 1). The experimental data in the right panel was measured by\nus for GdCo 5(crosses, with shaded background) or taken from Refs. [26], [29] and [58] (squares,\ndashed lines, circles) and Refs [28] and [51] (green diamonds and dashed lines, YCo 5).\nmoments [48, 57]. On the scale of the left panel both F1andF2give excellent \fts to the\ntorque calculations, especially up to moderate \felds. The plot of \u0011againstm2reveals some\ndi\u000berences with F2giving a marginally improved description of the data, but F1already\ncaptures the most important physics.\nWe also applied the FPMVB approach to YCo 5, using equation 2 and the model for F\nintroduced in Ref. [57]. Then, parameterizing the models [48] over the temperature range\n0{400 K, calculating Mvs.Bcurves and extracting Ke\u000busing the Sucksmith-Thompson\nplots gives the results shown in the left panel of Fig. 3. We also show \u00141of GdCo 5to\nemphasize the di\u000berence between FPMVB calculations and the \\standard\" ones of Fig. 1.\nComparing Ke\u000bto previously-published experimental measurements on GdCo 5raises\nsome issues. First, the three studies in the literature report anisotropy constants which di\u000ber\nby as much as 1 meV [26, 29, 58]. Indeed there was controversy over whether the observed\nresults were evidence of an anisotropic exchange interaction between Gd and Co [31, 32] or\n8an artefact of poor sample stoichiometry [33, 34]. Furthermore the only study performed\nabove room temperature [26] reports without comment some peculiar behavior where Ke\u000b\nof GdCo 5exceeds that of YCo 5at high temperature [28], despite conventional wisdom that\nthe half-\flled 4 fshell of Gd does not contribute to the anisotropy.\nOur calculations do in fact show an excess in the rigid-moment anisotropy of GdCo 5of\n16% at 0 K (Fig. 1) compared to YCo 5. The authors of Refs. [29, 31] \ftted their experimental\ndata with a much larger excess of 50%, while the high-\feld study of Ref. [33] found (11\n\u000615)%, with the authors of that work attributing the di\u000berence to an improved sample\nstoichiometry [34]. Our calculated excess at 0 K is formed from two major contributions:\nthe dipole interaction energy, which accounts for 0.31 meV/f.u., and K1;Gd(equation 4) which\nwe found to be 24% the size of K1;Co. The nonzero value of K1;Gdis due to the 5 delectrons,\nwhose presence is evident from the Gd magnetization (7.47 \u0016Bat 0 K). We did not \fnd a\nsigni\fcant contribution from anisotropic exchange, which we tested in two ways: \frst by\nattempting to \ft a term A(1\u0000p0sin2\u0012Co) cos(\u0012Gd\u0000\u0012Co) to our training set of calculations,\nand also by computing Curie temperatures with the (rigidly antiparallel) magnetization\ndirected either along the coraaxes. We found the magnitude of the anisotropy ( p0) to be\nsmaller than 0.5% and negative at 0 K, and to decrease in magnitude as the temperature\nis raised. Consistently the Curie temperature was found to be only 1 K higher for aaxis\nalignment, which we do not consider signi\fcant.\nHowever, our calculations do not predict the Ke\u000bvalue of GdCo 5to exceed YCo 5. Indeed,\nin Fig. 3\u00141of GdCo 5approaches that of YCo 5at high temperatures, which is signi\fcant\nbecause\u00141provides an upper bound for Ke\u000b[32]. To resolve this \fnal puzzle we performed\nour own measurements of Ke\u000bon the single crystal whose growth we reported recently [44].\nHard and easy axis magnetization curves up to 7 T were measured in a Quantum Design\nsuperconducting quantum interference device (SQUID) magnetometer, and the anisotropy\nconstants extracted from Sucksmith-Thompson plots [48]. The right panel of Fig. 3 shows\nour newly measured data as crosses. Previously reported measurements are shown in faint\nblue/green for GdCo 5[26, 29, 58]/YCo 5[28, 51].\nUp to 200 K, there is close agreement between the experiments of Ref. [26], our own ex-\nperiments, and the FPMVB calculations. Above this temperature our new experiments show\nthe expected drop in Ke\u000b, while the previously reported data show a continued rise [26]. We\nrepeated our measurements using di\u000berent protocols and found a reasonably large variation\n9in the extracted Ke\u000b[48]. Even taking this variation into account as the shaded area in\nFig. 3, the drop is still observed.\nWe therefore do not believe the high temperature behavior reported in Ref. [26] has\nan intrinsic origin. Possible extrinsic factors include the method of sample preparation,\ndegradation of the RCo 5phase at elevated temperatures [59], and potential systematic error\nwhen extracting Ke\u000b. We note that even the idealized theoretical curves in Fig. 2 show\ncurvature at higher temperature, making it more di\u000ecult to \fnd the intercept.\nIn conclusion, we have introduced the FPMVB approach to interpret experiments mea-\nsuring anisotropy of ferrimagnets, particularly RE-TM permanent magnets. We presented\nthe method in the context of our DLM formalism, but any electronic structure theory ca-\npable of calculating magnetic couplings relativistically [60{64] should be able to produce\nFPMVB curves, at least at zero temperature. However standard calculations which neglect\nthe external \feld should be used with care when comparing to experiments on ferrimagnets.\nSimilarly, the prototype GdCo 5serves as a reminder that a simple view of the anisotropy\nenergy does not fully describe the magnetization processes in ferrimagnets, which might have\nimplications in understanding e.g. magnetization reversal in nano-magnetic assemblies [65].\nOverall our work demonstrates the bene\ft of interconnected computational and experimen-\ntal research in this key area.\nThe present work forms part of the PRETAMAG project, funded by the UK Engineer-\ning and Physical Sciences Research Council (EPSRC), Grant no. EP/M028941/1. 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Lett. 110, 097202 (2013).\n[63] S. Ayaz Khan, P. Blaha, H. Ebert, J. Min\u0013 ar, and O. \u0014Sipr, Phys. Rev. B 94, 144436 (2016).\n[64] M. Ho\u000bmann, B. Zimmermann, G. P. M uller, D. Sch urho\u000b, N. S. Kiselev, C. Melcher, and\nS. Bl ugel, Nat. Commun. 8, 308 (2017).\n[65] Z. J. Guo, J. S. Jiang, J. E. Pearson, S. D. Bader, and J. P. Liu, Appl. Phys. Lett. 81, 2029\n(2002).\n13"    },    {        "title": "1906.12288v1.Spin_transport_in_an_insulating_ferrimagnetic_antiferromagnetic_ferrimagnetic_trilayer_as_a_function_of_temperature.pdf",        "content": "Spin transport in an insulating\nferrimagnetic-antiferromagnetic-ferrimagnetic trilayer as a\nfunction of temperature\nYizhang Chen,1Egecan Cogulu,1Debangsu Roy,1Jinjun Ding,2Jamileh Beik\nMohammadi,1Paul G. Kotula,3Nancy A. Missert,3Mingzhong Wu,2and Andrew D.\nKent1,a)\n1)Center for Quantum Phenomena, Department of Physics, New York University,\nNew York, New York 10003, USA\n2)Department of Physics, Colorado State University, Fort Collins, Colorado 80523,\nUSA\n3)Sandia National Laboratories, Albuquerque, New Mexico 87185,\nUSA\n(Dated: 1 July 2019)\nWe present a study of the transport properties of thermally generated spin currents\nin an insulating ferrimagnetic-antiferromagnetic-ferrimagnetic trilayer over a wide\nrange of temperature. Spin currents generated by the spin Seebeck e\u000bect (SSE) in a\nyttrium iron garnet (YIG) YIG/NiO/YIG trilayer on a gadolinium gallium garnet\n(GGG) substrate were detected using the inverse spin Hall e\u000bect in Pt. By studying\nsamples with di\u000berent NiO thicknesses, the NiO spin di\u000busion length was deter-\nmined to be 4.2 nm at room temperature. Interestingly, below 30 K, the inverse\nspin Hall signals are associated with the GGG substrate. The \feld dependence of\nthe signal follows a Brillouin function for a S=7/2 spin (Gd3+) at low temperature.\nSharp changes in the SSE signal at low \felds are due to switching of the YIG mag-\nnetization. A broad peak in the SSE response was observed around 100 K, which\nwe associate with an increase in the spin-di\u000busion length in YIG. These observa-\ntions are important in understanding the generation and transport properties of\nspin currents through magnetic insulators and the role of a paramagnetic substrate\nin spin current generation.\nKeywords: spin current, spin transport, spin Seebeck e\u000bect, spin valve\nI. INTRODUCTION\nA spin current, or a \row of spin angular momentum, can be carried by conduction\nelectrons1,2or spin waves3,4. In a material with large spin-orbit coupling, like Pt, a spin\ncurrent can be converted into a measurable voltage by the inverse spin Hall e\u000bect (ISHE)5.\nSpin currents can be generated by the spin Hall e\u000bect (SHE)6{8, spin pumping9, or the spin\nSeebeck e\u000bect10{13. The spin Seebeck e\u000bect refers to the generation of spin currents when\na temperature gradient is applied to a magnetic material and has potential applications in\nconverting waste heat into electricity.\nA conventional spin valve consists of two ferromagnetic metals separated by a non-\nmagnetic metal14,15. Recently, a new spin valve structure based on an antiferromagnetic\ninsulator (AFI) sandwiched between two ferromagnetic insulators (FI) was proposed16. An\nAFI can conduct both up and down spins due to the degeneracy of its magnon spectrum at\nzero \feld. The predicted valve e\u000bect associated with thermally induced spin currents has\nbeen observed by controlling the relative orientations of Y 3Fe5O12(YIG) magnetization in a\nYIG/NiO/YIG structure17. YIG is a ferrimagnetic insulator with low magnetic dissipation,\nhighly e\u000ecient spin current generation18, and long-distance magnon transport19. Nickel\na)Electronic mail: andy.kent@nyu.eduarXiv:1906.12288v1  [cond-mat.mes-hall]  28 Jun 20192\nFIG. 1. A schematic of the sample and cross-sectional characterization of the sample by scanning\ntransmission electron microscopy energy-dispersive X-ray spectroscopy (STEM-EDS). (a) Sample\ngeometry showing the layers, the electrical contacts and the applied magnetic \feld. ~jcis the density\nof the charge current applied in the x-direction. Vxyis the voltage measured in the transverse\ndirection, and 'is the angle between the applied magnetic \feld and the current. GGG, YIG, NiO,\nand Pt are represented as purple, yellow, green, and grey, respectively. (b) Sample cross section\ncharacterized by STEM-EDS. GGG, YIG, NiO, and Pt layers are colored in blue, red, green, and\ndark gray, respectively.\nFIG. 2. Angular dependence and NiO thickness dependence of V2!\nxymeasured with an in-plane\nmagnetic \feld of 0 :4 T at room temperature. (a) Angular dependence of V2!\nxywith an AC density\nofjAC= 1:5\u00021010A=m2. \u0001V2!\nxyis extracted by \ftting the curve with a cosine function. (b) \u0001 V2!\nxy\nas a function of the NiO thickness. The curve is \ftted with V=V0e\u0000t=\u0015NiO, whereV0= 182 \u000644 nV\nand the spin di\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm.\nOxide (NiO) is an antiferromagnetic insulator used to decouple the two ferrimagnetic layers\nwhile conducting thermally generated spin currents.\nTo understand the generation, transmission, and detection of spin currents through a mul-\ntilayer consisting of di\u000berent magnetic insulators, transport measurements were performed\nin samples consisting of GGG(500 \u0016m)/YIG(20 nm)/NiO(t nm)/YIG(15 nm)/Pt(5 nm).\nHere GGG (Gd 3Ga5O12) is the standard substrate used to grow epitaxial YIG. Above the\nspin-glass transition temperature ( \u0018\u00000:18 K), GGG is paramagnetic with no long-range\nmagnetic order20,21. In addition, GGG has been shown to have a SSE, with a magnitude\ncomparable to the SSE that of YIG at low temperatures22.\nIn this article, room-temperature measurements were \frst performed to characterize the\nspin di\u000busion length of NiO. Then experiments were conducted over a broad range of tem-\nperature from 5 to 300 K. These revealed a strong enhancement of the SSE below 30 K that\noriginates from the GGG substrate. Further, \feld-dependent experiments show behavior3\nFIG. 3. Second harmonic response V2!\nxymeasured at several temperatures with an applied magnetic\n\feld\u00160H= 1:0 T for NiO thicknesses of 2.5, 5, and 10 nm. (a) Angular dependence of V2!\nxy\nmeasured from 5 to 300 K. Note that the angle 'is de\fned in Fig. 1(a). (b) \u0001 V2!\nxymeasured as a\nfunction of the temperature. Inset: a broad peak is observed around 100 K for the sample with a\n5 nm NiO thickness.\nassociated with switching of YIG magnetization and paramagnetism of GGG. Furthermore,\na broad peak in the SSE response around 100 K was observed, which may originate from\nthe temperature dependence of the spin di\u000busion length in YIG.\nII. SAMPLE FABRICATION AND MEASUREMENT TECHNIQUES\nThe sample was fabricated in the following way. First, a 20 nm YIG layer was grown\nepitaxially on a (111)-oriented GGG substrate (500 \u0016m) at room temperature and annealed\nin O 2at high-temperature23. An Ar plasma was used to clean the surface of the samples\nbefore depositing NiO via radio frequency (RF) sputtering in another chamber. Afterward,\na 15 nm YIG layer was grown on top with the same growth conditions of the \frst layer. Then\nthe sample was capped with a 5 nm Pt layer. For transport and SSE measurements, the\nPt was patterned into Hall bar structures using electron beam lithography and Ar plasma\netching. The Hall bar has a width of 4 \u0016m and the length between the two longitudinal\ncontacts is 130 \u0016m. An alternating current (AC) with a frequency of 953 Hz was used. As\nthe temperature gradient induced by the AC oscillates at twice the frequency, the second\nharmonic Hall voltage V2!\nxymeasured by a lock-in ampli\fer is proportional to the amplitude\nof the SSE-produced spin current24,25. Room-temperature measurements were performed\nwith a 0.4 T magnetic \feld applied in-plane. Temperature-dependent measurements are\ncarried out in the Quantum Design PPMS system also with an in-plane applied magnetic\n\feld.\nIII. EXPERIMENTAL RESULTS\nFigure 1(a) is a schematic of the GGG/YIG/NiO/YIG/Pt sample. A magnetic \feld is\napplied in-plane at an angle 'with respect to the current. The cross section of the sample\nis characterized by scanning transmission electron microscopy with energy-dispersive X-ray\nspectroscopy, shown in Fig. 1(b). Both the top and bottom YIG layers are crystalline, with\nthickness of 15 nm and 20 nm. NiO is polycrystalline, with a thickness of 5 nm for this\nsample (see Fig. S1 in the supplemental materials).\nFirst,V2!\nxywas measured as a function of 'for samples with NiO thicknesses of 2.5, 5,\n7.5, and 10 nm. V2!\nxyreaches a maximum at '= 180oand minimum at '= 0o. This is\nconsistent with the ISHE symmetry of the Hall voltage VISHE/~js\u0002^\u001b/rT\u0002^m/cos('),4\nwhere~jsis the spin current, ^ \u001bis the spin polarization direction, rTis the temperature\ngradient, and ^ mis a unit vector in the direction of magnetization. The angular dependence\nof theV2!\nxywas \ftted with a cosine function and the amplitude \u0001 V2!\nxyis plotted as a function\nof the NiO thickness (Fig. 2). \u0001 V2!\nxydecays rapidly as the NiO thickness increases and\nis \ftted to an exponentially decaying function V=V0e\u0000t=\u0015NiO. The characteristic spin\ndi\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm, close to what has been found in previous\nwork on YIG/NiO/Pt structures26.\nTo further understand the generation and transport of thermally generated spin currents\nthrough the heterostructure, the angular dependence of V2!\nxywas measured from 5 to 300 K\nwith an applied magnetic \feld of 1.0 T (Fig. 3(a)). The amplitude \u0001 V2!\nxyis extracted by the\nsame method discussed above and is plotted as a function of the temperature (Fig. 3(b)).\nFor the 5 nm thick NiO sample, as temperature decreases from 300 to 100 K, \u0001 V2!\nxyincreases\nsteadily from 150 to 271 nV. From 100 to 50 K, \u0001 V2!\nxyslightly decreases to 257 nV, forming\na broad peak around 100 K, shown in the inset of Fig. 3(b). However, as temperature\ndecreases below 30 K, \u0001 V2!\nxyincreases dramatically from 297 to 994 nV. The enhancement\nbelow 30 K was observed for all samples.\nAs has been previously noted, the SSE depends on the magnon population, the spin\ndi\u000busion length, and the interfacial spin-mixing conductance in the heterostructure. In\norder to understand the correlation between the SSE signal and the magnetization of the\nsamples, \feld-dependent measurements of V2!\nxywere performed in the sample with 2.5 nm\nthick NiO. Fig. 4(a) shows V2!\nxyas a function of the applied magnetic \feld between -5.0\nand 5.0 T with temperature ranging from 5 to 50 K. At 50 K, as the applied \feld goes\nfrom -5.0 to -1.0 T, V2!\nxy\u0019530 nV, almost independent of the applied \feld. As the applied\n\feld increases from -1.0 T to 20 mT, V2!\nxydecreases slowly to 108 nV. Then V2!\nxydrops\nsharply to -156 nV as the applied \feld increases from 20 to 100 mT. As the applied \feld\nincreases to 1.0 T, V2!\nxydecreases slowly to -540 nV and again is nearly constant thereafter.\nThe sharp switching steps observed in the V2!\nxy\u0000Hcurves around\u000650 mT occur at the\ncoercive \feld of the YIG, which is smaller than 50 mT at room temperature (see Fig. S2\nin the supplemental materials). Only one magnetization reversal can be identi\fed between\n-200 and 200 mT in the V2!\nxy\u0000Hcurves. As temperature decreases from 50 to 5 K, V2!\nxy\nincreases from 535 to 1970 nV, while the low-\feld step does not change signi\fcantly. A\nclear correlation between the V2!\nxyand the magnetization of a paramagnet can be seen by\ncomparing the V2!\nxy\u0000Hcurves with the Brillouin function of a S = +7/2 spin (Gd3+),\nshown in Fig. 4(b).\nIV. DISCUSSION\nThe SSE voltages decay rapidly as NiO thickness increases, as presented in Fig. 2. This\nindicates that spin currents were generated not only from the top YIG layer but also from\nthe bottom YIG or GGG layer. The NiO spin di\u000busion length is close to what has been\nfound before at room temperature in YIG/NiO/Pt structures26.\nA dramatic enhancement of the SSE voltages has been observed below 30 K, which\nis likely associated with GGG. The same enhancement has been observed in GGG(500\n\u0016m)/YIG(20 nm)/Pt(5 nm), shown in Fig. S3 in the supplemental materials. A previous\nstudy has shown that the spin current jsgenerated by paramagnetic SSE in GGG/Pt bilayer\nhas aT\u00001temperature dependence, associated with GGG susceptibility, which follows the\nCurie-Weiss law \u001f=C=(T\u0000\u0002CW), whereCis the Curie constant and \u0002 CWCurie-\nWeiss temperature22. At low temperatures, the GGG thermal conductivity kGGG has a\nT3temperature dependence. Therefore, the temperature gradient generated by a constant\npower isrT/1=kGGG/1=T\u00003. The resulting SSE voltage goes as VSSE/js\u0001rT/T\u00004.\nIn addition, a broad peak of the SSE signal observed around 100 K in the YIG/Pt structure\nsuggests that the spin di\u000busion length in YIG has a strong temperature dependence27. As\nspin currents generated and transmitted through YIG layers, the temperature-dependent5\nFIG. 4. (a) Field dependence of the V2!\nxymeasured between 5 and 50 K, with '= 0o. The sample\nis GGG (500 \u0016m)/YIG(40 nm)/NiO(2.5 nm)/YIG(20 nm)/Pt (5 nm). The magnetic \feld is swept\nbetween -5 and +5 T. The o\u000bset of V2!\nxyhas been removed. (b) Brillouin function of an S=+7/2\nspin.\nspin di\u000busion length in YIG would have a signi\fcant e\u000bect on the ISHE voltage generated\nin Pt. So the broad peak observed around 100 K in Fig. 3 may be associated with the\ntemperature dependence of spin di\u000busion length in YIG. However, further experiments\nare needed to understand how spin currents are transmitted through bulk GGG, NiO,\nGGG/YIG, YIG/NiO, and NiO/YIG interfaces at di\u000berent temperatures.\nComparing the \feld dependence of SSE voltages and the Brillouin function from 5 to 50\nK, it is clear that there is a contribution to SSE from GGG at low temperatures. At 5 K,\nthe SSE voltage follows the Brillouin function as the magnetic \feld swept from -5.0 to 5.0\nT. As temperature increases, the SSE voltages start deviating from the Brillouin function\n(Fig. 4 and Fig. S4). The underlying physics is not yet fully understood, since the role\nplayed by GGG, YIG, NiO and their corresponding interfaces vary with temperature.\nV. SUMMARY\nIn summary, the spin transport properties of an insulating trilayer based on two ferrimag-\nnetic insulators separated by a thin antiferromagnetic insulator were presented. The spin\ndi\u000busion length of NiO was found to be \u0015NiO'4:2 nm at room temperature. In addition,\na large increase of the SSE signal was observed below 30 K, revealing the dramatic e\u000bects\nof paramagnetic SSE from the GGG substrate. The \feld dependence of the SSE shows the\nswitching of YIG magnetization at low \feld as well as paramagnetic behavior associated\nwith GGG. Furthermore, the SSE voltages show a broad peak around 100 K, a feature\nthat may be related to the temperature dependence of spin di\u000busion length in YIG. This\nexperimental study provides information on how spins can be generated, transported and\ndetected in a heterostructure consisting of paramagnetic, ferrimagnetic and antiferromag-\nnetic insulators.\nSUPPLEMENTARY MATERIAL\nThe supplementary material provides the details of sample characterization by scanning\ntransmission electron microscopy (SEM), Vibrating Sample Magnetometer (VSM), trans-\nport measurements of a GGG/YIG/Pt sample, and \feld-dependent measurements of a\nGGG/YIG/NiO/YIG/Pt sample above 50 K.6\nACKNOWLEDGEMENTS\nThis work was supported partially by the MRSEC Program of the National Science\nFoundation under Award Number DMR-1420073. The instrumentation used in this research\nwas support in part by the Gordon and Betty Moore Foundations EPiQS Initiative through\nGrant GBMF4838 and in part by the National Science Foundation under award NSF-DMR-\n1531664. ADK received support from the National Science Foundation under Grant No.\nDMR-1610416. JD and MW were supported by the U. S. National Science Foundation under\nGrants No. EFMA-1641989 and No. ECCS-1915849. Sandia National Laboratories is a\nmulti-program laboratory managed and operated by National Technology and Engineering\nSolutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for\nthe U.S. Department of Energy's National Nuclear Security Administration under contract\nDE-NA-0003525. The views and conclusions contained herein are those of the authors and\nshould not be interpreted as necessarily representing the o\u000ecial policies or endorsements,\neither expressed or implied, of the U.S. Government. We also would like to express our\nthanks to Pradeep Manandhar from Spin Memory Inc. for TEM characterization.\n1M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich,\nand J. Chazelas, \\Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices\", Phys. Rev. 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X 6(3), 031012 (2016)."    },    {        "title": "1910.01405v2.Current_driven_domain_wall_dynamics_in_ferrimagnetic_strips_explained_by_means_of_a_two_interacting_sublattices_model.pdf",        "content": "Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model AIP/123-QED\nCurrent driven domain wall dynamics in ferrimagnetic strips explained by means of a\ntwo interacting sublattices model\nEduardo Mart \u0013 \u0010nez,1V \u0013 \u0010ctor Raposo,1and \u0013Oscar Alejos2\n1)Dpto. F\u0013 \u0010sica Aplicada, Universidad de Salamanca, 37008 Salamanca,\nSpain\n2)Dpto. Electricidad y Electr\u0013 onica. Universidad de Valladolid. 47011 Valladolid,\nSpaina)\n(Dated: 23 October 2019)\nThe current-driven domain wall dynamics along ferrimagnetic elements are here the-\noretically analyzed as a function of temperature by means of micromagnetic simula-\ntions and a one dimensional model. Contrarily to conventional e\u000bective approaches,\nour model takes into account the two coupled ferromagnetic sublattices forming the\nferrimagnetic element. Although the model is suitable for elements with asymmet-\nric exchange interaction and spin-orbit coupling e\u000bects due to adjacent heavy metal\nlayers, we here focus our attention on the case of single-layer ferrimagnetic strips\nwhere domain walls adopt achiral Bloch con\fgurations at rest. Such domain walls\ncan be driven by either out-of-plane \felds or spin transfer torques upon bulk current\ninjection. Our results indicate that the domain wall velocity is optimized at the an-\ngular compensation temperature for both \feld-driven and current-driven cases. Our\nadvanced models allow us to infer that the precession of the internal domain wall\nmoments is suppressed at such compensation temperature, and they will be useful to\ninterpret state-of-the art experiments on these elements.\na)Electronic mail: oscar.alejos@uva.es.\n1arXiv:1910.01405v2  [physics.app-ph]  22 Oct 2019Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\nI. INTRODUCTION\nA great e\u000bort is being devoted to the \fnding of optimal systems permitting fast displace-\nment of domain walls (DWs) along racetrack elements.1As recent experiments demonstrate,\nDW velocities in the order of 1km\nscan be achieved along ferrimagnetic (FiM) strips,2,3with\na linear relationship between DW velocities and the magnitude of applied stimuli.2{4\nHere we provide a theoretical description of DW dynamics in FiM strips based on an\nextended collective coordinates model (1DM).5,6Di\u000berently from other approaches, based\non e\u000bective parameters, our model considers such elements as formed by two ferromagnetic\nsublattices, and coupled by means of an interlattice exchange interaction. Full micromag-\nnetic (\u0016M) simulations have been performed also to back up those drawn by the 1DM.\nImportantly, our approaches allow to infer results not achievable from e\u000bective models, and\nto provide insights and interesting predictions of the current-driven dynamics of DWs along\nFiM \flms.\nFig.1.(a) schematizes the local orientation of magnetic moments in the ferrimagnet.\n~ mi(i= 1;2) represent the orientations of the respective magnetic moments of each ferro-\nmagnetic sublattice. The magnetization of each sublattice is temperature dependent, so that\nmagnetization of each sublattice vanishes at Curie temperature ( TC), with a magnetization\ncompensation temperature TM, as it is shown in Fig.1.(b). The temperature dependence\ncan be described by the analytical functions: Ms;i(T) =M0\ns;i\u0010\n1\u0000T\nTC\u0011ai,M0\ns;ibeing the\nrespective magnetizations at zero temperature, and aibeing dependent on the sublattice\ncomponents.\nThe model can be applied to two di\u000berent architectures. As a \frst architecture (Fig.1.(c)),\na FiM strip on top of a heavy metal (HM) can be considered. The FiM/HM interface\npromotes interfacial asymmetric exchange, resulting in N\u0013 eel type DWs and current driven\ndomain wall motion (CDDWM) due to spin orbit torques (SOT), with rigid DWs. At the\nangular momentum compensation temperature ( TA), di\u000bering from TMdue to the distinct\nLand\u0013 e factors gifor each sublattice, DW magnetic moments keep aligned with the current,\nleading to a linear increase of DW velocities. Thus, DW velocities are maximized at TA.\nThis \frst architecture has already been adequately discussed from both the experimental3\nand theoretical3,6points of view, in particular, by using the model to be here recalled6.\nIn the second architecture (Fig.1.(d)), the FiM does not lie on a HM, and so interfacial\n2Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)\nX YZ\n𝑚2\n𝜓1𝑚1 𝜓2\n𝜓1>0𝜓2>0 (b)\nTC\nMs,i\nT emp eratureMs,1(T)\nMs,2(T)\nTMTA\n(c)\nX\nYZ\n\u0004\u0005\u0006\u0007 \n(d)\nX\nYZ\n\u0004\u0005\u0006\u0007 \nFIG. 1. Two sublattices constitute the FiM: (a) magnetizations are represented by the unit vectors\n~ m1and~ m2, with in-plane orientation angles  1and 2, respectively, (b) temperature dependence\nof the magnetization of each sublattice, (c) magnetic DW of N\u0013 eel type, and (d) magnetic DW of\nBloch type amidst two domains oriented out of plane (the strip width wis here shown).\nasymmetric exchange vanishes. CDDWM is dominated by the spin transfer torques (STT),\nand DW precessional regimes emerge, due to reduced magnetostic interactions, resulting in\nDW velocities proportional to current magnitudes. Again, DW velocities have been found\nto maximize at TA, when precession freezes, leading to a CDDWM characterized by rigid\nDWs, what is to be shown along this text.\nII. TWO-SUBLATTICE MODEL OF FERRIMAGNETS\nThe description of the DW dynamics by means of a 1DM starts from the application of\nvariational principles to the \u0016M equation, i.e, the Landau-Lifshitz-Gilbert (LLG) equation.7,8\nThis procedure is then augmented to study the magnetization dynamics in FiMs by posing\ntwo coupled LLG equations, that is, a two-sublattice model (TSLM). Details on the deriva-\n3Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\ntion of the 1DM equations for the TSLM are given in Ref.6, so here we will only recall the\nrequired model parameters.\nWithin the model, the respective Gilbert constants of each sublattice are represented\nby the values \u000bi. The e\u000bective \felds are the sum of the external \feld, the demagnetizing\n(magnetostatic) \felds, the anisotropy \felds, the isotropic exchange \felds and the asymmetric\nexchange \felds. The external \feld have components ( Bx;By;Bz). The demagnetizing term\npossesses out-of-plane and in-plane components, given by the e\u000bective anisotropy constants\nKeff;iandKsh;i. The asymmetric exchange provides a chiral character to some magnetic\ntextures, whereas the isotropic one can be reduced on \frst approach to the sum of an\nintra-sublattice exchange \feld, given by the exchange sti\u000bness Ai, and an inter-sublattice\ninteraction due to the misalignment of both sublattices. The latter is accounted for by\na parameter B12>0 (<0), which promotes the antiparallel (parallel) alignment of the\nsublattices. Finally, LLG equations also include the torques due to spin polarized currents,\ni.e., the STT7and the SOT8. Here, we focus our attention on the STT, consisting of\nadiabatic interactions and their non-adiabatic counterparts. The adiabatic interactions are\nde\fned by values ui, proportional to the electric density current Jx\rowing along the element,\nand calculated as ui=1\n2gi\u0016BP\neMs;iJx, with\u0016Bbeing Bohr's magneton, ethe electron charge,\nandPthe degree of polarization of the spin current. The non-adiabatic interactions are\nproportional to the adiabatic ones by factors \fi.\nThe derivation of the 1DM requires the DW pro\fle to be described in terms of the DW\npositionq, width \u0001 and transition type Q. In the TSLM, the DW is considered to be\ncomposed of two transitions, one for each sublattice, which share the same q, and the same\n\u0001 (see Fig.1.(c) and (d)), but Qi=\u00061 establishes the transition type for each sublattice.\nQi= +1 ( \u00001) means up-down (down-up) transition. Due to the antiferro coupling between\nsublattices, it follows that Q1=\u0000Q2.\nIII. RESULTS AND DISCUSSION\nWhen FiMs, such as GdFeCo or Mn 4N, are grown on top of certain substrates, the ab-\nsence of interfacial asymmetric exchange2,9results in the formation of achiral DWs. The\norientation of DW internal moments at rest is then dependent on purely geometrical as-\npects. In particular, for thin strips su\u000eciently wide, magnetostatic interactions determine\n4Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\nthe formation of Bloch-type walls. Importantly, due to the low net magnetization of FiMs as\ncompared with ferromagnets, the magnetostatic interactions are rather low. If some paral-\nlelism between ferro- and ferrimagnets is made, Walker breakdown in FiMs is then expected\nto occur for rather low applied \felds10or currents11,12in the temperature range around TM.\nConsequently, the DW dynamics for moderate \felds or currents is ruled by the precession\nof DW magnetic moments.\nThe case of the \feld-driven DW dynamics in ferrimagnetic GdFeCo alloys can be recalled\nat this point. This has been the subject of recent experimental work,2where fast \feld-driven\nantiferromagnetic spin dynamics is realized in FiMs at TA. This behavior has been found\nto be reproducible with the TSLM. Our simulations have been carried out with a set of\nparameters similar to those considered in previous works,3,6but adapted as to take into\naccount the absence of interfacial asymmetric exchange and SOTs. The parameters are:\nAi= 70pJ\nm,Keff;i\u0019Ku;i= 1:4MJ\nm3,Ku;ibeing the magnetic uniaxial anisotropy constant\nof the FiM sublattices. With these parameters, DW width is \u0001 \u00196nm. Besides, \u000bi=\n0:02. Due to the low net magnetization in the temperature range of interest, Ksh;i\u0019\n0. The antiferromagnetic coupling is accounted for by the parameter B12= 9MJ\nm3.13The\ngyromagnetic ratios ( \ri=gi\u0016B\n~) are di\u000berent due to distinct Land\u0013 e factors: g1= 2:2 and\ng2= 2:0.2The Curie temperature is set to TC= 450K, and M0\ns;1= 1:4MA\nmandM0\ns;2=\n1:71MA\nm, witha1= 0:5 anda2= 0:76. According to these values, TM\u0019241:5K, and\nTA\u0019305K. The dimensions of the FiM strips are w\u0002tFiM= 512nm \u00026nm.\nFig.2.(a) presents the dependence of the DW terminal velocity, computed as vst=\nq(\u0001t)\u0000q(0)\n\u0001t, with \u0001t= 2ns, on the out-of-plane applied \feld Bzat di\u000berent temperatures.\nIn agreement with experiments,2vstincrease linearly with Bz, and the slope reaches a max-\nimun atTA. This fact is made clear in Fig.2.(b) where terminal velocity is represented as\na function of temperature with Bzas a parameter. In all shown cases, no dynamics occurs\natTMsince the net magnetization vanishes, whereas the highest speeds are found close to\nTA. The clue for this behavior can be found in DW precession, represented as a function\nof temperature in Fig.2.(c). Precession frequencies are obtained as \u0017=_ i(\u0001t)\n2\u0019(i= 1;2),\nsince _ 1(\u0001t)\u0019_ 2(\u0001t). The results demonstrate that during the dynamics, DW magnetic\nmoments precess except at temperatures around TMandTA, where precession freezes and\nthe orientation of DW magnetic moments during the whole dynamics holds.\nPrevious \feld-driven analysis serves as a starting point to also understand the CDDWM in\n5Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)\n050100150200250300350\n0 20 40 60 80 100 120\nvst/parenleftbig\nm ·s−1/parenrightbig\nBz(mT)T= 200K\nT= 241 .5K\nT= 270K\nT= 305K\nT= 340K (d)\n020406080100120\n0 0 .5 1 1 .5 2\n|vst|/parenleftbig\nm·s−1/parenrightbig\nJx/parenleftbig\nTA·m−2/parenrightbigT= 241 .5K\nT= 270K\nT= 300K\nT= 340K (g)\n050100150200\n0 0 .5 1 1 .5 2\n|vst|/parenleftbig\nm·s−1/parenrightbig\nJx/parenleftbig\nTA·m−2/parenrightbigT= 241 .5K\nT= 270K\nT= 300K\nT= 350K\n(b)\n050100150200250300350\n220 240 260 280 300 320 340 360\nvst/parenleftbig\nm ·s−1/parenrightbig\nT(K)Bz= 20mT\nBz= 40mT\nBz= 80mT\nBz= 120mTTM TA (e)\n020406080100120\n220 240 260 280 300 320 340 360\n|vst|/parenleftbig\nm·s−1/parenrightbig\nT(K)Jx= 0.5TA·m−2\nJx= 1.0TA·m−2\nJx= 1.5TA·m−2\nJx= 2.0TA·m−2TM TA (h)\n050100150200250\n220 240 260 280 300 320 340 360\n|vst|/parenleftbig\nm·s−1/parenrightbig\nT(K)Jx= 0.5TA·m−2\nJx= 1.0TA·m−2\nJx= 1.5TA·m−2\nJx= 2.0TA·m−2TM TA\n(c)\n−1−0.500.511.522.5\n200 250 300 350 400\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTMTA (f)\n−1−0.8−0.6−0.4−0.200.20.40.60.8\n220 240 260 280 300 320 340 360\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTM TA (i)\n−2−1.5−1−0.500.511.5\n220 240 260 280 300 320 340 360\nν(GHz)\nT(K)sublattice 1\nsublattice 2\nTM TA\nFIG. 2. Field-driven an current-driven dynamics in a FiM strip: (a) terminal velocity as a function\nofBzwith temperature as a parameter, (b) terminal velocity with Bzas a parameter and (c)\nprecessional frecuencies of DWs for Bz= 40mT as functions of temperature, (d) and (g) terminal\nvelocity as a function of Jxwith temperature as a parameter, (e) and (h) terminal velocity with\nJxas a parameter and (f) and (i) precessional frecuency of DWs for Jx= 1TA\nm2as functions of\ntemperature. \fi=\u000bifor (d), (e) and (f), whereas \fi= 2\u000bifor (g), (h) and (i). Dots and\ncontinuous lines correspond respectively to full \u0016M simulations and the 1DM results.\nthese elements. This dynamics is purely governed by STT because DWs move contrary to the\ncurrent direction.9Fig.2.(d) and (g) present the dependence of the absolute terminal velocity\nas a function of the current Jxwith the temperature as a parameter. The polarization has\nbeen set to P= 0:7, and the non-adiabatic transfer torque parameters have been chosen\nas (d)\fi=\u000bi(also for \fgures (e) and (f)), and (g) \fi= 2\u000bi(also for \fgures (h) and (i)).\nDi\u000berently from the results obtained in the \feld-driven case, the CDDWM at TMis not\nnull, since the STT pushes the transitions in each sublattice in the same direction (and not\nin opposite directions as it occurs in the \feld-driven case). However, the maximum slope is\nagain found at TA, when the precessional frequency vanishes.\nTo show in more detail this behavior, Fig.3 presents the snapshots of the CDDWM at\ntwo representative temperatures, for the case \fi=\u000bi. The two sublattices composing the\nFiM are presented superposed, as to simplify the view, so one sublattice is on top of the\n6Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\n(a)T < T A (b)T=TA\n(b) up -down , \n0 ns 𝑡(ns)\n𝑥\n0.5 ns\n1 ns\n𝐽𝑥=1TA\nm2\n𝑚𝑥𝑚𝑦\n(b) up -down , \n0 ns 𝑡(ns)\n𝑥\n0.5 ns\n1 ns\nFIG. 3. Snapshots of the CDDWM in a FiM strip with \fi=\u000biat (a)T <TA, and (b)T=TA.\nother. The images in (a) correspond to the dynamics at T < TA. In this case, the DW\ninternal moments precess, and a turn of approximately 180otakes place within the 1ns-\ninterval passing from the image on top to the bottom image. However, no precession takes\nplace atT=TA, as shown in (b). The distinct distances run by the DWs can be also\ncompared.\nDi\u000berently from the behavior of magnetic moments in pure ferromagnets, where STT\ncompensates damping when \fi=\u000bi, FiMs seem to present these precessing magnetic mo-\nments even in this case. Such precession would be associated with the torque due to the\ncoupling between the two sublattices and freezes at TA, a result that would not be in any\ncase explainable by means of e\u000bective models.\nIV. CONCLUSIONS\nThe aim of this work has been \frst to highlight the capacities of the TSLM and, par-\nticularly, the 1DM based on it, to reproduce recent experimental work on DW dynamics in\nFiMs. Di\u000berently from previous approaches, the TSLM does not require the use of e\u000bective\nparameters, but experimentally determined ones, which allows providing insightful details\nabout the dynamics.\nThe work has been devoted to FiMs structured so that DWs adopt achiral Bloch con\fg-\nurations at rest, and the main conclusions of this work are as following. The DW dynamics\n7Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\nin FiMs is characterized by DW precession, which freezes at TA. Because of that, the DW\nvelocity at TAis enhanced, both for the \feld- and for the current-driven cases. Our results\nare also in good qualitative agreement with recent experimental observations: Ref.2 for the\n\feld-driven case, and Ref.9 for the current-driven one. Finally, the physical origin or the\nfundamental reasons behind these observations can only be achieved by adopting models\nwhich consider the independent but antiferromagnetically coupled nature of the two sublat-\ntices forming the FiM. Therefore, our models will be useful to understand state-of-the-art\nexperiments and also to develop and optimize future DW-based devices.\nV. ACKNOWLEDGEMENT\nThis work was partially supported by Project No. MAT2017-87072-C4-1-P from the\n(Ministerio de Econom\u0013 \u0010a y Competitividad) Spanish Government and Project No. SA299P18\nfrom the (Consejer\u0013 \u0010a de Educaci\u0013 on) of Junta de Castilla y Le\u0013 on.\nVI. BIBLIOGRAPHY\nREFERENCES\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, \\Magnetic domain wall racetrack memory,\"\nScience 320, 190 (2008).\n2K.-J. Kim, S. K. Kim, Y. Hirata, S.-H. Oh, T. Tono, D.-H. Kim, T. Okuno, W. S. Ham,\nS. Kim, G. Go, Y. Tserkovnyak, A. Tsukamoto, T. Moriyama, K.-J. Lee, and T. Ono, \\Fast\ndomain wall motion in the vicinity of the angular momentum compensation temperature\nof ferrimagnets,\" Nature Materials 16, 1187{1192 (2017).\n3L. Caretta, M. Mann, F. B uttner, K. Ueda, B. Pfau, C. M. G unther, P. Hessing,\nA. Churikova, C. Klose, M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik,\nS. Eisebitt, and G. S. D. Beach, \\Fast current-driven domain walls and small skyrmions\nin a compensated ferrimagnet,\" Nature Nanotechnology 3(2018).\n4S. A. Siddiqui, J. Han, J. T. Finley, C. A. Ross, and L. Liu, \\Current-induced domain\nwall motion in a compensated ferrimagnet,\" Physical Review Letters 121, 057701 (2018).\n5\u0013O. Alejos, V. Raposo, L. Sanchez-Tejerina, R. Tomasello, G. Finocchio, and E. Martinez,\n\\Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-\n8Current driven domain wall dynamics in ferrimagnetic strips explained by means of a two interacting sublattices model\ncoupled by a spacer: A micromagnetic study,\" Journal of Applied Physics 123(1) , 013901\n(2018).\n6E. Mart\u0013 \u0010nez, V. Raposo, and \u0013O. Alejos, \\Current-driven domain wall dynamics in ferrimag-\nnets: Micromagnetic approach and collective coordinates model,\" Journal of Magnetism\nand Magnetic Materials 491, 165545 (2019).\n7P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans,\nNature Materials 12, 299 (2013).\n8S. Zhang and Z. Li, Physical Review Letters 93, 1 (2004).\n9T. Gushi, M. Jovi\u0014 cevi\u0013 c Klug, J. Pe~ na Garc\u0013 \u0010a, H. Okuno, J. Vogel, J. P. Attan\u0013 e, T. Suemasu,\nS. Pizzini, and L. Vila, \\ mn4nferrimagnetic thin \flms for sustainable spintronics,\" (2019),\narXiv:1901.06868 [cond-mat.mtrl-sci].\n10A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferr\u0013 e, \\Domain wall mobility,\nstability and walker breakdown in magnetic nanowires,\" Europhysics Letters 5, 57007\n(2007).\n11A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, Journal of Applied Physics 95(11) ,\n7049 (2004).\n12A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhysics Letters 69, 990 (2005).\n13C. T. Ma, X. Li, and S. J. Poon, \\Micromagnetic simulation of ferrimagnetic tbfeco \flms\nwith exchange coupled nanophases,\" Journal of Magnetism and Magnetic Materials 417,\n197{202 (2016).\n9"    },    {        "title": "2312.00367v2.Compensated_Ferrimagnets_with_Colossal_Spin_Splitting_in_Organic_Compounds.pdf",        "content": "Compensated Ferrimagnets with Colossal Spin Splitting in Organic Compounds\nTaiki Kawamura1, Kazuyoshi Yoshimi2, Kenichiro Hashimoto3, Akito Kobayashi1, and Takahiro Misawa2∗\n1Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan\n2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n3Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan\nThe study of the magnetic order has recently been invigorated by the discovery of exotic collinear\nantiferromagnets with time-reversal symmetry breaking. Examples include altermagnetism and\ncompensated ferrimagnets, which show spin splittings of the electronic band structures even at zero\nnet magnetization, leading to several unique transport phenomena, notably spin-current generation.\nAltermagnets demonstrate anisotropic spin splitting, such as d-wave, in momentum space, whereas\ncompensated ferrimagnets exhibit isotropic spin splitting. However, methods to realize compensated\nferrimagnets are limited. Here, we demonstrate a method to realize a fully compensated ferrimagnet\nwith isotropic spin splitting utilizing the dimer structures inherent in organic compounds. Moreover,\nbased on ab initio calculations, we find that this ferrimagnet can be realized in the recently discovered\norganic compound (EDO-TTF-I) 2ClO 4. Our findings provide an unprecedented strategy for using\nthe dimer degrees of freedom in organic compounds to realize fully compensated ferrimagnets with\ncolossal spin splitting.\nIntroduction.— Collinear antiferromagnets have tradi-\ntionally been viewed as conventional magnetic orderings\nthat lack unique phenomena such as spin current gener-\nation. However, recent theoretical advances have identi-\nfied exotic collinear antiferromagnets with time-reversal\nbreaking, notably altermagnets [1–3] and compensated\nferrimagnets [4]. These magnetic states exhibit spin\nsplitting in their electronic band structures even with-\nout net magnetization. Because spin splitting may drive\nspin-dependent novel transport phenomena and uncon-\nventional superconducting phases, these distinctive anti-\nferromagnets have attracted considerable attention.\nSeveral materials have been proposed as candidates\nfor altermagnetism [3, 5–13]. For example, κ-ET-\ntype organic compounds [6] and transition metal oxide\nRuO 2[8, 10, 14] exhibit altermagnetism with anisotropic\nspin splitting in electronic band structures. These mate-\nrials are expected to exhibit spin-dependent transport\nand anomalous Hall effects owing to anisotropic spin\nsplitting. In contrast, compensated ferrimagnets offer\nisotropic spin splitting, increasing efficiency for spin-\ncurrent generation. The concept of metallic compen-\nsated ferrimagnetism (half-metallic antiferromagnetism)\nwas introduced by van Leuken and de Groot [15]. Since\nthen, several candidate materials have been suggested us-\ningab initio calculations [16, 17]. More recently, mono-\nlayer MnF 2was proposed as an insulating compensated\nferrimagnet [18]. Since the magnetic moment of compen-\nsated insulating ferrimagnets is strictly zero due to the\nLuttinger theorem [4], small perturbations do not change\nthe compensation condition. In addition, compensated\nferrimagnets have lower crystal symmetry than altermag-\nnets [4]. Therefore, compensated ferrimagnets have ad-\nvantages over altermagnets, leading to various potential\napplications, such as thin-film synthesis. However, the\nnumber of compensated ferrimagnets discovered experi-\nmentally is limited [19–22]. To harness the compensated\n(a)\nA\nA’B\nB’t1s\nt2\n 0  0.5  1up spin\ndown spin\nk/2π(b)\n-3-2-1 0 1 2 3 4\ns s\ns s sFIG. 1. (a) Schematic of a one-dimensional model showing\nthe compensated ferrimagnetism. The broken lines show a\nunit cell. The inter-dimer hopping integral s, the intra-dimer\nhopping integrals t1andt2are represented by the horizontal\nthin black lines, the vertical thick red line, and the vertical\nthin blue line, respectively. The up (down) spin polariza-\ntions are described by the yellow upward (purple downward)\narrows. (b) Band structures of the Hamiltonian defined in\nEq. (2). We take t1= 1.0,t2= 0.6,s= 0.6,δ= 0.2, and\n∆ = 1 .5. For comparison, we also show the band structures\nof the equivalent dimer case with the broken curves, whose\nparameters are given by t1=t2= 1.0,s= 0.6,δ= 0.0, and\n∆ = 1 .5.\nferrimagnets, a simple method to realize them is neces-\nsary.\nIn this Letter, we present a path towards fully com-\npensated ferrimagnets with colossal spin splitting us-\ning typical dimer structures in organic compounds. Us-\ning a simple one-dimensional model, we demonstrate\nthat a collinear antiferromagnetic order with inequiva-\nlent dimers can induce fully compensated ferrimagnets.\nFurthermore, we find that the recently discovered or-\nganic compound (EDO-TTF-I) 2ClO 4[EDO-TTF-I=4,5-\nethylenedioxy-4′-iodotetrathiafulvalene] [23] can realize\nthis mechanism based on ab initio calculations. The ex-\nperiments demonstrate that (EDO-TTF-I) 2ClO 4under-arXiv:2312.00367v2  [cond-mat.mtrl-sci]  21 Mar 20242\ngoes a structural transition with anionic ordering at 190\nK. Below the structural transition, unit-cell doubling oc-\ncurs with the extended unit cell containing two inequiv-\nalent dimers. By deriving and solving the ab initio effec-\ntive Hamiltonian for the low-temperature phase of (EDO-\nTTF-I) 2ClO 4, we find that the ground state is a collinear\nantiferromagnet with isotropic spin splitting, i.e., a fully\ncompensated ferrimagnet.\nSimple model.— First, to show the key idea for real-\nizing the compensated ferrimagnets, we consider a one-\ndimensional model whose unit cell contains two inequiv-\nalent dimers. A schematic of the model is illustrated in\nFig. 1 (a). The inequivalence between the two dimers\nis characterized using difference between the intra-dimer\nhoppings t1andt2and chemical potential difference δ.\nWe also consider the collinear dimer antiferromagnetic\n(DAF) state, where the up-(down-)spin electrons are lo-\ncated on A and A′(B and B′). This DAF state is not in-\nvariant to any combination of time reversal with transla-\ntion/rotation operations because of dimer inequivalence.\nThus, isotropic spin splitting is expected in this dimer\ncollinear DAF state.\nTo determine the mechanism of spin splitting in this\nmodel, we consider the following tight-binding Hamilto-\nnian for the DAF state:\nH=X\nk,σc†\nkσHσ(k)ckσ, (1)\nHσ(k) =\nσ∆ t1 A(k) 0\nt1 σ∆ 0 A(k)\nA(k)∗0 −σ∆ +δ t 2\n0 A(k)∗t2 −σ∆ +δ\n,\n(2)\nwhere c†\nkσ= (c†\nAkσ, c†\nA′kσ, c†\nBkσ, c†\nB′kσ),A(k) = s(1 +\ne−ik) and ∆ denotes the gap induced by the DAF or-\nder. The spin index σtakes +1 and −1 for the up- and\ndown-spins, respectively. The eigenvalues of this Hamil-\ntonian are given by\nE0,σ,±(k) =δ/2+t+±[(σ∆ +t−−δ/2)2+ 2s2C(k)]1/2,\n(3)\nE1,σ,±(k) =δ/2−t+±[(σ∆−t−−δ/2)2+ 2s2C(k)]1/2,\n(4)\nwhere t±= (t1±t2)/2 and C(k) = (1+cos k). Thus, spin\nsplitting of the bands is induced by t−= (t1−t2)/2 and δ,\ni.e., inequivalence of the dimers. From these expressions,\nit is evident that the differences in intra-dimer hoppings\nhave the similar gap-opening effect as the differences in\nthe chemical potentials. Because t−andδare indepen-\ndent of the wave number, spin splitting is isotropic.\nFigure 1(b) shows the electronic band structures of\nthe Hamiltonian defined in Eq. (2) for a typical param-\neter set. We also plot the band structures when the two\ndimers are equivalent ( t1=t2andδ= 0). As expected,\nFIG. 2. (a) Crystal structure of (EDO-TTF-I) 2ClO 4atT=\n100 K and the real-space distribution of maximally localized\nWannier functions (MLWFs) drawn by VESTA [24]. Anions\nClO 4are represented by balls and rods. Colors of anions rep-\nresent their orientations. Due to the different configurations\nof anions around the dimers, two dimers (A-A′and B-B′) be-\ncome inequivalent. Black lines show the unit cell including\nfour molecules. (b) Energy band structure obtained by the\nDFT calculation (red lines) and the MLWFs (blue circles) for\nthe paramagnetic states. The Fermi energy is set to zero.\nisotropic spin splitting occurs in the band structures. At\ncommensurate filling (i.e., three-quarter, half, and quar-\nter filling), the DAF state is insulating, and the net mag-\nnetization is zero because the number of up- and down-\nbands below the Fermi energy are the same. Thus, the\nDAF state at the commensurate filling is the fully com-\npensated ferrimagnets with isotropic spin splitting.\nCrystal and electronic structures of (EDO-TTF-\nI)2ClO 4.— First, we summarize the crystal structure\nof (EDO-TTF-I) 2ClO 4. (EDO-TTF-I) 2ClO 4consists of\nEDO-TTF-I molecule with +1 /2 charge (3 /4 filling) and\nanion ClO 4layers. Above 190 K, the unit cell contains\ntwo EDO-TTF-I molecules and space inversion symme-\ntry is macroscopically protected because of the random\norientation of ClO 4. By lowering the temperature, the\nstructural phase transition with anion ordering occurs at\napproximately T= 190 K, which induces unit-cell dou-\nbling, as evidenced by the X-ray analysis. As shown in\nFig. 2 (a), each unit cell contains four molecules in the\nlow-temperature phase. A and A′(B and B′) molecules\nin a unit cell form a dimer, referred to as dimer I (dimer\nII). The inversion centers are at the centers of each\ndimer. The inequivalence of the dimers I and II can\nalso be understood from the partial density of states\n(PDOS), which are shown in the Supplemental Mate-\nrial [25]. Because dimers I and II are inequivalent af-\nter the structural phase transition, (EDO-TTF-I) 2ClO 4\nwill exhibit compensated ferrimagnetism if appropriate\nantiferromagnetic order occurs. We note the related or-\nganic compound (EDO-TTF-I) 2PF6[26] has similar crys-\ntal structures; however, all the dimers are equivalent.3\nThis indicates that the ordering of ClO 4plays an essen-\ntial role in realizing inequivalent dimers.\nNext, we summarize the low-temperature electronic\nstructure of (EDO-TTF-I) 2ClO 4. Although the resis-\ntivity exhibits semimetallic behavior immediately after\nthe structural phase transition, a metal-insulator phase\ntransition occurs at approximately T= 95 K [23]. It\nis confirmed that the metal-insulator transition does not\naccompany a structural change. As explained later, ab\ninitio analysis suggests that the DAF ordering is the ori-\ngin of the insulating phase although the origin of the\nmetal-insulator transition is not experimentally clarified\nyet.\nAb initio effective Hamiltonian for (EDO-TTF-\nI)2ClO 4.—To investigate the electronic structures of the\nlow-temperature phase in (EDO-TTF-I) 2ClO 4, we derive\ntheab initio low-energy effective Hamiltonian, based on\nband structures obtained by the density functional the-\nory (DFT). We use Quantum ESPRESSO [27] to ob-\ntain the DFT bands. In this study, we employ the opti-\nmized norm-conserving Vanderbilt pseudopotentials and\nplane-wave basis sets [28, 29]. The exchange-correlation\nfunctional used in this study is the generalized gradient\napproximation proposed by Perdew, Burke, and Ernzer-\nhof [30]. The cut-off energy of the wave functions and\ncharge densities are set to be 80 Ry and 320 Ry, respec-\ntively. During the self-consistent loop process, a 5 ×5×3\nuniform k-point grid and Methfessel-Paxton smearing\nmethod are used [31]. We use the crystal structure data\nat 100 K [23]. We perform structural optimization for\nhydrogen atoms and use the optimized structure in the\nfollowing analyses. In the DFT calculations, we only con-\nsider the paramagnetic solutions. Figure 2 (b) shows the\nobtained energy band structures of (EDO-TTF-I) 2ClO 4.\nThe four bands around the Fermi level, which are isolated\nfrom the other bands, mainly consist of the highest occu-\npied molecular orbitals of the four EDO-TTF-I molecules\nin the unit cell (A,A′,B, and B′). We select these four\nbands as the low-energy degrees of freedom and use them\nto construct the maximally localized Wannier functions\n(MLWFs) using RESPACK [32]. Isosurfaces of the ML-\nWFs are shown in Fig.2 (a). In addition, we confirm\nthat the bands interpolated by the MLWFs accurately\nreproduce the DFT band structures (Fig.2 (b)).\nAfter constructing MLWFs, we derive low-energy ef-\nfective Hamiltonian, which is given by\nHEDO=H0+Hint,\nH0=X\ni,j,α,β,σtiαjβc†\niασcjβσ,\nHint=X\ni,αUiαniα↑niα↓+1\n2X\ni,j,α,βViαjβNiαNjβ,(5)\nwhere c†\niασ(ciασ) is a creation (annihilation) operator\nfor an electron in the i-th unit cell with orbitals α= A,A′, B, B′and spin σ. The number operators are de-\nfined as niασ=c†\niασciασandNiα=niα↑+niα↓. The\ntransfer integrals tiαjβare evaluated using the MLWFs.\nWe also evaluate the screened Coulomb interactions Uiα\nandViαjβusing the constrained random phase approxi-\nmation [33, 34] implemented in RESPACK [32]. We set\nthe cutoff energy of the polarization function to 5 .0 Ry.\nDetails of the transfer integrals and interaction parame-\nters are summarized in the Supplemental Material [25].\nWe note that the difference in the intra-dimer hoppings\nt1andt2and the existence of the site potential δindi-\ncate the inequivalence of the dimers. In actual calcula-\ntions, we subtract a constant value ∆ DDFfrom onsite and\noffsite Coulomb interactions to consider the interlayer\nscreening [35, 36]. Following previous studies [36–39],\nwe employ ∆ DDF= 0.2 eV. We confirm that the results\nare insensitive to ∆ DDF. We also perform an electron-\nhole transformation to reduce the numerical cost.\nmany-variable variational Monte Carlo (mVMC)\nanalysis.— The effective Hamiltonian is solved using the\nmVMC method [40, 41], which can take into account\nquantum fluctuations and spatial correlations seriously.\nThe trial wave function used in this study is given by\n|ψ⟩=PGPJLS|ϕpair⟩, (6)\nPG= exp\"X\nigini↑ni↓#\n, (7)\nPJ= exp\n1\n2X\ni̸=jvijNiNj\n, (8)\n|ϕpair⟩=\nNsiteX\ni,jfijc†\ni↑c†\nj↓\nNe/2\n|0⟩, (9)\nwhere PG,PJ, and LSare the Gutzwiller factor [42],\nlong-range Jastrow factor [43, 44], and total spin projec-\ntor [45, 46], respectively [41, 47]. NeandNsiteindicate\nthe number of electrons and sites, respectively. We im-\npose a 1 ×4 sublattice structure on variational parame-\nters. We use the Hartree-Fock approximation results as\nthe initial fijvalues. The spin-singlet projection ( S= 0)\nis used in ground-state calculations for La=Lb≤8,\nwhile it is not used for La=Lb≥10 to reduce the nu-\nmerical costs. We confirm that the spin projection does\nnot largely affect physical quantities, such as the spin\nstructure factors. All variational parameters are simul-\ntaneously optimized using the stochastic reconfiguration\nmethod [48].\nAs detailed in the Supplementary Materials, the\nHartree-Fock approximation shows that the DAF state\n(Fig.3 (a)) and antiferromagnetic states with charge or-\ndering (AF+CO) are ground state candidates. Within\nthe Hartree-Fock approximation, the ground state of the\neffective Hamiltonian is the AF+CO state. However, us-\ning the mVMC method, we find that the DAF state be-4\nFIG. 3. (a) Schematic of the DAF state and representative\nhopping integrals ( t1-t6) in the conducting layer of (EDO-\nTTF-I) 2ClO 4. (b) Spin structure factor obtained by the\nmVMC method for La=Lb= 12. Sharp peaks appear at\nq= (0, π/2),(0,3π/2), which correspond to the DAF state.\n(c) Size dependence of the peak value of the spin structure\nfactors. (d) Doping dependence of the chemical potential.\ncomes the ground state of the effective Hamiltonian. We\nalso find that the AF+CO state converges to the DAF\nstate after optimization, even when using the variational\nparameters of AF+CO state as the initial state. This\nresult indicates that correlation effects beyond the mean-\nfield approximation are important in stabilizing the DAF\nstate.\nTo examine the existence of long-range antiferromag-\nnetic order, we calculate the spin-correlation functions of\nthe ground state, defined as\nS(q) =1\n(Nsite)2X\ni,j⟨Si·Sj⟩eiq·(ri−rj), (10)\nwhere the original lattice structure is mapped to the\nequivalent La×4Lbsquare lattice for simplicity. Fig-\nure 3 (b) shows S(q) in the momentum space, with sharp\nBragg peaks at ( qx, qy)=(0, π/2) and (0 ,3π/2). As shown\nin Fig. 3 (c), we confirm that the peak values of S(q) re-\nmain finite in the thermodynamic limit. We also confirm\nthat charge densities are uniform and there is no signa-\nture of a charge-ordered state. These results show that\nthe ground state of the effective Hamiltonian is the DAF\nstate. In addition, we calculate the charge gap ∆ c, given\nby ∆ c=µ(Ne+ 1)−µ(Ne−1), where the chemical po-\ntential is defined as µ(Ne+ 1) = [ E(Ne+ 2)−E(Ne)]/2.\nFigure 3 (d) shows the doping rate γd=Ne/Nsite−1.5\ndependence of the chemical potential. From this plot, we\nestimate the charge gap [∆ c=µ(Ne+ 1)−µ(Ne−1)] as\n∆c∼0.4 eV. This result indicates that the ground state\nFIG. 4. (a) Density of state (DOS) of the low-energy effective\nHamiltonian (EDO-TTF-I) 2ClO 4for the DAF state obtained\nby the Hartree-Fock approximation. DOS for up (down) spin\nis described by the red (blue) lines. (b) Band dispersions of\nthe DAF state obtained by the Hartree-Fock approximation.\nThe red (blue) surfaces describe up-spin (down-spin) band\ndispersions.\nof (EDO-TTF-I) 2ClO 4is the DAF insulator.\nSpin splitting.— Based on the results obtained using\nthe mVMC method, we analyze spin splitting in the DAF\nstate using the Hartree-Fock approximation (for more de-\ntails, see Ref. [25]). We assume the DAF order and scale\nthe interaction parameters to reproduce the charge gap\n∆c∼0.4 eV obtained from the mVMC calculations. The\nscaling ratio λ, which monotonically scales the onsite and\noffsite Coulomb interactions, is estimated to be λ= 0.7.\nThe details of the Hartree-Fock calculations are provided\nin the Supplemental Material.\nUsing the Hartree-Fock approximation, we calculate\nthe DOS defined as Dσ(ω)=(πLaLb)−1P\nk,nIm(ω−iη−\nEk,n,σ+µ)−1, where µandηare the chemical poten-\ntial and the smearing factor, respectively. Ek,n,σdenotes\nthen-th eigenvalue of the mean-field Hamiltonian at mo-\nmentum k. We set η= 0.002 eV. As shown in Fig. 4 (a),\nspin splitting occurs in the DAF order ( D↑(ω)̸=D↓(ω)).\nThe electronic band dispersions in Fig. 4(b) also show\nisotropic spin splitting over the entire Brillouin zone.\nThis demonstrates that (EDO-TTF-I) 2ClO 4can be fully\ncompensated ferrimagnets if a DAF order occurs.\nHere, we analyze the origin of spin splitting in (EDO-\nTTF-I) 2ClO 4. As in the case of the simple model,\nt−= (t1−t2)/2 and δcan induce spin splitting. From\ntheab initio calculations, we find that t−= 0.036 eV\nis comparable to δ= 0.047 eV. Thus, spin splitting in\n(EDO-TTF-I) 2ClO 4is induced by both t−andδ. One\nmight think that finite δwould make the total magneti-\nzation finite; however, a charge gap guarantees that the\ntotal magnetization is robust against perturbations. In\nthis case, the total magnetization is zero at δ= 0 and\nremains zero even if δis added, provided that δis signif-\nicantly smaller than the charge gap.\nSummary and discussion.— In this Letter, we present\na simple method to realize fully compensated ferrimag-\nnetism using the dimer degrees of freedom, which are5\ntypical of organic compounds. Using a simple model,\nwe demonstrate that the inequivalence of the two dimers\nand DAF order can induce fully compensated ferrimag-\nnets at commensurate filling. Furthermore, ab initio cal-\nculations suggest that the ground state of (EDO-TTF-\nI)2ClO 4is a DAF insulator with inequivalent dimers. As\na result, the DAF order induces isotropic spin splitting\nin the electronic band structure. Our study shows that\nthe key to realizing compensated ferrimagnetism lies in\ninequivalent dimer structures induced by anion ordering.\nThis finding offers an unanticipated direction in materi-\nals design, where exotic magnetism can be achieved by\nselecting and modifying anions to exhibit anion order-\ning. The discovery of compensated ferrimagnetism in\ninequivalent dimer structures, as well as the potential for\nmaterials design using anion ordering, demonstrates that\norganic compounds offer a versatile platform for realiz-\ning exotic magnetism. An intriguing future issue would\nbe to examine the doping effects in the DAF insulating\nstate. Because the lowest unoccupied band is fully po-\nlarized and its DOS is large (Fig.4 (a)), unconventional\nsuperconductivity, such as triplet superconductivity, is\nexpected [49]. Further experimental and theoretical in-\nvestigations in this direction are desirable.\nThe authors thank Y. Nakano, M. Ishikawa, H.\nYamochi, and A. Otsuka for the fruitful discussions. This\nwork was financially supported by Grants-in-Aid for Sci-\nentific Research (KAKENHI) (Grant Nos. 23H03818,\n23KJ1065, 15K05166, 22K18683 21H01793), Grant-in-\nAid for Scientific Research for Transformative Research\nAreas (A) “Condensed Conjugation” (No. JP20H05869)\nfrom Japan Society for the Promotion of Science (JSPS),\nand JST SPRING (Grant No. JPMJSP2125). The com-\nputations were performed using the facilities at the Su-\npercomputer Center, Institute for Solid State Physics,\nUniversity of Tokyo.6\nSupplemental Material for “Compen-\nsated Ferrimagnets with Colossal Spin\nSplitting in Organic Compounds”\nPARAMETERS IN AB INITIO LOW-ENERGY\nEFFECTIVE HAMILTONIANS\nWe calculate the electronic band structures by\nthe density functional theory (DFT) using Quantum\nESPRESSO [27] and evaluate the transfer integrals\nby the maximally localized Wannier functions (ML-\nWFs). Then, we evaluate Coulomb interaction by the\nconstraint random phase approximation (cRPA) using\nRESPACK [32]. The values of the transfer integrals\nlarger than 0 .020 eV and the Coulomb interactions are\nshown in Table I. We show schematic illustrations of the\ntransfer integrals and the Coulomb interactions in Fig. 5.\nAll input and output files are uploaded to the ISSP data\nrepository [50].\nTransfer integrals [eV] Coulomb interactions [eV]\nδ 0.047 UA=UA′ 2.094\nUB=UB′ 2.076\nt1 0.252 V1 0.903\nt2 0.179 V2 0.884\nt3 0.128 V3 0.880\nt4 0.112 V4 0.700\nt5 0.084 V5 0.681\nt6 0.058 V6 0.614\nt7 - V7 0.659\nt8 - V8 0.628\nTABLE I. Transfer integrals and screened Coulomb interac-\ntions of the ab initio low-energy effective Hamiltonians for\n(EDO-TTF-I) 2ClO 4.\nFIG. 5. Schematic of the transfer integrals and the off-site\nCoulomb interactions in the conduction layer.7\nPARTIAL DENSITIES OF STATES\nTo see the inequivalence of the dimers, we calculate\nthe partial densities of states (PDOS) using the one-body\npart of the low-energy effective Hamiltonians for (EDO-\nTTF-I) 2ClO 4in the momentum space, which is given by\nH0\nσ(k) =X\nkX\n∆r,α,βt∆r,αβeik·∆rc†\nk,α,σck,β,σ\n=X\nkX\n∆r,α,βH0\nαβ,σ(k)c†\nk,α,σck,β,σ.(11)\nHere, t∆r,αβis the transfer integrals obtained by\nRESPACK and ∆ rrepresents the translational vector.\nThe Hamiltonian H0\nσ(k) satisfies the following eigenvalue\nequation:\nH0\nσ(k)|k, n, σ⟩=En,σ(k)|k, n, σ⟩, (12)\n|k, n, σ⟩=\ndA,n,σ(k)\ndA′,n,σ(k)\ndB,n,σ(k)\ndB′,n,σ(k)\n, (13)\nwhere En,σ(k) and|k, n, σ⟩are the eigenvalue and eigen-\nvectors of H0. The band and spin indices are represented\nbynandσ, respectively. Using En,σ(k) and dα,n,σ(k),\nthe PDOS Dα(ω) can be calculated as\nDα(ω) =1\nπLaLbX\nk,n,σIm|dα,n,σ(k)|2\nω−iη−Ek,n,σ+µ,(14)\nwhere ηtakes the positive infinitesimal value and µis the\nchemical potential determined to set the electron num-\nber to 6. We set η= 0.002 eV in the numerical calcu-\nlation. Figure 6 shows the PDOS Dα(ω). We consider\nthe paramagnetic state in this calculation. The inequiva-\nlence in DA(ω) and DB(ω) indicates the inequivalence of\nthe dimer I and II. We note that the relations DA(ω) =\nDA′(ω) and DB(ω) =DB′(ω) are satisfied due to space-\ninversion symmetry.\nFIG. 6. PDOS obtained by the tight-binding model. Inequiv-\nalence of the dimers appears in the PDOS ( DA(ω) ̸=DB(ω)).8\nDETAILS OF THE HARTREE-FOCK\nAPPROXIMATION\nTo examine the ground state candidates of (EDO-\nTTF-I) 2ClO 4, we perform the Hartree-Fock (HF) ap-\nproximation to the ab initio low-energy effective Hamil-\ntonians defined in Eq. (5) in the main text. We use the\nunrestricted HF code implemented in mVMC [41]. Us-\ning the results of HF calculations, we generate the initial\nvariational parameters for the mVMC method. In this\nstudy, we examine the ordered states with q= 0, i.e.,\nwe consider the symmetry-broken states within the unit\ncell. As illustrated in Fig. 7, we consider five initial\nstates: (i) the paramagnetic (PM) state, (ii) the ferro-\nmagnetic (FM) state, (iii) the dimer antiferromagnetic\n(DAF) state, (iv) the AF state, (v) the AF state with\ncharge ordering (AF+CO). To examine the correlation ef-\nfects, we introduce the parameter λ, which monotonically\nscales the Coulomb interactions as ˜Uiα≡λ(Uiα−∆DDF)\nand˜Viαjβ≡λ(Viαjβ−∆DDF). Here, ∆ DDFdenotes the\nconstant shift that takes into account the screening ef-\nfects between conduction layers [35, 36].\nFigures 8 (a) and (b) show the charge density\nnC\ni\u000b\n=\n⟨ni↑+ni↓⟩and the spin density\nnS\ni\u000b\n=⟨ni↑−ni↓⟩at the\nith (=A, A′, B, B′) site in the ground states obtained\nby the HF approximation for ∆ DDF = 0.2 eV, respec-\ntively. We find that the PM state [ ⟨nC\nA(B)⟩=⟨nC\nA′(B′)⟩\nand⟨nS\ni⟩= 0] is the ground state below λ∼0.3. In\n0.3≲λ≲0.5, DAF state [ ⟨nC\nA(B)⟩=⟨nC\nA′(B′)⟩and\n⟨nS\nA⟩=⟨nS\nA′⟩=−⟨nS\nB⟩=−⟨nS\nB′⟩] becomes the ground\nstate. For λ≳0.5, the AF state with the CO [ ⟨nC\nA(B′)⟩>\n⟨nC\nA′(B)⟩and⟨nS\nA′(B′)⟩>⟨nS\nA(B)⟩] becomes the ground\nstate. Figure 8 (c) shows the λ-∆DDF phase diagram.\nSince amplitudes of the on-site and off-site Coulomb in-\nteractions become small with increasing ∆ DDF, the phase\nboundary between the DAF state and the AF+CO state\nslightly shifts to large λregion by increasing ∆ DDF.9\nFIG. 7. Schematic illustrations of initial states used in the HF approximation. The up and down arrows indicate the spin-up\nand spin-down states, respectively. The orange and blue ellipses represent the dimer states constructed by A-A′molecules and\nB-B′molecules, respectively. The green circles represent the charge-rich sites.10\nFIG. 8. λdependence of (a) the charge density and (b) the spin density. (c) λ-∆DDF phase diagram obtained by the HF\napproximation.11\nMVMC ANALYSIS OF THE GROUND-STATE\nPHASE DIAGRAM\nBased on the results obtained by the HF calcula-\ntions, we investigate the ground states using the mVMC\nmethod, which can treat correlation effects more ac-\ncurately. Following previous studies [36–39], we take\n∆DDF= 0.20 eV. Figure 9(a) shows the phase diagram\nas a function of λ. By increasing λ, the phase transi-\ntion between the DAF state and the PM state occurs\naround λ= 0.5. Above λ∼2.2, the AF+CO state be-\ncomes the ground state due to the off-site Coulomb inter-\nactions. Figure 9(b) shows the energy difference between\nthe PM state (the AF+CO state) and the DAF state,\ni.e., ∆ E1=EPM−EDAF(∆E2=EAF+CO −EDAF) as\na function of λforLa=Lb= 6 lattice. The AF+CO\nstate is a quasi-stable state for λ≳1.4 and becomes the\nground state for λ≳2.2. We note that the AF+CO state\nis not stabilized even when we select the AF+CO state\nas an initial state for λ≲1.2. Figure 10 shows the spin\nand charge density structure factors of the PM, the DAF,\nand the AF+CO states. The charge structure factor is\ndefined by\nN(q) =1\n(Nsite)2X\ni,j\n(Ni−¯N)·(Nj−¯N)\u000b\neiq·(ri−rj),\n(15)\n¯N=1\nNsiteX\ni⟨Ni⟩. (16)\nThe ordering wave vectors q= (0, π/2),(0,3π/2) in S(q)\n(Fig. 10 (b)) correspond to the DAF state. Meanwhile,\nthe ordering wave vectors q= (0, π/2),(0,3π/2) in S(q)\nandq= (0, π) inN(q) (Fig. 10(c) and (f)) correspond\nto the AF+CO state.12\nFIG. 9. (a) The ground state phase diagram as a function of λobtained by the mVMC calculation. (b) λ-dependencies of the\nenergy difference between the PM state and the DAF state (∆ E1) and the one between the AF+CO state and the DAF state\n(∆E2).13\nFIG. 10. Spin structure factors S(q) [charge structure factors N(q) ] of the PM state ( λ= 0.3), the DAF state ( λ= 1), and\nthe AF+CO state ( λ= 2.4) in (a), (b), and (c) [(d), (e), and (f)], respectively.14\nMEAN-FILED HAMILTONIAN IN THE\nMOMENTUM SPACE\nTo see the spin splitting of the DAF states, we calcu-\nlate band dispersions and DOS using the one-body Green\nfunctions obtained by the HF approximation. Here, we\nsetλ= 0.7, which reproduces the charge gap estimated\nby the mVMC method. By performing the Fourier trans-\nformation for the mean-field Hamiltonian in the real\nspace, we obtain the mean-field Hamiltonian in the mo-\nmentum space, which is given by\nH=X\nkX\n∆r,α,β,σt∆r,αβeik·∆rc†\nk,α,σck,β,σ\n+X\nkX\nα,σUα⟨nα,¯σ⟩c†\nk,α,σck,α,σ\n+X\nkX\n∆r,α,β,σV∆r,αβ⟨Nβ⟩c†\nk,α,σck,α,σ\n−X\nkX\n∆r,α,β,σV∆r,αβD\nc†\nr0+∆r,β,σcr0,α,σE\neik·∆rc†\nk,α,σck,β,σ.\n(17)\nHere, ∆ rdenotes the translational vector and ¯ σ=−σ.\nOff-diagonal one-body Green functions in the real space\nare represented by ⟨c†\nr0+∆r,β,σcr0,α,σ⟩. We take r0=0as\nthe representative coordinate of r0because r0is an ar-\nbitrary coordinate due to translational symmetry. Using\nthe mean-field Hamiltonians in the momentum space, we\ncalculate the band dispersions and the DOS. All values of\nthe one-body Green functions are uploaded to the ISSP\ndata repository [50].\n∗tmisawa@issp.u-tokyo.ac.jp\n[1] L. ˇSmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji,\nand T. 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Itoh\nCollege of Science and Technology, Nihon University, 24-1\nNarashinodai 7-chome, Funabashi-shi, Chiba 274-8501, Japan\n(Dated: March 10, 2022)\nAbstract\nUsing photo-emission electron microscopy with X-ray magnetic circular dichroism as a contrast\nmechanism, new insights into the all-optical magnetization switching (AOS) phenomenon in GdFe\nbased rare-earth transition metal ferrimagnetic alloys are provided. From a sequence of static\nimages taken after single linearly polarized laser pulse excitation, the repeatability of AOS can be\nmeasured with a correlation coe\u000ecient. It is found that low coercivity enables thermally activated\ndomain wall motion, limiting in turn the repeatability of the switching. Time-resolved measurement\nof the magnetization dynamics reveal that while AOS occurs below and above the magnetization\ncompensation temperature TM, it is not observed in GdFe samples where TMis absent. Finally,\nAOS is experimentally demonstrated against an applied magnetic \feld of up to 180 mT.\nPACS numbers: 75.78.Jp, 68.37.Yz, 75.70.Kw, 75.50.Gg\n1arXiv:1412.0396v1  [cond-mat.mtrl-sci]  1 Dec 2014I. INTRODUCTION\nControlling magnetism on the ultrashort time scale of sub-100 ps has become an im-\nportant research subject, not only for the potential applications in novel high density and\nhigh speed magnetic recording technologies but also for the unique opportunity to investi-\ngate magnetism on the fundamental time scales of the interactions between electrons, spins\nand lattice.1The demonstration in 19962of a rather unexpected ultrafast sub-ps demag-\nnetization in a thin Ni \flm upon femtosecond laser excitation inspired a large number of\nfollowing studies.3Of particular importance was the surprising demonstration of a deter-\nministic magnetization reversal by the sole action of a single 40 fs laser pulse in GdFeCo\nrare earth-transition metal (RE-TM) alloys.4The microscopic mechanism responsible for\nthis phenomenon, now referred to as all-optical switching (AOS), remains debated. Element\nselective studies of the ultrafast demagnetization in GdFeCo alloys led to the interpreta-\ntion that AOS is driven by the heating from the laser pulse and is therefore independent of\nthe laser polarization and largely insensitive to any applied magnetic \feld.5,6The helicity\ndependent AOS reported earlier could then be understood in terms of a di\u000berential light\nabsorption induced by the magnetic circular dichroism in the magnetic alloy.7Finally, as\nthese RE-TM alloys usually display chemical inhomogeneities, the role of super-di\u000busive\nspin currents is also being discussed.8\nWhile early studies concentrated on GdFeCo alloys, magnetization switching by laser\npulses has now been reported in a growing range of systems, namely other RE-TM alloys9and\nmulti-layers,10RE-free synthetic ferrimagnets10,11and granular ferromagnets.12These recent\ndevelopments are raising a number of crucial questions for the understanding of the AOS\nphenomenon and its transfer to real world applications. Among these, the exact role played\nby the magnetization compensation temperature TMat which the magnetization of the two\nsub-lattices cancel each other remains a puzzle. On the one hand, strong changes in the\nmagnetization dynamics upon crossing TMhave been observed13{15and AOS seems to occur\npreferably for alloys displaying a TMwhich can be reached through laser excitation.9,10,16On\nthe other hand, atomistic simulations of the spin dynamics as well as experiments have shown\nthat AOS is feasible below and above TM.6In addition, helicity dependent magnetization\nswitching in granular ferromagnets where no TMexists has been reported.12Finally, in view\nof potential applications, it is crucial to be able to characterize to which extent AOS is a\n2deterministic process.\nIn this article, we investigate all-optical magnetization switching in GdFe based alloys\nusing photo-emission electron microscopy (PEEM) with X-ray magnetic circular dichroism\n(XMCD) as a contrast mechanism, allowing imaging of the magnetic domain con\fguration\nwith a spatial resolution of approximately 100 nm. Single linearly polarized laser pulses\nwere used to excite a multi-domain con\fguration at temperatures below and above the\nmagnetization compensation temperature TMof the alloys. Introducing the Pearson product-\nmoment correlation coe\u000ecient on series of XMCD images allows us to report a nearly purely\ndeterministic AOS in both cases. Extrinsic pulse to pulse laser pointing stability and intrinsic\n\fnite domain sizes and thermally activated domain wall motion are found to be the main\nlimiting factors for a purely deterministic AOS. Using time-resolved XMCD PEEM imaging\nof the magnetization dynamics upon femtosecond laser excitation with 70 ps time resolution\nand approximately 200 nm spatial resolution, it is found that AOS can even be achieved\nagainst a 180 mT applied magnetic \feld. Finally, strong reduction of the switching window\naboveTMis observed and is partly related with the proximity of the Curie temperature TC\nof the sample.\nII. METHODS\nA. Time-resolved XMCD PEEM\nIn order to resolve the magnetic domain con\fguration and its dynamics upon AOS, the\nElmitec photoemission electron microscope (PEEM) at the Surface/Interface: Microscopy\n(SIM) beamline17at the Swiss Light Source (SLS) was used. Employing the X-ray magnetic\ncircular dichroism (XMCD) e\u000bect at the Fe L 3edge at 708 eV, a quantitative determination\nof the Fe spin orientation with a 100 nm spatial resolution is possible.18From two images\nrecorded with opposite X-ray helicity, an asymmetry image is computed which contains only\nnormalized magnetic contrast information. Such image typically shows white or black regions\ncorresponding to magnetic domains with magnetizations of opposite directions with respect\nto the X-ray propagation vector.19Time-resolved measurements of the sample magnetization\nwere performed by taking advantage of the pulsed nature of the X-rays produced by the SLS\nvia the gating of the detection in synchronization to an isolated X-ray pulse. This scheme,\n3presented in detail in Ref. 20, allows stroboscopic pump-probe imaging of the sample with\na time resolution determined by the 70 ps Full Width at Half Maximum (FWHM) temporal\nX-ray pulse length. At this time scale, both TM and RE magnetizations are in equilibrium\nsuch that measuring the Fe sub-lattice is su\u000ecient to characterize the sample magnetization\norientation. The pump laser pulses were produced by an XL-500 oscillator from Femtolasers\nProduktions GmbH which are characterized by a wavelength of \u0015= 800 nm, a pulse duration\nof\u001c= 50 fs with an energy of 500 nJ per pulse at a 5.2 MHz repetition rate. This repetition\nrate is then reduced by a Pockels cell in combination with a crossed polarizer to match the\n1.04 MHz repetition rate of the isolated X-ray probe pulses. The linearly p-polarized laser\npump pulses were focused on the sample at a grazing incidence of 16\u000eto a spot size of about\n30\u0002100\u0016m2FWHM. The time overlap ( t= 0) between the laser and the X-ray pulse is\nunambiguously determined to better than \u000615 ps by the sudden space charging21,22which\nis induced by the laser pump pulse which reduces signi\fcantly the amount of photo-emitted\nelectrons collected by the microscope. Finally, the sample could be cooled down with a\n\row of liquid nitrogen and the temperature measured with a thermocouple attached to the\nsample holder.\nB. Samples\nThe samples are grown on Si substrates to achieve fast cooling time during MHz rep-\netition rate experiments23and are capped with a 3 nm Si 3N4layer to prevent oxidation.\nThree di\u000berent samples have been used for this study. The \frst sample of composition Si/\nAlTi(10 nm)/Si 3N4(5 nm)/Gd 25Fe65:6Co9:4(20 nm)/Si 3N4(3 nm) has a TMof 260 K. The two\nother samples are GdFe alloys of composition Si/Si 3N4(5 nm)/Gd 20Fe80(30 nm)/Si 3N4(3 nm)\nwith aTMbelow 10 K and Si/Si 3N4(5 nm)/Gd 24Fe76(30 nm)/Si 3N4(3 nm) with a TMabove\n500 K. In the rest of the paper, each sample is referred to by a reduced notation consisting\nof the Gd content like for example Gd25FeCo or Gd20Fe.\n4III. RESULTS\nA. Single laser pulse excitation\nIn view of potential applications, the question of the repeatability of AOS is essential.\nAOS was therefore studied on a multi-domain con\fguration were one laser pulse excites\nseveral di\u000berent magnetic domains at once. The magnetic domain con\fguration before and\nafter single linearly polarized laser pulse exposure was recorded with static XMCD PEEM\nimaging. Sequences of such Ipimages taken at the Fe L3edge for the Gd25FeCo sample at\na temperature above and below TMin the absence of any applied magnetic \feld are shown\nin Figs. 1 (a) and (c) respectively. In those images, white (black) contrast corresponds to\nmagnetic domains whose out-of-plane magnetization has a positive (negative) projection on\nthe X-ray direction, as indicated by the gray scale in Figs. 1 (a) and (c). In both cases,\nbelow and above TM, changes in the magnetic domains in the center of the images are\nseen. To better emphasise the changes or the lack of them occurring in these multi-domains\ncon\fguration, the pixel by pixel product between two successive images separated by a single\nlinearly polarized laser pulse excitation Ip\u00001Ipis computed and shown in Fig. 1(b) and (d).\nIrrespective of the initial magnetic domain orientation, in the Ip\u00001Ipimage, a black contrast\ncorresponds to a magnetization switching (SW), a gray contrast to a domain wall (DW)\nand a white contrast to an absence of changes, i.e.no switching (NS), as indicated by the\ngray scale in the inset. Visible in the product of successive images Ip\u00001Ipshown in Fig. 1(b)\nand (d) is a black elongated elliptical region at the center surrounded by a white region\nuna\u000bected by the laser pulses. This elongated elliptical shape corresponds to the laser spot\nsize seen at the 16\u000egrazing incidence used in this experiment. This black elongated region\nclearly corresponds to a laser induced switching occurring equally for both magnetic domain\norientations enclosed in the laser spot size. Since this AOS seems to occur with every\nlaser pulse, it appears to be purely deterministic. To better quantify how deterministic\nthis phenomenon of AOS really is, we introduce the pixel-by-pixel Pearson product-moment\ncorrelation coe\u000ecient rfor a sequence of XMCD images as:\nr=Pn\np=1Ip\u00001IpqPn\np=1I2\np\u00001qPn\np=1I2\np;\n5whereIpis the XMCD image after plaser pulses in the sequence. In the case of purely\ndeterministic switching, this correlation coe\u000ecient ris -1, while in the absence of changes,\ni.e.no switching, r= +1. In the event of an unrelated domain con\fguration after every\nsingle laser pulse, such as in the case of heating above TC,r= 0. Such correlation coe\u000ecient\nimagesrcalculated from the measured sequences are shown in Figs. 1(e) and (f) for a\nsample temperature above and below TMrespectively. The darkest region in these images\ncorresponds indeed to a correlation coe\u000ecient of r= -1, i.e.a purely deterministic switching\nwith each of the 10 laser pulse of the sequence, occurring both below and above TM. It is\nalso evident that the spatial extent of this r= -1 region is limited by the spatial extent with\nwhich these 10 laser pulses overlap. Therefore, the pulse to pulse pointing stability is the\nonly extrinsic limitation to a somewhat purely deterministic AOS.\nHowever, there can also be intrinsic limitation such as domain walls, in particular at the\nboundary between the switching and non switching region of each laser pulse. For example,\nin the case of the sample temperature above TMshown in Fig. 1(b), the domain wall at\nthe bottom of the laser pulse region is nearly continuously moving in the same direction\nbetween successive images, as indicated by the red arrows as well as the dashed ellipse in\nFig. 1(e). As this domain wall is clearly outside the elongated elliptical region where AOS\noccurs, we know that the laser \ruence is too low to induce a deterministic AOS. In fact,\nin the XMCD PEEM images I1andI10shown in Fig. 1(a), one can even see the domain\nwall motion occurring during the imaging which results in an extended gray region rather\nthan a either completely black or completely white region. This is indicative of a very low\ncoercivity of the domain walls at this temperature which favors thermally activated domain\nwall movements in the otherwise non switching region and should be regarded as intrinsically\nlimiting the repeatability of the AOS. Comparing the domain size above and below TM, as\nshown in Figs. 1(a) and (c), one can immediately realize that the coercivity is higher in the\nsecond case as the magnetic domains are smaller, and thus more stable. Nevertheless, here\nsome changes in the domain con\fguration can also be seen at the edges of the AOS region, as\nindicated by the blue arrow in the I9I10image shown in Fig. 1(d). The small protuberance\ncorresponds to a small black domain outside the AOS region which disappeared between\nthe images I9andI10shown in Fig. 1(c). This is likely the collapse of a too small domain\nformed by the intersection of the existent domain pattern and the AOS region created by\nthe laser pulse. These processes of domain collapse and thermally activated domain wall\n6hopping should not be confused with AOS. In fact, they lower the repeatability of AOS.\nInside the r= -1 region, all magnetic domains are switching with every laser pulse.\nHowever, it is unclear what is happening for the domain wall separating them since the\ncorrelation coe\u000ecient ris unde\fned there. To visualize the various domain wall position\nduring the sequence of laser pulses, it is best to look at the low intensity part of the average\nof the squared image hI2\npishown in Fig. 1(g) and (h) for the sample temperature above and\nbelowTM, respectively. In those hI2\npiimages, the darker the domain wall, the less it moved\nduring the sequence of laser pulses. In the case of the sample at a temperature above TM\nshown in Fig. 1(g), some changes are visible at the domain wall inside the switching region,\nas indicated by the red arrow. In the case below TMshown in Fig. 1(h), no changes are\nvisible, meaning that the domain wall stayed in place within the 100 nm spatial resolution\nof the instrument. Considering the low coercivity of this material, this is a rather surprising\nand noteworthy feature of AOS. Nevertheless, evidences for potential domain wall hopping\nwell inside the r= -1 region are seen at least in one case, limiting the repeatability of the\nAOS. Overall, apart from the di\u000berence in coercivity, very little di\u000berences are seen between\nAOS below and above TM.\nB. Time-resolved dynamics around TM\nTo gain more insight into the AOS and in particular into the role played by TM, the\nmagnetization dynamics in this sample was investigated around TM. For this, time-resolved\nXMCD PEEM measurements were performed and the results are shown in Fig. 2, for a\nsample temperature (a) above and (c) below TM, and for a strong H = 180 mT and a weak\nH = 30 mT out-of-plane magnetic \feld. The magnetic \feld is used to reset the sample\nmagnetization to a well de\fned initial state, allowing for stroboscopic measurement of the\ndynamics. The \frst thing to notice is that at negative time delay t, i.e. before the laser\npulse, the sample is saturated for both applied magnetic \felds, and that the orientation\nof the Fe sub-lattice magnetization reverses between Fig. 2(a) and (c), meaning that the\nsample is e\u000bectively on either side of the magnetization compensation temperature TMat\nthe temperature used. From the time-resolved XMCD images, the magnetization dynamics\nat the center can be extracted and is shown in Figs. 2(b) and (d), for a sample temperature\nabove and below TM, respectively. In both cases, magnetization switching occurs right\n7after the laser pulse excites the sample. Thus, within the 70 ps time resolution of the\nexperiments, no di\u000berence is seen in the switching dynamics for either low or high magnetic\n\feld and either below or above TM. On the other hand, the relaxation towards the \fnal state\nis strongly in\ruenced by both the applied magnetic \feld and the sample base temperature.\nAt a temperature above TM, as shown in Fig. 2(b), the reversed state is instable against\nthe applied magnetic \feld, leading to a fast relaxation towards the initial state, the faster\nthe higher the \feld. It is worth noting here that switching with a laser pulse against a\n\feld of 180 mT is thus possible, even though the relaxation is very fast, demonstrating the\nimpetuous by which this AOS occurs.6Due to this fast relaxation and the 70 ps long X-ray\nprobe pulse length, a saturated switched state is not observed. At temperatures below TMas\nshown in Fig. 2(d), the reversed state is now stable within the illuminated area, indicating\nthat the temperature in this region is now above TM. In this case, after the laser pulse,\nthe applied magnetic \feld is now stabilizing the reversed domain, leading to a very long life\ntime.\nTime-resolved XMCD PEEM images taken at the same \fxed time delay of t= +230 ps\nafter the laser pulse on the same Gd25FeCo sample are shown in Fig. 3(a) above and (b)\nbelowTM, as a function of the laser pump \ruence. A small static out-of-plane magnetic\n\feld of H = 30 mT was applied to reset the sample after switching. This 30 mT magnetic\n\feld is small enough to not hinder the AOS at this time scale as can been seen in Fig. 2(b).\nWhile below TM, the laser \ruence can be increased signi\fcantly without losing the AOS, the\nsame is not true above TM. There, a small 10% increase from 2.7 to 3.0 mJ \u0001cm\u00002is enough\nto bring the central region of the laser spot into a demagnetized state. This e\u000bect is most\nstriking at the \ruence of F= 3.5 mJ\u0001cm\u00002in Fig. 3(a), where the switched region forms a\nvery thin 2 \u0016m wide ring around the laser pulse. The AOS \ruence switching window is thus\nreduced above TM, and this asymmetry of the switching window around TMis consistent\nwith literature.13{15Part of this e\u000bect might be attributed to the proximity with the Curie\ntemperature TC.\nC. Time-resolved dynamics far from TM\nDue to the limited accessible temperature range in the PEEM, investigation of the AOS\nfar fromTMrequires samples with di\u000berent compositions. For this, time-resolved XMCD\n8PEEM measurement were thus performed on Gd20Fe with TMaround 0 K and Gd24Fe with\nTMaround 500 K, under a small static out-of-plane magnetic \feld of 30 mT. The results\nare shown in Fig. 4. For both samples, a time resolution limited demagnetization process\noccurs. The samples then stay demagnetized for about 500 ps which is then followed by a\nslow dynamics on a time scale of around 10 ns, towards the initial state for Gd20Fe and\ntowards the reversed state for Gd24Fe. This reversal in the Gd24Fe sample shows that\nthere is an accessible magnetization compensation temperature TMin this sample below TC,\nallowing the applied 30 mT out-of-plane magnetic \feld to reverse the sample magnetization\non a slow few nanoseconds long time scale and eventually back to the initial state at even\nlonger time scale after cooling down. For the Gd20Fe sample, the temperature is already\naboveTMbefore the laser pulse, and therefore no magnetic \feld assisted switching occurs.\nLooking at the XMCD PEEM images taken at \fx time delay and shown in Fig. 4(a), it\ncan be seen that in the case of the Gd20Fe sample, the demagnetized region has a di\u000buse\nboundary, meaning that no magnetic domain is actually formed. On the other hand, for\nGd24Fe, at around 750 ps after the laser pulse, a clear boundary appears in the heated\nregion, which is seen in Fig. 4(a) at t = 3.1 ns. This very late formation of the reversed\ndomain in Gd24Fe and the absence of switching in Gd20Fe allow us to conclude that no\nAOS window exists far from TM.\nIV. DISCUSSION\nDetermining if a system can display all-optical magnetization switching and to which\nextent this AOS is deterministic are two questions of crucial importance, for a better un-\nderstanding of the phenomenon as well as in view of its potential applications. In this\ncontext, sequences of XMCD PEEM images separated by single linearly polarized laser\npulse excitation on a multi-domain con\fguration such as shown in Fig.1 can provide valu-\nable information. First of all, since linearly p-polarized laser pulses are equally absorbed by\neach domain orientation, a direct comparison between what happens inside each domain is\npossible.7This is in contrast with multiple circularly polarized laser pulses used in recent\nstudies such as in Refs. 10 and 12, where such a comparison can only be made after care-\nfully taking into account the magnetic circular dichroism of the material. Second, randomly\ndemagnetized initial states are better than saturated or arti\fcially created domain states\n9since no stray \feld is created which could in\ruence the switching. Third, the reversed do-\nmain con\fguration in such case is known to be stable as well, therefore a collapse of the\nreversed domain state because of too low coercivity or too high net magnetization is not to\nbe expected.24Finally, from such a sequence of images, the actual reproducibility of AOS\ncan be measured using the Pearson product-moment correlation coe\u000ecient ras introduced.\nFrom our analysis, it follows that the purely deterministic AOS observed in the GdFeCo\nsamples is limited by a number of extrinsic and intrinsic e\u000bects. The largest limitation we\nobserve in Fig. 1(e) and (f) is the pulse to pulse laser pointing stability which is extrinsic\nin nature to the switching phenomenon itself. The second limitation observed is related\nto the stability of the domain con\fguration. For example, at the edge of the laser pulse,\nthe overlap of the r= -1 switching region with the preexistent domain con\fguration can\ncreate domains which are too small to be stable, as seen in Fig. 1(d) I9I10. In addition,\nthermal activation of domain walls can occur outside as well as inside the r= -1 switching\nregion, as seen in Fig. 1(g) and indicated by the arrow and dashed ellipse. Since these two\ne\u000bects are related to the coercivity of the material, this constitutes an intrinsic limitation to\nthe repeatability of the AOS. However, by understanding these limitations, we can envisage\nthat engineered materials can potentially alleviate these limitations. For example, in pat-\nterned materials where each structure preferably host a single magnetic domain, a purely\ndeterministic switching would be maintained.\nRegarding the role played by the magnetization compensation temperature TMon the\nAOS, we \frst of all con\frm previous studies in that AOS occurs below and above TM.6,14\nSingle shot laser pulse experiments shown in Fig. 1 as well as time-resolved measurements\nof the magnetization dynamics shown in Fig. 2 both reveal AOS below as well as above TM.\nHowever, there exists a clear di\u000berence between switching below and above TM, as shown\nin the \ruence-dependent patterns observed at t= 230 ps in Fig. 3. In addition, for GdFe\nsamples with no or far from their TM, no switching is observed, as shown in Fig. 4. This leads\nto the conclusion that while the existence of a reachable TMduring the laser excitation is not\na strict requirement to observe AOS, sample compositions with TMnear room temperature\nare preferred. It must be noted that in addition to TM, an angular momentum compensation\ntemperature TAalso exists at a slightly higher temperature.25However, our experimental\ngeometry with out-of-plane magnetic \feld does not allow magnetization precession dynamics\nto be observed, precluding any investigation of the e\u000bect of TAon AOS. Finally, AOS is a\n10very robust switching mechanism as it can be realized against an opposing applied magnetic\n\feld6, as demonstrated experimentally here in the case of a 180 mT \feld in Fig. 2(b).\nV. CONCLUSIONS\nIn conclusion, using static and time-resolved PEEM microscopy with XMCD to probe\nthe sample magnetization upon laser excitation, important aspects of the AOS have been\nrevealed. Sequences of images after single linearly polarized laser pulse excitation on a\nmulti-domain con\fguration allow for the study of the repeatability of the process by using\nthe correlation coe\u000ecient as its measure. It is found the AOS in the Gd25FeCo sample\nstudied is nearly purely deterministic. Moreover, intrinsic limitation from the low coercivity\nof the material leading to thermally activated domain wall hopping could be alleviated in\npatterned media. From the time-resolved measurement of the magnetization dynamics, it\nis found that AOS occurs below and above TM, while on the other hand, no AOS occurs for\nsample temperatures far from it. Strong reduction of the \ruence switching window occurs\naboveTMand is likely related to the proximity with the Curie temperature TC. Finally,\nAOS against an applied magnetic \feld of 180 mT is demonstrated, illustrating the impetus\nby which AOS occurs.\nACKNOWLEDGMENTS\nWe thank the European Research Council under the European Unions Seventh Frame-\nwork Programme FP7/2007{2013 (grants NMP3-SL-2008-214469 (UltraMagnetron), FP7-\nNMP-2011-SMALL- 281043 (FEMTOSPIN) and 214810 (FANTOMAS)) for part of the\n\fnancial supports as well as the MEXT-Supported Program for the Strategic Research\nFoundation at Private Universities, 2013{2017. Part of this work was performed at the\nSwiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. We thank J. Honegger for\nhis support.\n\u0003email: loic.le guyader@helmholtz-berlin.de\n11yPresent address: College of Science and Technology, Nihon University, 24-1 Narashinodai 7-\nchome, Funabashi-shi, Chiba 274-8501, Japan\nzPresent address: Institute for Quantum Electronics, Physics Department, ETH Zurich, CH-8093\nZurich, Switzerland\n1J. St ohr and H. C. 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B 73, 220402 (2006).\n14(a)\n(h)\n (e)(b)\n(c)\n(d)\n(f) (g)\nr\n-1 0 +1I₀ I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉ I₁₀\nI₀I₁ I₁I₂I₂I₃ I₃I₄I₄I₅ I₅I₆I₆I₇ I₇I₈I₈I₉I₉I₁₀-M+M\n0\nSWNS\nDW\n-M+M\n0\nSWNS\nDW\nr\n-1 0 +1I₀ I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉ I₁₀\nI₀I₁ I₁I₂I₂I₃ I₃I₄I₄I₅ I₅I₆I₆I₇ I₇I₈I₈I₉I₉I₁₀FIG. 1. (Color online)(a) Sequences of XMCD PEEM images Iptaken after psingle laser pulse\nexcitation above at T = 300 K and (c) below at T = 160 K the magnetization compensation\ntemperature TM= 260 K of the Gd25FeCo sample. The gray scale in the inset on the right\nindicates the out-of-plane magnetization orientation. (b) Sequences of image product Ip\u00001Ipabove\nand (d) below TM. The gray scale in the inset on the right indicates which gray level corresponds to\nmagnetization switching (SW), no switching (NS) or domain wall (DW).(e) Correlation coe\u000ecient\nimagesrderived from the sequences of single laser pulse excitation above and (f) below TM. (g)\nAverage imagehI2\npishowing the domain wall positions above and (h) below TM. Arrows and dashed\nellipses indicate magnetization switching not related to AOS and are discussed in the text. All\nscale bars are 20 \u0016m.\n15(a)\nt = -0.12H = 180, 30 mT\n0.06 0.18 0.44 1.63 ns\n(b)\n(c)\nt = -0.12H = 180, 30 mT0.06 0.18 1.7 12.7 ns\n(d)XMCD (%)\nM ⊙-1.5 1.5\n⊗XMCD (%)\nM ⊙-1.5 1.5\n⊗FIG. 2. (Color online)(a) Time-resolved XMCD PEEM images on Gd25FeCo sample at di\u000berent\ntime delays, for two di\u000berent applied out-of-plane magnetic \feld of 30 mT and 180 mT, measured\nat the Fe L 3edge, at a temperature above TMat T = 300 K and (b) the extracted magnetiza-\ntion dynamics for each applied out-of-plane magnetic \feld. (c) and (d) the same for a sample\ntemperature below TMat T = 160 K. The scale bars are 20 \u0016m.\n16(a)\nF = 0.0 2.3 2.7 3.0 3.5 mJ·cm ⁻²\n(b)\nF = 0.0 2.6 2.8 3.1 3.6 mJ·cm ⁻²\nXMCD (%)\nM ⊙-1.5 1.5\n⊗FIG. 3. (a) Time-resolved XMCD PEEM images taken at t= 230 ps after the laser pulse on\nGd25FeCo sample above at T = 300 K and (b) below at T = 160 K the magnetization compensation\ntemperature TM= 260 K, as a function of the laser pump \ruence. The static out-of-plane magnetic\n\feld was 30 mT. The scale bars are 20 \u0016m.\n17(b)(a)\nGd20Fe Gd24Fet = -4.4 0.3 0.75 3.1 8.5 ns\nXMCD (%)\nM ⊙-2.5 2.5\n⊗FIG. 4. (Color online)(a) Time-resolved XMCD PEEM images at various \fxed time delays and (b)\nextracted magnetization dynamics on Gd20Fe ( TMaround 0 K,F= 5.7 mJ\u0001cm\u00002) and Gd24Fe\n(TMaround 500 K,F= 3.9 mJ\u0001cm\u00002) samples at room temperature with a 30 mT of out-of-plane\nmagnetic \feld. The scale bar is 20 \u0016m.\n18"    },    {        "title": "1011.2486v1.Ab_initio_prediction_of_ferrimagnetism__exchange_interactions_and_Curie_temperatures_in_Mn2TiZ_Heusler_compounds.pdf",        "content": "arXiv:1011.2486v1  [cond-mat.mtrl-sci]  10 Nov 2010Ab initio prediction of ferrimagnetism, exchange\ninteractions and Curie temperatures in Mn 2TiZ\nHeusler compounds\nM Meinert, J M Schmalhorst, and G Reiss\nDepartment of Physics, Bielefeld University, 33501 Bielef eld, Germany\nE-mail:meinert@physik.uni-bielefeld.de\nAbstract. The Heusler compounds Mn 2TiZ(Z= Al, Ga, In, Si, Ge, Sn, P, As,\nSb) are of large interest due to their potential ferrimagnet ic properties and high\nspin polarization. Here, we present calculations of the str uctural and magnetic\nproperties of these materials. Their magnetic moment follo ws the Slater-Pauling\nrulem=NV−24. None of them is actually a perfect half-metallic ferrima gnet,\nbut some exhibit more than 90% spin polarization and Curie te mperatures well\nabove room temperature. The exchange interactions are comp lex, direct and\nindirect exchange contributions are identified. The Curie t emperature scales with\nthe total magnetic moment, and it has a positive pressure dep endence. The role\nof theZelement is investigated: it influences the properties of the compounds\nmainly via its valence electron number and its atomic radius , which determines\nthe lattice parameter. Based on these results, Mn 2TiSi, Mn 2TiGe, and Mn 2TiSn\nare proposed as candidates for spintronic applications.Ferrimagnetism in Mn 2TiZ Heusler compounds 2\n1. Introduction\nA very interesting class of Heusler compounds that has received co nsiderable\ntheoretical, but only few experimental attention to date, are the half-metallic\nferrimagnetsMn 2YZ,whereY=V,Cr, Mn, Fe, Co, Ni, Cuand ZisagroupIII,IV, or\nV element [1, 2, 3, 4, 5, 6, 7, 8, 9]. Half-metallic compounds are chara cterized by a gap\nfor either the spin-down or the spin-up density of states (DOS) at the Fermi energy,\nso that an electric current has purely up or down electrons. This pr operty makes\nthem highly interesting for applications in spintronics. A half-metallic f errimagnet\nhas advantages over the well-known half-metallic ferromagnets: d ue to the internal\nspin compensation it has rather low magnetic moment, while the Curie t emperature\nremains fairly high. A low magnetic moment gives rise to low stray fields, which is\ndesiredforspintronics,asisahighCurietemperatureandthusago odthermalstability\nof the compound [10]. The most prominent compound out of this class is Mn2VAl,\nwhich has been studied thoroughly by experiment and theory [11, 1 2, 13, 14, 15].\nTogether with numerous other compounds in the Mn 2VZseries it has been predicted\nto be a half-metallic ferrimagnet [1, 16]. Its low magnetic moment of ab out 2µBper\nformula unit (f.u.) and the high Curie temperature of 760K make it a pr omising\ncompound for spintronics [13]. Several other materials classes ha ve been proposed\nto be half-metallic ferrimagnets, e.g., Cr 0.75Mn0.25Se and Cr 0.75Mn0.25Te in the zinc\nblende structure [17], or Cr antisites in CrAs, CrSb, CrSe, and CrTe , having the zinc\nblende structure [18].\nIdeally, an electrode material for spintronics would be a half-metal with zero\nnet moment. This can not be achieved with antiferromagnets becau se of the spin-\nrotational symmetry (resulting in zero polarization), but well chos en half-metallic\nferrimagnets can be tuned to zero moment. This property is also kn own as half-\nmetallic antiferromagnetism, and has been first predicted for Mn an d In doped\nFeVSb [19]. Among others, La 2VMnO 6and related double perovskites [20] and\ncertain diluted magnetic semiconductors have been later predicted to be half-\nmetallic antiferromagnets as well [21]. Finally, the ferrimagnetic Heus ler compounds\nMn2VAl and Mn 2VSi have been proposed as a starting point for doping with Co\nto achieve the full compensation [23]. However, it should be noted th at the half-\nmetallic antiferromagnetism is limited to zero temperature and a small macroscopic\nnet moment is expected at elevated temperature—in particular nea r the Curie\ntemperature—because of the inequivalent magnetic sublattices [22 ].\nFollowing the Slater-Pauling rule connecting the magnetic moment mand the\nnumber of valence electrons NVviam=NV−24 in the half-metallic Heusler\ncompounds [24], it is expected to find another series of ferrimagnet ic half-metals\nin the Mn 2TiZsystem with −3 to−1µB/f.u. The negative moment indicates\nthat the half-metallic gap would appear for the majority states. Th ese compounds\ncould—if they are half-metals—provide another series of potential electrodes for\nspin-dependent applications and could also become a starting point f or half-metallic\nantiferromagnetism.\nIn this paper, we discuss ab initio calculations of the properties of the\n(hypothetical)Mn 2TiZcompounds, crystallizedintheL2 1structure. Noexperimental\ndata are available for this system, and only Mn 2TiAl has been studied theoretically\nbefore [25]. However, it is expected that parts of this series will exis t in the L2 1\nstructure, seeing that Mn 2VAl and Mn 2VGa, as well as parts of the Co 2TiZseries\nhave been prepared [26, 27, 28].Ferrimagnetism in Mn 2TiZ Heusler compounds 3\n2. Calculational approach\nThe calculations presented in this study were performed within two d ifferent density\nfunctional theory-based band structure codes: the full-poten tial linearized augmented\nplane waves (FLAPW) package Elk [29] and the full-potential Korrin ga-Kohn-\nRostoker Munich SPRKKR [30] package. Although both methods are in principle\nequivalent for crystalline systems, there are subtle differences as sociated with their\nnumerical implementations, and thus it is worth to compare both met hods on the\nrather complex intermetallic system Mn 2TiZ.\nElk was used to determine the theoretical lattice parameters and t he total energy\ndifferences between ferrimagnetic and nonmagnetic states. Thes e calculations were\ncarried out on a 12 ×12×12kpoint mesh (72 points in the irreducible wedge of the\nBrillouin zone). The muffin-tin radii of all atoms were set to 2.0 a.u. to a void overlaps\nat small lattice parameters. The equilibrium lattice parameters awere determined\nusing a third-degree polynomial fit to the total energies. To obtain accurate magnetic\nmoments and densities of states, the calculations were performed at the equilibrium\nlattice parameter using a 16 ×16×16k-mesh (145 points in the irreducible wedge)\nand nearly touching muffin-tin spheres.\nThe SPRKKR calculations were performed on the theoretical equilibr ium lattice\nparametersdeterminedwithElk. Thecalculationswerecarriedoutin thefull-potential\nmode with an angular momentum cutoff of lmax= 3 on a 22 ×22×22kpoint mesh\n(289 points in the irreducible wedge of the Brillouin zone). Both the fu ll potential\nas well as the increased angular momentum cutoff are necessary to ensure accurate\nresults. The DOS were calculated on a denser mesh of 1145 kpoints with 0.5mRy\nadded as the imaginary part to the energy.\nThe exchange-correlation potential was modeled within the genera lized gradient\napproximationofPerdew,Burke,andErnzerhofinbothschemes[ 31]. Thecalculations\nwere converged to about 0.1 meV. All calculations were carried out in the scalar-\nrelativisticrepresentationofthe valencestates, thusneglecting the spin-orbitcoupling.\nSPRKKR allows to calculate the Heisenberg exchange coupling parame tersJij\nwithin a real-space approach using an expression proposed by Liech tenstein et al.\n[32]. Using the Jijthe Curie temperatures were calculated within the mean field\napproximation (MFA). For a single-lattice system the Curie tempera ture is given\nwithin the MFA by\n3\n2kBTMFA\nC=J0=/summationdisplay\njJ0j. (1)\nIn a multi-sublattice system—as, e.g., the Heusler compounds with fo ur sublattices—\none has to solve the coupled equations\n3\n2kBTMFA\nC/angbracketlefteµ/angbracketright=/summationdisplay\nνJµν\n0/angbracketlefteν/angbracketright (2)\nJµν\n0 =/summationdisplay\nR/negationslash=0Jµν\n0R,\nwhere/angbracketlefteν/angbracketrightis the average zcomponent of the unit vector eν\nRpointing in the direction\nof the magnetic moment at site ( ν,R). The coupled equations can be rewritten as an\neigenvalue problem:\n(Θ−TI)E= 0 (3)\n3\n2kBΘµν=Jµν\n0Ferrimagnetism in Mn 2TiZ Heusler compounds 4\n/s65/s115\n/s68/s69/s32/s61/s32/s48/s46/s48/s49/s52/s32/s101/s86/s71/s101\n/s68/s69/s32/s61/s32/s48/s46/s49/s57/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s71/s97\n/s68/s69/s32/s61/s32/s48/s46/s53/s50/s32/s101/s86/s80\n/s68/s69/s32/s61/s32/s48/s46/s48/s48/s52/s52/s32/s101/s86/s83/s105\n/s68/s69/s32/s61/s32/s48/s46/s49/s55/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48/s65/s108\n/s68/s69/s32/s61/s32/s48/s46/s53/s48/s32/s101/s86\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48/s83/s98\n/s68/s69/s32/s61/s32/s48/s46/s48/s51/s57/s32/s101/s86\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s83/s110\n/s68/s69/s32/s61/s32/s48/s46/s50/s53/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48/s73/s110\n/s68/s69/s32/s61/s32/s48/s46/s55/s48/s32/s101/s86\nFigure 1. Total energies of the investigated compounds in dependence of their\nlattice parameters. The results for the ferrimagnetic and t he non-magnetic states\nare represented with + and ×, respectively.\nwith a unit matrix Iand the vector Eν=/angbracketlefteν/angbracketright. The largest eigenvalueof the Θmatrix\ngives the Curie temperature [16, 33]. In order to separate the two Mn lattices, the\ncalculations were run in F ¯43m space group, in which the Mn atoms are not equivalent\nby symmetry. The R-summation in Eq. (2) was taken to a radius of Rmax= 3.0a,\nwhich has been shown to be sufficient for half-metallic Heusler compou nds [34, 35].\n3. Results\n3.1. Energy minimization and lattice parameters\nThree types of magnetic configurations were tested: ferro-, fe rri-, and nonmagnetic.\nIt was found for all compounds that the ferromagnetic configura tions were unstable\nand converged into the ferrimagnetic state. Fig. 1 displays the tot al energies of\nthe ferrimagnetic and the nonmagnetic configurations in dependen ce on the lattice\nparameters a. We find that the ferrimagnetic state has always lower energy than the\nnon-magnetic state; the difference in total energy reduces with in creasing number of\nvalence electrons, but it increases within the groups with the atomic number. The\nlattice parameters follow roughly a linear dependence on the atomic r adius of the Z\nelement with the correlation coefficient of r= 0.92 (Fig. 2 (a)). Some compounds\nshow a strong asymmetry of the total energy curve in the ferrima gnetic configuration\nand even kinks in the curves for very large a. This is caused by a steep increase of the\nmagnetic moments for increasing awhich causes a stronger binding. However, thisFerrimagnetism in Mn 2TiZ Heusler compounds 5\nTable 1. Results of the ground state properties calculations with El k and\nSPRKKR. The total magnetic moments are given in µBper formula unit, the\natomic magnetic moments are given in µBper atom. The SPRKKR results for\nMn2TiAs were obtained with a= 5.95˚A (see text).\nElk SPRKKR\nMn2TiZa(˚A)m m MnmTiP (%) m m MnmTiP (%)\nAl 5.96 2.98 1.83 -0.57 21 2.98 1.76 -0.49 82\nGa 5.95 2.95 1.84 -0.60 45 2.97 1.77 -0.53 79\nIn 6.23 3.17 2.17 -0.86 7 3.08 1.98 -0.82 32\nSi 5.78 1.98 1.16 -0.31 94 1.98 1.13 -0.26 87\nGe 5.87 1.97 1.20 -0.37 94 1.97 1.16 -0.33 89\nSn 6.14 1.97 1.32 -0.51 97 2.00 1.25 -0.48 93\nP 5.68 0.30 0.18 -0.05 -3 — — — —\nAs 5.82 0.94 0.59 -0.20 84 0.97 0.61 -0.22 58\nSb 6.07 0.97 0.65 -0.25 88 0.98 0.62 -0.24 79\neffect is never strong enough to shift the equilibrium lattice paramet er to such a high-\nmstate. The equilibrium lattice parameters are summarized in Table 1. T ypically we\nfind the equilibrium lattice parameters of Heusler compounds obtaine d with Elk to be\naccurate within ±0.5% compared to experiment.\n3.2. Magnetic moments and densities of states\nThe results discussed in this subsection are summarized in Table 1 and Fig. 3.\n3.2.1. Mn 2TiAl, Mn 2TiGa, Mn 2TiInFrom the rule m=NV−24 we expect to find\na magnetic moment of 3 µB/f.u.for these compounds. The FLAPW calculations show\nsmall deviations from this rule, indicating that the compounds are no t perfect half-\nmetals. This is confirmed by the DOS, which show spin polarizations at t he Fermi\nlevel below 50%, and in particular only 7% for Mn 2TiIn, where the magnetic moment\nis enhanced to 3 .17µB/f.u.. This arises from the large lattice parameter and the fact\nthat all three compounds do not form a gap in the DOS. The Fermi lev el for Mn 2TiAl\nand Mn 2TiGa is in a region with low DOS for both spin channels (see insets in Fig.\n3), but both of them have a very large empty spin-down DOS right ab oveEF. Small\nvariations of the lattice parameter would thus lead to strong variat ions of the spin\npolarization.\nThe calculations performed with SPRKKR reproduce the magnetic mo ments\nobtained in Elk very well. Although the total moments are practically e qual, a larger\ndeviation is found for the atom-resolved moments. The Fermi ener gy is found at\nslightly different positions in the DOS, and the detailed structures ob served in Elk\naroundEFare less pronounced, especially the dip in the spin-down states at EF. This\nleads to significantly higher spin polarization values in SPRKKR. Howeve r, the trend\nthat Mn 2TiIn has the lowest polarization within this group is reproduced.Ferrimagnetism in Mn 2TiZ Heusler compounds 6\n/s48/s46/s55/s53\n/s48/s46/s55/s48\n/s48/s46/s54/s53\n/s48/s46/s54/s48\n/s48/s46/s53/s53/s109/s77/s110/s32/s47/s32/s109\n/s54/s46/s50 /s54/s46/s49 /s54/s46/s48 /s53/s46/s57 /s53/s46/s56 /s53/s46/s55\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s109/s84/s105/s32/s47/s32/s109/s80/s73/s110/s83/s110/s83/s98\n/s83/s105/s65/s115\n/s71/s101/s71/s97\n/s65/s108\n/s40/s98/s41/s54/s46/s51\n/s54/s46/s50\n/s54/s46/s49\n/s54/s46/s48\n/s53/s46/s57\n/s53/s46/s56\n/s53/s46/s55\n/s53/s46/s54/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41\n/s49/s54/s48 /s49/s52/s48 /s49/s50/s48 /s49/s48/s48\n/s90/s32/s97/s116/s111/s109/s105/s99/s32/s114/s97/s100/s105/s117/s115/s32/s40/s112/s109/s41/s80/s83/s105/s71/s97\n/s71/s101\n/s65/s115/s83/s110\n/s65/s108/s83/s98/s73/s110\n/s40/s97/s41\nFigure 2. (a): Dependence of the lattice parameter aon the atomic radius of\ntheZelement. (b): Normalized magnetic moments of Mn and Ti in dep endence\nof the lattice parameter.\n3.2.2. Mn 2TiSi, Mn 2TiGe, Mn 2TiSnAccording to the “rule of 24” a total magnetic\nmoment of 2 µB/f.u.is expected. Again, small deviations from this rule are observed;\nall moments are lower by about 1.5%. In Elk, the three compounds ar e found to\nform a half-metallic gap in the spin-up states slightly above EF. The gap onset above\nEF(width) is 0.16eV (0.49eV) for Si, 0.24eV (0.25eV) for Ge, and 0.19eV ( 0.01eV)\nfor Sn. Nevertheless, the spin polarization is above 90% in these calc ulations. The\nstructure of the DOS around EFleads to a stable spin polarization and magnetic\nmoment upon isotropic lattice compression or expansion. For this se ries, having the\nsame valence electron counts and nearly half-metallic DOS, one can o bserve clearly\na narrowing of the bands, i.e., the DOS are contracted towards EF, while the Fermi\nlevel itself moves upwards. This is directly associated with the gradu ally increasing\nlattice parameter in this series, which reduces the overlap of the 3d orbitals and\ntherebyreducesthe itinerancyofthe system. An increasedlocaliz ationofthe electrons\nprovides also an explanation for the increasing atomic magnetic mome nts along this\nseries. Similar behavior has been observed earlier for Co 2MnZ, withZ=Si, Ge, Sn\n[36, 37] and Ni 2MnSn [38]. In the first case the Mn moment is increased and the Co\nmoment is lowered along the series, keeping the total moment intege r. Calculations\non Co2MnSi with increased lattice parameter reproduced this behavior. I n the second\ncase, the pressure dependence of the moments was studied. Und er increasing pressure,\ni.e., with reduced lattice parameter, both the Ni and the Mn moment d ecrease, and\nthus the total moment decreases. However, Ni 2MnSn is not a half-metal, hence the\ntotal moment is not restricted to an integer value. Consequently, both observations\non quite different ferromagnetic Heusler compounds are in accord w ith our case of\n(nearly) half-metallic ferrimagnetic Heusler compounds.\nWe note, that the magnetic moments and DOS from SPRKKR are in ver y good\nagreement with the ones obtained from Elk. However, the Fermi lev el is found at a\nlower position, giving rise to the slightly reduced polarization values.\n3.2.3. Mn 2TiP, Mn 2TiAs, Mn 2TiSbIn these cases a total magnetic moment of only\n1µB/f.u.is expected. Because of the very small lattice parameter of Mn 2TiP, itsFerrimagnetism in Mn 2TiZ Heusler compounds 7\n/s49/s48\n/s53\n/s48\n/s45/s53\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s83/s98/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s65/s115/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s80\n/s45/s56/s45/s52/s48/s52/s56\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s110/s45/s52/s48/s52\n/s71/s101/s45/s52/s48/s52\n/s83/s105\n/s49/s48\n/s53\n/s48\n/s45/s53\n/s45/s49/s48/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s47/s32/s101/s86/s32/s47/s32/s117/s110/s105/s116/s32/s99/s101/s108/s108/s41/s45/s48/s46/s49 /s48/s46/s49\n/s71/s97\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s73/s110/s56\n/s52\n/s48\n/s45/s52/s45/s48/s46/s49 /s48/s46/s49\n/s65/s108\nFigure 3. Densities of states calculated with Elk. The spin-up DOS is p ointing\nup, the spin-down DOS is pointing down. The insets for Al and G a show the\nregion around the Fermi energy.\nspin-splitting is small with only 0 .3µB/f.u.in the Elk calculation. The situation of\nMn2TiAs and Mn 2TiSb is similar to that of Mn 2TiSi and Mn 2TiGe. A spin-up gap\nis formed above the Fermi level with onset (width) of 0.29eV (0.53eV ) for As and\n0.19eV (0.44eV) for Sb. Though not being half-metallic, both compou nds have spin\npolarizations of more than 80%.\nFinally, the magnetic moments of Mn 2TiSb in SPRKKR agree very well with\nthose obtained with Elk. But again, the Fermi level is lower and the sp in polarization\nis reduced. ForMn 2TiP and Mn 2TiAs the situation is quite different. They can not be\nconverged into ferrimagnetic states at the equilibrium lattice param eters determined\nby Elk; instead, they are found to be nonmagnetic. This is caused by the tiny\nenergy difference between the ferrimagnetic and the nonmagnetic configuration, which\nleads to a numerical instability of the ferrimagnetic state. By increa sing the lattice\nparameterofMn 2TiAsbyabout2%to5.95 ˚A,theseparationisincreasedartificiallyto\nabout 30meV/f.u. and the calculation convergesinto the ferrimagn etic state. Because\nof this, the properties obtained with SPRKKR for this compound hav e to be taken\nwith care: in all other cases the individual atomic moments are slightly lower in\nSPRKKR than those from Elk; here instead, larger moments are fou nd. However, the\nsame procedure can not be applied to Mn 2TiP, within a reasonable range of latticeFerrimagnetism in Mn 2TiZ Heusler compounds 8\nparameters.\n3.2.4. General remarks It is worth to note that the magnetic moments of the Z\ncomponent are always below 0.06 µBand that they are always parallel to the Ti\nmoment. In detail, the values are Al 0.044 µB, Ga 0.052 µB, In 0.058 µB, Si 0.034 µB,\nGe 0.035 µB, Sn 0.034 µB, P 0.0062 µB, As 0.018 µB, and Sb 0.017 µB.\nAnother property worth noting is the fact that the ratios mMn/mandmTi/m\nfollow a linear dependence (with correlationcoefficients of r≈0.9 in both casesfor the\nElkdata)onthelatticeparameter(andhencetheinteratomicdista nces)independently\non theZtype, see Fig. 2 (b). As mentioned above, with increasing lattice par ameter\nthe itinerantcharacterofthesystemisreducedandlocalizesthem omentsgraduallyon\nthe atoms. Therefore, the influence of the Zcomponent in Mn 2TiZis twofold. First,\nit determines the lattice parameter of the compound and following fr om that, the\ndegree of electron localization. And second, the total magnetic mo ment is determined\nvia the number of electrons supplied, if the lattice parameterdoes n ot exceed a certain\nrange (which is not the case for P and In).\n3.3. Exchange interactions and Curie temperatures\nThe exchange interactions are investigated here for Mn 2TiGa, Mn 2TiGe, and\nMn2TiSb, which are representative compounds for their respective Zgroup. Fig.\n4 (a) displays the Jijcalculated for the intra-sublattice interaction Mn1(2)-Mn1(2)and\nthe inter-sublattice interactions Mn1(2)-Mn2(1)and Mn-Ti of the three compounds.\nAll other interactions are very small and can be neglected for the f ollowing discussion.\nIn all three cases it is clear that the Mn1(2)-Mn2(1)inter-sublattice interaction\nprovides the largest contribution to the exchange. Further, the nearest neighbor\ninteraction of Mn-Ti is always negative, hence all compounds are fe rrimagnets. All\ninteractions are mostly confined within a radius of 1 .5a. Apart from these similarities,\nthere are many interesting differences.\nFirst, we discuss the details of the dominating inter-sublattice inter action Mn1(2)-\nMn2(1). The first and second nearest neighbors provide a large, positive e xchange.\nThe second nearest neighbors have two different values of Jij. This is a feature that\nis not observed in frozen-magnon calculations (see, e.g. [16]), beca use the Fourier\ntransform that is necessary to obtain the exchange parameters involves a spherical\naveraging. Instead, with the real-space approach used here we o bserve a difference for\nMn atoms with a Ti atom or a Zatom in between. We found larger values on the\nMn atoms mediated via Ti and lower values on the Zmediated ones. The nearest Mn\nneighborshaveadistanceofabout2.95 ˚A,andthe exchangeisapparentlyindirect. For\ndirect exchange, one would expect a scaling with the magnetic momen ts, which is not\nobserved here. It rather oscillates with the sp electron number. A similar result has\nbeen obtained earlier on other half and full Heusler compounds [39]. The ratio of the\nnearest and second nearest neighbor coupling is significantly reduc ed with increasing\nelectron concentration, and the nearest neighbor interaction do minates in Mn 2TiSb.\nThe antiferromagnetic Mn-Ti interaction is only significant for the n earest\nneighbors. Accordingly, the interaction between Mn and Ti, which ha ve a distance of\nabout 2.55 ˚A is essentially given by direct exchange coupling and the scaling with th e\nTi moment corroborates this assumption.\nThe intra-sublattice interaction of Mn1(2)-Mn1(2)exhibits a notable oscillatory\nbehavior. In the two cases with odd valence electron number it is pos itive for theFerrimagnetism in Mn 2TiZ Heusler compounds 9\n/s51\n/s50\n/s49\n/s48\n/s45/s49 /s77/s110/s50/s84/s105/s83/s98/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105/s54\n/s52\n/s50\n/s48\n/s45/s50 /s77/s110/s50/s84/s105/s71/s101/s71/s101/s32/s109/s101/s100/s105/s97/s116/s101/s100/s84/s105/s32/s109/s101/s100/s105/s97/s116/s101/s100 /s56\n/s52\n/s48\n/s45/s52/s74/s105/s106/s32/s40/s109/s101/s86/s41\n/s77/s110/s50/s84/s105/s71/s97\n/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s54/s46/s48/s55/s32/s197/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s114/s32/s47/s32/s97/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s53/s46/s56/s55/s32/s197/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s74/s105/s106/s32/s40/s109/s101/s86/s41\n/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s53/s46/s54/s55/s32/s197/s40/s97/s41\n/s40/s98/s41\nFigure 4. Heisenberg exchange parameters Jijin dependence on the normalized\ndistance r/a. (a):Jijfor Mn 2TiGa, Mn 2TiGe, Mn 2TiSb for their respective\nequilibrium lattice parameters. (b): Jijfor Mn 2TiGe with different lattice\nparameters. Note the different scales of the vertical axes in the top row.\nnearest neighbors, negative for the second, and again positive fo r the third nearest\nneighbors. For Mn 2TiGe with its even electron count the first two neighbors have\nnegative and the third neighbor has positive interaction. So in the lat ter case, the\ntotal Mn-Mn intra-sublattice interaction is effectively antiferroma gnetic.\nIn order to study the dependence of Jijon the lattice parameter as a possible\nexplanation for the differences discussed above, additional calcula tions on Mn 2TiGe\nhave been performed with lattice parameters of (5 .87±0.2)˚A. This compound was\nchosen because of the wide (pseudo-)gap for the spin-up states , which warrants a\nstable total magnetic moment and minimal band structure effects o ver the range of a\nused here.\nThe resultsfrom these calculationsare givenin Fig. 4 (b). Obviously, the changes\nherearerathersubtleandcannotaccountforthethe largediffer encesdiscussedabove.\nHowever, we note a reduction of the nearest neighbor Mn1(2)-Mn2(1)interaction and\nof the Ti mediated second nearest Mn1(2)-Mn2(1)neighbor. Meanwhile, the Mn-Ti\ninteraction increases, in agreement with increased Mn and Ti momen ts.\nThe strong confinement of the exchange interactions to a sphere with a radius of\nabout 1.5ais reflected in the Curie temperature calculated as a function of the cluster\nradius which is nearly converged at r/greaterorsimilar1.5a, see Fig. 5 (a). At larger radii a weak\noscillation of TMFA\nCis observed, indicating long-ranged RKKY-like behaviour.\nA deeper discussion of the exchange interaction is beyond the scop e of this\npaper. However, it was recently shown for numerous half and full H eusler compounds\nthat various exchange mechanisms—such as RKKY, superexchang e and Anderson s-d\nmixing—contribute to the indirect exchange interactions [39].Ferrimagnetism in Mn 2TiZ Heusler compounds 10\n/s56/s48/s48\n/s55/s48/s48\n/s54/s48/s48\n/s53/s48/s48\n/s52/s48/s48\n/s51/s48/s48\n/s50/s48/s48\n/s49/s48/s48\n/s48/s84/s67/s77/s70/s65/s32/s40/s75/s41\n/s51/s46/s48 /s50/s46/s53 /s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s114/s32/s47/s32/s97/s32/s65/s108 /s32/s83/s105\n/s32/s71/s97 /s32/s71/s101 /s32/s65/s115\n/s32/s73/s110 /s32/s83/s110 /s32/s83/s98\n/s40/s97/s41/s56/s48\n/s55/s48\n/s54/s48\n/s53/s48\n/s52/s48\n/s51/s48\n/s50/s48\n/s49/s48\n/s48\n/s45/s49/s48\n/s45/s50/s48\n/s45/s51/s48\n/s45/s52/s48\n/s45/s53/s48/s74/s48/s32/s40/s109/s101/s86/s41\n/s65/s108/s71/s97 /s73/s110/s83/s105/s71/s101/s83/s110 /s80/s65/s115/s83/s98/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105\n/s40/s98/s41\nFigure 5. (a): The Curie temperature TMFA\nCin dependence on the normalized\ncluster radius r/ataken into the summation. (b): R-summed exchange coupling\nparameters J0.\nThe relevant contributions to the J0matrix in Eq. (2) are displayed in Fig. 5\n(b). In agreement with the previous discussion it is found that the in ter-sublattice\ninteraction Mn1(2)-Mn2(1)provides the largest contribution, followed by the Mn-Ti\ninteraction, which can become as large as the Mn1(2)-Mn2(1)interaction in Mn 2TiIn.\nThe intra-sublattice interaction Mn1(2)-Mn1(2)is generally weak, positive for Al, Ga,\nIn, and negative for Si, Ge, Sn. All other inter- and intra-sublattic e contributions are\nbelow 1meV. A negative intra-sublattice contribution means that th e interaction acts\nagainstthe ferromagneticorderonthis latticeandthusreducest heCurietemperature.\nTo estimate the accuracyofour method for the Curie temperatur e determination,\nwe calculated the Curie temperatures of some Heusler compounds a t their respective\nexperimental lattice parameters. The calculated (experimental) v alues are: Co 2MnSi\n1049K (985K)[40], Co 2TiSn 383K (355K)[41], Mn 2VAl 605K (760K)[13] and\nMn2VGa 560K (783K)[26]. Further values, obtained using the same meth od, can\nbe found in Ref. [35]. For the Co-based ferromagnetic compounds, the calculated\nmean-field values are in good agreement with experiment. However, in the case of the\ntwo ferrimagnetic Mn-based compounds, the MFA Curie temperatu re is about 25%\nlower than the experimental one.\nTable 2summarizesourcalculated Curietemperatures. Theyarewe llaboveroom\ntemperature for the compounds with 21 and 22 valence electrons, but considerably\nlower for Mn 2TiAs and Mn 2TiSb. The Curie temperature scales roughly linear\nTable 2. Curie temperatures TMFA\nCcalculated in the mean-field approximation.\nMn2TiZ Al Ga In Si Ge Sn P As Sb\nTMFA\nC(K) 665 663 630 424 398 354 — 132 156Ferrimagnetism in Mn 2TiZ Heusler compounds 11\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53/s109/s32/s40/s109/s66/s41\n/s54/s46/s48/s55 /s53/s46/s57/s55 /s53/s46/s56/s55 /s53/s46/s55/s55 /s53/s46/s54/s55\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s40/s99/s41\n/s32/s109/s77/s110\n/s32/s109/s84/s105/s52/s50/s48\n/s52/s48/s48\n/s51/s56/s48\n/s51/s54/s48\n/s51/s52/s48/s84/s67/s77/s70/s65/s32/s40/s75/s41/s40/s98/s41/s54/s48\n/s52/s48\n/s50/s48\n/s48\n/s45/s50/s48/s74/s48/s32/s40/s109/s101/s86/s41/s40/s97/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105\nFigure 6. Dependence of J0(a),TMFA\nC(b) and magnetic moments (c) on the\nlattice parameter in Mn 2TiGe. Markers in (b) are the same as in (a). Magnetic\nmoments in (d) are mMn(⊓ ⊔) andmTi(△).\nwith the total magnetic moment. Within one group, the Curie temper atures are\ncomparable, though a trend to decrease with increasing atomic num ber of the Z\ncomponent is clear for 21 and 22 valence electrons.\nThe Curie temperatures of Mn 2TiAl, Mn 2TiGa and Mn 2TiIn are quite similar.\nThe slightly reduced TMFA\nCof Mn 2TiIn is caused by the steep reduction of the\nMn1(2)-Mn2(1)interaction. On the other hand, a simultaneous increase of the\nMn-Ti interaction stabilizes TMFA\nCat a still high level. In the series Mn 2TiSi –\nMn2TiGe – Mn 2TiSn the Mn1(2)-Mn2(1)decreases, but here the increase of the Mn-Ti\ninteraction can not compensate this and hence the Curie temperat ure decreases. In\nany case, the Mn1(2)-Mn2(1)interaction provides the dominant contribution to TMFA\nC,\nonly in Mn 2TiIn the Mn-Ti interaction is dominant. The significantly lower Curie\ntemperature of Mn 2TiAs with respect to Mn 2TiSb can be attributed to the artificially\nincreased lattice parameter used in the calculation.\nThe dependence of the exchange parameters and TMFA\nCon the lattice constant\nwas studied above for Mn 2TiGe. The corresponding terms of the J0matrix, the\nCurie temperature and the magnetic moments are presented in Fig. 6 (a)-(c). A\ndecrease of the Mn1(2)-Mn2(1)interaction and simultaneously of TMFA\nCwith increasing\nais observed, although both mMnandmTiincrease. Obviously, the individual\nmoments play only a minor role in the exchange and the interatomic dist ances are\nmoreimportant. The Mn-TiaswellastheMn1(2)-Mn1(2)interactionsbecomestrongerFerrimagnetism in Mn 2TiZ Heusler compounds 12\nwith increasing a, but they nearly compensate each other. In agreement with a dire ct\nexchange coupling, the Mn-Ti interaction scales with the magnetic m oments. The\nchanges in J0reproduce very well the changes observed in Fig. 5 (b) for the Si – Ge\n– Sn series.\nPut in terms of a pressure dependence, we observe d TC/dp >0, i.e., the\nCurie temperature increases with increasing pressure. Kanomata et al.proposed\nan empirical interaction curve for Ni 2MnZand Pd 2MnZfull Heusler compounds that\nsuggestes d TC/dp >0 for these compounds [42]. The origin of this behavior is\nattributed to the Mn-Mn distance and the indirect exchange betwe en the Mn atoms,\nwhich fully carry the magnetism of the compounds. Hence, all other interactions\ncan be neglected. A numerical confirmation by first principles of this interaction\ncurve was given recently [38]. For half-metallic Heusler compounds of type Co 2YZ\nK¨ ubleret al.analyzed the dependence of TCon the valence electron number, which\nis approximately linear, and scales thus with the total magnetic mome nt [43]. Further\nit was also proposed for Co 2MnZcompounds to have d TC/dp >0, although the Co\natom participates significantly in the exchange interactions [37]. Exp erimentally this\ndependence on the lattice parameter was even observed for the C o2TiZseries (with Z\n= Si, Ge, Sn), where the Ti atoms have nearly vanishing magnetic mom ent [27].\nInterestingly, the magnetic moments of Mn and Ti in Mn 2TiGe vary within the\nsamerangeasthemomentsfordifferentcompoundsshowninFig. 2( b), whilethe total\nmoment remains fixed at 2 µB/f.u. These findings demonstrate the stronginfluence of\nthe lattice parameter, while the details of the electronic structure of theZelement are\nless important. Consequently, the Zelement influences the properties of the Mn 2TiZ\ncompound mainly via its number of valence electrons and its atomic rad ius, which\ndetermines the equilibrium lattice parameter.\n4. Conclusion\nOur results suggest that the Mn 2TiZHeusler compound series with Z= Al, Ga,\nIn, Si, Ge, Sn, P, As, Sb, can exhibit ferrimagnetism in accordance w ith the rule\nm=NV−24. Most of the compounds have large spin polarization and a spin-up gap\nforms above the Fermi energy. The Curie temperatures calculate d within the mean-\nfield approximation indicate that the compounds with 21 and 22 valenc e electrons will\nbe ferrimagnetic at room temperature. A thorough understandin g of the influence of\ntheZcomponent on the properties of the compounds has been establish ed on the\nbasis of ab initio band structure and exchange coupling calculations. It was found\nthat the pressure dependence of TCis positive, in agreement with ferromagentic full\nHeusler compounds. Because of their large and stable spin polarizat ions and their\nhigh Curie temperatures we propose in particular Mn 2TiSi, Mn 2TiGe, and Mn 2TiSn\nas candidates for spintronic applications.\nAcknowledgements\nThis work has been supported by the German Bundesministerium f¨ u r Bildung und\nForschung (BMBF) under contract number 13N9910. Helpful disc ussions with Prof.\nAndrei Postnikov are acknowledged.Ferrimagnetism in Mn 2TiZ Heusler compounds 13\nReferences\n[1]¨Ozdo˜ gan K, Galanakis I, S ¸a¸ sioglu E and Akta¸ s B 2006 J. Phys.: Condens. Matter 182905\n[2] Fujii S, Okada M, Ishida S and Asano S 2008 J. Phys. Soc. Jpn 77074702\n[3] Luo H, Zhu Z, Liu G, Xu S, Wu G, Liu H, Qu J and Li Y 2008 J. Magn. Magn. Mater. 320421\n[4] Wurmehl S, Kandpal H C, Fecher G H and Felser C 2006 J. Phys.: Condens. Matter 186171\n[5] Luo H Z, Zhang H W, Zhu Z Y, Ma L, Xu S F, Wu G H, Zhu X X, Jiang C B a nd Xu H B 2008\nJ. Appl. 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Ostler,1Roy W. Chantrell,1Ilie Radu,2and Theo Rasing3\n1Department of Physics, University of York, Heslington, York YO10 5DD United Kingdom.\n2Institut f ur Methoden und Instrumentierung der Forschung mit Synchrotronstrahlung,\nHelmholtz-Zentrum Berlin f ur Materialien und Energie,\nGmbH, Albert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n3Radboud University Nijmegen, Institute for Molecules and Materials,\nHeyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.\nSynthetic ferrimagnets are composite magnetic structures formed from two or more anti-\nferromagnetically coupled magnetic sublattices with di\u000berent magnetic moments. Here we report on\natomistic spin simulations of the laser-induced magnetization dynamics on such synthetic ferrimag-\nnets, and demonstrate that the application of ultrashort laser pulses leads to sub-picoscond magne-\ntization dynamics and all-optical switching in a similar manner as in ferrimagnetic alloys. Moreover,\nwe present the essential material properties for successful laser-induced switching, demonstrating\nthe feasibility of using a synthetic ferrimagnet as a high density magnetic storage element without\nthe need of a write \feld.\nThe dynamic response of magnetic materials to ultra-\nshort laser pulses is currently an area of fundamental\nand practical importance that is attracting a lot of atten-\ntion. Since the pioneering work of Beaurepaire et al [1] it\nhas been known that the magnetization can respond to\na femtosecond laser pulse on a sub-picosecond timescale.\nHowever studies of magnetic switching are more recent.\nIn this context an especially intriguing phenomenon is\nthat of all-optical switching, which uses the interaction of\nshort, intense pulses of light with a magnetic material to\nalter its magnetic state without the application of an ex-\nternal magnetic \feld[2, 3]. Recent experiments [4{6] and\ntheoretical calculations[5, 7{9] have demonstrated that\nthe origin of all-optical switching in ferrimagnetic alloys\nis due to ultrafast heating of the spin system. The mag-\nnetic switching arises due to a transfer of angular momen-\ntum between the two sublattices within the material[7, 8]\nand the resulting exchange-\feld induced precession[7].\nRemarkably, this e\u000bect occurs in the absence of any sym-\nmetry breaking magnetic \feld [5], and can be considered\nas Thermally Induced Magnetic Switching (TIMS). So\nfar TIMS has only been demonstrated experimentally in\nthe rare-earth transition metal (RE-TM) alloys GdFeCo\nand TbCo which, in addition to their strong magneto-\noptical response, have two essential properties for heat-\ninduced switching: antiferromagnetic coupling between\nthe RE and TM sublattices[10] and distinct demagneti-\nzation times of the two sublattices[4]. The antiferromag-\nnetic coupling allows for inertial magnetization dynam-\nics, while the distinct demagnetization times under the\naction of a heat pulse allow a transient imbalance in the\nangular momentum of the two sublattices, which initi-\nates a mutual high speed precession enabling ultrafast\nswitching to occur.\nAlthough GdFeCo has excellent switching properties,\nits potential use in magnetic data storage is limited by\nits low anisotropy and amorphous structure, precluding\nthe use of single magnetic domains typically less than10 nm in size, required for future high density magnetic\nrecording media. One intriguing possibility, and the fo-\ncus of this paper, would be the use of a synthetic fer-\nrimagnet (SFiM), consisting of two transition metal fer-\nromagnets anti-ferromagnetically exchange coupled by a\nnon-magnetic spacer[11], shown schematically in Fig 1.\nThe important but as yet unanswered question is whether\nall-optical switching would also work in such an arti\fcial\nstructure and what essential physical properties of the\ndesign are required. Such a composite magnet also has a\nnumber of distinct advantages over intrinsic rare-earth-\ntransition metal ferrimagnets: the dynamic properties of\neach sublattice may be separately selected by choice of\nmaterial, nano-patterning is possible in the sub-10 nm\nsize range due to their crystalline nature and the omis-\nsion of costly rare-earth metals. Importantly the compos-\nite design has the advantage of allowing the use of high\nanisotropy materials such as FePt or CoPt to enhance\nthe thermal stability of the medium. These advantages\ncould make such synthetic structures very promising can-\ndidates for magnetic data storage applications.\nIn this letter we present dynamic studies of such a syn-\nthetic ferrimagnet using an atomistic spin model. We in-\nvestigate the dynamic properties of the separate layers\nand show that the demagnetization time is determined\nprimarily by the local atomic spin moment and the in-\ntrinsic Gilbert damping of the material. We \fnally con-\nsider an exchange-coupled Fe/FePt synthetic ferrimagnet\nand show that a short heat-pulse is su\u000ecient to induce\nultrafast heat-induced switching of the material.\nThe dynamic properties of the SFiM are studied us-\ning an atomistic spin model using the vampire software\npackage[12, 13]. The energetics of the system are de-\nscribed using a Heisenberg spin Hamiltonian, which in\ncondensed form reads:\nH=\u0000X\ni<jJijSi\u0001Sj\u0000X\nikuS2\ni;z (1)arXiv:1310.5170v2  [cond-mat.mtrl-sci]  8 Jun 20142\nFeExchange separationlayerFePt\nFIG. 1. Visualization of a synthetic ferrimagnetic structure\nconsisting of two ferromagnetic layers separated by a non-\nmagnetic spacer layer to engineer anti-ferromagnetic coupling\nbetween them. (Color Online).\nwhereJijis the exchange energy between nearest neigh-\nboring spins, SiandSjare unit vectors describing the\nspin directions for local sites iand nearest neighbor sites\njrespectively, and kuis the uniaxial anisotropy constant.\nThere are three distinct Jijexchange interactions in the\nSFiM system, arising from the intralayer and interlayer\ncontributions, detailed in Tab. I. Since we are consider-\ning heat-induced switching no external \feld is applied\nduring the simulations. The system of coupled spins\nis integrated using the Landau-Lifshitz-Gilbert equation\nwith the Langevin dynamics formalism at the atomistic\nlevel using the Heun numerical scheme[13]. More de-\ntails of the model are provided in the Supplementary\nInformation[14].\nFor a synthetic ferrimagnet to exhibit thermally in-\nduced magnetic switching it is essential to consider the\nphysical requirements of the structure analogous to those\nof intrinsic RE-TM ferrimagnets. The \frst property is\nthe anti-ferromagnetic exchange coupling of the compo-\nnent layers of the synthetic ferrimagnet, which can be\nengineered by a suitable choice and thickness of material\nsuch as Ir or Ru[11]. The second criterion is the existence\nof distinct magnetization dynamics for the two compo-\nnent layers which allows the formation of a transient fer-\nromagnetic state and drives the switching process[4, 5].\nLet us start with a simplistic scenario, where the ef-\nfects of exchange are ignored and the demagnetization\ntime\u001cdof a ferromagnetic material is given by the ratio\nof the atomic magnetic moment and damping[15], such\nthat\u001cd\u0018\u0016s=\u000b. In this simplistic view, a variation of the\ndamping parameter or the local atomic spin moment will\nlead to a straightforward variation of the demagnetiza-\ntion time. To test this assertion, we have simulated a fer-\nromagnetic material, with an exchange coupling the same\nas FePt, but with freely varied local atomic spin moment\n\u0016sand damping parameter, \u000b, to which a step-function\nincrease to a temperature above the Curie temperature is\napplied. The sudden increase in temperature leads to a\ndemagnetization of the material, with characteristic be-\n  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8\n0.0 0.5 1.0 1.5 2.0Normalized magnetization\nTime (ps)a\nα = 0.01\n0.02\n0.05\n0.10\n0.50\n0.01 0.1   1  10\n 1  10  100Demagnetization time (ps)\nTime constant µs / α (µB)b\nτ1 (µs)\nτ1 (α)\nτ2 (µs)\nτ2 (α)FIG. 2. ( a) Simulated demagnetization of a ferromagnet un-\nder the action of a heat pulse (points) for di\u000berent values of\nthe intrinsic damping constant, \u000b, \ftted to Eq. 2 (lines). ( b)\nFitted demagnetization time constants \u001c1and\u001c2as a func-\ntion of the ratio \u0016s=\u000bfor a ferromagnetic material. The leg-\nend shows the origin of the calculated demagnetization time\nfor variable \u000b(\u0016s= 1:5\u0016B) or for variable \u0016s(\u000b= 0:1) re-\nspectively. The data show that the the demagnetization time\nconstants for the same exchange interaction depend only on\nthe ratio of \u0016s=\u000b, and not the individual values of \u0016sand\u000b.\n(Color Online).\nhavior shown in Fig. 2(a) for di\u000berent values of the damp-\ning parameter. The demagnetization dynamics are not\ngenerally describable by a single demagnetization time\n\u001cd, but usually it is su\u000ecient to describe the demagne-\ntization dynamics by two leading contributions[16]. We\ntherefore \ft the time-dependent demagnetization dynam-\nics with the function[4, 17]\nm(t) =A1exp\u0012\n\u0000t\n\u001c1\u0013\n+A2exp\u0012\n\u0000t\n\u001c2\u0013\n+ const:(2)\nwhere the demagnetization time constants \u001c1and\u001c2are\nintrinsic timescales and A1,A2and the constant are \ft-\nting parameters. Since the \fnal temperature is above the\nCurie point the constant is close to zero but is treated as\na free parameter of the \ftting.\nA systematic variation of the Gilbert damping param-\neter and local spin moment leads to a range of demagne-3\n0.00.20.40.60.81.0\n 0  0.2  0.4  0.6  0.8  1Normalized magnetization\nTime (ps)FePt\nFe\nFIG. 3. Simulated magnetization dynamics for Fe and FePt\nferromagnets under action of an ultrafast heat pulse Gaussian\nin time with a width of 20 fs. The magnetization is normal-\nized to the 0K saturation value, and so the di\u000berent Curie\ntemperatures lead to di\u000berent reduced magnetization at the\nstarting temperature of T=300K. The heat pulse is su\u000ecient\nto demagnetize the FePt layer, but the Fe layer remains or-\ndered. (Color Online).\ntization times (by \ftting to Eq. 2), shown in Fig. 2(b).\nThe essential result is that the demagnetization time con-\nstants, for a \fxed exchange constant, depends only on\nthe ratio\u0016s=\u000b. Thus a certain demagnetization time can\nbe engineered through any combination of the local spin\nmoment (changeable by varying the composition in FeCo\nalloys for example), and the damping parameter (for ex-\nample by using high anisotropy materials or by introduc-\ning disorder in the form of defects and impurities).\nIn addition to the requirements of antiferromagnetic\ncoupling and distinct demagnetization times, the thick-\nness of the two layers is also important for synthetic fer-\nrimagnets. In RE-TM alloys the antiferromagnetic ex-\nchange interactions are a bulk e\u000bect existing between\nneighboring atoms. For the synthetic structure however\nthe exchange is an interface e\u000bect, so the e\u000bective ex-\nchange \feld on the layer is inversely proportional to the\nlayer thickness. Hence for large e\u000bective coupling thin\nlayers are preferred. For thermal stability larger volumes\nare preferred, and so there must be a balance between\nthis and the e\u000bective exchange \feld. In general it ap-\npears to be preferable for the high anisotropy material to\nhave a larger volume, while the other layer can be thin\nto maximize the e\u000bective exchange \feld.\nHaving de\fned the essential physical parameters for\nthermally induced switching in synthetic ferrimagnets,\nwe now consider speci\fc materials which may be suit-\nable for applications. For devices, strong magnetocrys-\ntalline anisotropy is essential, and so the obvious choice\n(excluding rare-earth metals) are CoPt and FePt based\nalloys, currently used in conventional magnetic recording\nmedia. The dynamic properties of such materials canbe inferred from theoretical calculations of atomic mo-\nments and experimental results of the e\u000bective damping.\nIn L1 0FePt the magnetic moments are not evenly dis-\ntributed between the Fe and Pt atoms, since pure Pt is\nnon-magnetic[18]. Ab-initio calculations[18] show that\nthe Fe moments in FePt are signi\fcantly enhanced over\nbulk Fe, with an e\u000bective moment of 3 :2\u0016B, and so one\nwould normally assume relatively slow dynamics for the\nFe moments in FePt. However, the strong spin-orbit cou-\npling in FePt also leads to large Gilbert damping around\n0.1[19], and so FePt as a whole exhibits `fast' dynam-\nics. The relatively low Curie temperature of \u0018700K is\nalso suitable for laser induced switching as the threshold\n\ruence is lower. For heat-induced switching the two lay-\ners must exhibit distinct magnetization dynamics, and so\nthe other layer must be `slower' than FePt. The obvious\nchoices are elemental Fe or Co, due to their lower damp-\ning and high moments. Here we choose Fe due to its\nlower Curie temperature of 1043K, and larger magnetic\nmoment of 2.2 \u0016B.\nBefore considering the switching properties of the\nFe/FePt synthetic ferrimagnet, we \frst address the de-\nmagnetization dynamics of the two uncoupled layers indi-\nvidually. The system consists of a 5 nm thick FePt layer\nand 1 nm thick Fe layer with a lateral size of 8 nm in the\nshape of a cylinder. The temporal temperature pro\fle\narising from a 20 fs laser pulse is calculated using the two\ntemperature model treating the electron and phonon sys-\ntems separately and coupling the magnetic system to the\nelectrons[20]. The parameters for the two temperature\nmodel are the same as those in Ref. 10 and further de-\ntails are provided in the Supplementary Information[14].\nThe resulting demagnetization dynamics for the Fe and\nFePt layers are presented in Fig. 3. As expected, the\nFePt and Fe layers exhibit `fast' and `slow' dynamics re-\nspectively, but the di\u000berent Curie temperatures of the\ntwo layers leads to di\u000berent levels of demagnetization.\nWe \fnally consider ultrafast all-optical switching in the\nSFiM nanostructure. Heat-induced switching is driven\nby an intricate process involving ultrafast demagnetiza-\ntion, transfer of angular momentum between the sub-\nlattices, and high frequency precession via the exchange\nmode[7], shown schematically in Fig. 4. The strength\nof the exchange interaction determines the timescale of\nthe switching process and the rate of transfer of angu-\nlar momentum between the sublattices and so should be\nas strong as possible. Depending on the system there is\nsome disparity between the values of exchange measured\nexperimentally[11, 21] and calculated theoretically[22],\nand so we assume an intermediate value of the interlayer\nexchange interaction of \u00181=5 that of the bulk FePt ex-\nchange (JFePt\u0000Fe\nij =\u00001:635\u000210\u000021J/link). The calcu-\nlated sublattice magnetization dynamics resulting from a\n20 fs heat pulse is shown in Fig. 5.\nThe FePt layer is initially demagnetized by the heat\npulse and switches direction under the in\ruence of the4\n(a)(b)FePtFe(c)(d)\nFIG. 4. Schematic illustration of the switching process for\nall-optical heat-induced magnetic switching. ( a) The two sub-\nlattices are initially aligned anti-parallel, and application of\na laser pulse causes rapid demagnetization of the FePt sub-\nlattice. ( b) Thermal \ructuations lead to a small transverse\ncomponent which initiates a rapid precession of the FePt sub-\nlattice in the exchange \feld of the Fe layer and relaxation\ntoward the Fe sublattice. ( c) As the FePt layer precesses the\nFe layer responds to the laser pulse and begins mutual pre-\ncession leading to a transient ferromagnetic state. ( d) After\nthe system has reached thermal equilibrium the sublattices\nare aligned antiparallel and relax to the easy axis direction,\ncompleting the switching process. (Color Online.)\nexchange \feld from the Fe, while the Fe layer itself re-\ntains a higher degree of order. As the electron system\ncools the FePt recovers its order along the - zdirection i.e.\nit switches its magnetization. Simultaneously the Fe sub-\nlattice begins to precess in the increasing exchange \feld\nof the FePt layer, and switches precessionally, as seen\nfrom the oscillations in the Fe magnetization in Fig. 5.\nFinally the system relaxes to the usual (but reversed) an-\ntiferromagnetic state, with some \fnal relaxation towards\nthe easy axis direction on a longer timescale. The simula-\ntions demonstrate the feasibility of ultrafast heat-induced\nswitching in synthetic ferrimagnetic nanostructures with\nthe correct physical characteristics.\nAlthough the switching is a thermally driven pro-\ncess, the mechanism is itself deterministic and quite ro-\nbust. To demonstrate this, we present similar switching\nfor a CoPt/Fe bilayer structure in the Supplementary\nInformation[14]. The exchange coupling between the two\nlayers is an essential component for the switching. For\nfast dynamics this should ideally be as large as possi-\nble, but switching also occurs for typical values of \u00186\nmJ/m2reported experimentally[11], as presented in the\nSupplementary Information[14].\nTo conclude, we have investigated the laser induced dy-\nnamic properties of synthetic ferrimagnets, and demon-\nstrated the possibility of all-optical magnetic switching\nin Fe/FePt nano structures. Material combinations other\nthan Fe/FePt are possible, including other alloys based\non Ni, Fe and Co ferromagnets such as Fe/CoPt. The\ninterlayer exchange coupling, which drives the switching\nprocess, can also be engineered by using di\u000berent ma-\nterials such as Ir or Si, though the optimal coupling de-\npends on the thickness of the layers. This allows the tun-\ning of properties such as the Curie temperature, magne-\ntocrystalline anisotropy and Gilbert damping to achieve\n-1.0-0.50.00.51.0\n 0  5  10  15  20Normalized magnetization\nTime (ps)FePt\nFeFIG. 5. Calculated magnetization dynamics for the Fe/FePt\nbilayer in response to an ultrafast heat pulse. The FePt layer\nrapidly demagnetizes and develops a transverse component,\ninducing rapid precession leading to a transient ferromagnetic\nstate. The mutual precession of the Fe and FePt sublattices\nleads to switching of the Fe sublattice, while the FePt sublat-\ntice is stabilized by the high magnetocrystalline anisotropy.\n(Color Online).\nthe desired dynamic behavior and switching properties,\nand opens the possibility of engineering high performance\nmagnetic data storage devices. This is highly signi\fcant\nfor the following reasons. Firstly, the removal of the ne-\ncessity for the write \feld in magnetic recording write\ntransducers would lead to a dramatic reduction in the\ncomplexity of transducer design and the number of op-\nerations required for their production. Secondly, it has\nbeen shown that that the magnitude of the write \feld is\na major factor limiting the ultimate density in magnetic\nrecording[23]. Essentially, the \feld during writing must\nbe su\u000eciently large to overcome thermally driven back-\nswitching of the magnetization, a factor termed the ther-\nmal writability. In Ref. [23] it was shown that thermal\nwritability is a more important factor than the thermal\nstability criterion in determining the limiting recording\ndensity. The e\u000bective \feld in the TIMS process is the\nexchange \feld between the sublattices, which is signi\f-\ncantly larger than the values accessible by today's induc-\ntive technology, which would essentially remove the ther-\nmal writability as a limiting factor in magnetic recording.\nTABLE I. Summary table of model parameters and their units\nFe FePt Unit\nJij 6:75\u000210\u0000214.5\u000210\u000021J/link\n\u0016s 2.2 3.2 \u0016B\nku 0.0 1 :61\u000210\u000022J/atom\n\u000b 0.01 0.1 -\nFinancial support from the EU Seventh Framework\nProgramme under grant agreement No. 281043 fem-5\ntospin is gratefully acknowledged.\n\u0003richard.evans@york.ac.uk\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Phys. Rev. Lett. 76, 4250 (1996).\n[2] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pis-\narev, A. M. Balbashov, and T. Rasing, Nature 435, 655\n(2005).\n[3] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n[4] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. Durr, T. A. Ostler, J. Barker, R. F. L. Evans,\nR. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk,\nT. Rasing, and A. V. Kimel, Nature 472, 205 (2011).\n[5] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,\nL. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolt-\ning, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov,\nA. M. Kalashnikova, K. Vahaplar, J. Mentink, A. Kiri-\nlyuk, T. Rasing, and A. V. Kimel, Nat. Commun. 3, 666\n(2012).\n[6] A. R. Khorsand, M. Savoini, A. Kirilyuk, A. V. Kimel,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n108, 127205 (2012).\n[7] U. Atxitia, T. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, and O. Chubykalo-Fesenko, Phys. Rev. B 87,\n224417 (2013).\n[8] J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,\nA. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand T. Rasing, Phys. Rev. Lett. 108, 057202 (2012).\n[9] J. Barker, U. Atxitia, T. A. Ostler, O. Hovorka, O.Chubykalo-Fesenko and R. W. Chantrell, Sci. Rep. 3,\n3262 (2013).\n[10] T. A. Ostler, R. F. L. Evans, R. W. Chantrell, U. Atxitia,\nO. Chubykalo-Fesenko, I. Radu, R. Abrudan, F. Radu,\nA. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and\nA. Kimel, Phys. Rev. B 84, 024407 (2011).\n[11] S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n[12]vampire software package, version 3. Available from\nhttp://vampire.york.ac.uk/.\n[13] R. F. L. Evans, W. J. Fan, P. Chureemart, M. O. A. Ellis,\nT. A. Ostler and R. W. Chantrell, J. Phys. Condens.\nMatt. (2014). [In press]\n[14] See supplementary material at [URL will be inserted by\nAIP] for additional model details and extended results.\n[15] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and\nW. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 (2005)\n.\n[16] D. A. Garanin, Phys. Rev. E 54, 3250-3256 (1996).\n[17] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. Fhnle, T. Roth, M. Cinchetti and M. Aeschlimann,\nNat. Mater. 9, 259265 (2010).\n[18] O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W.\nChantrell, Euro. Phys. Lett. 69, 805 (2005).\n[19] Z. Zhang, B. Cui, G. Wang, B. Ma, Q. Y. Jin, and\nY. Liu, Appl. Phys. Lett. 97, 172508 (2010).\n[20] S. Anisimov, B. Kapeliovich, and T. Perelman, Sov.\nPhys. JETP 39, 375377 (1974).\n[21] J. Jiang, N. Tezuka, and K. Inomata, Journal of Applied\nPhysics 98, 063902 (2005).\n[22] H. C. Herper, P. Weinberger, L. Szunyogh, and C. Som-\nmers, Phys. Rev. B 66, 064426 (2002).\n[23] R. F. L. Evans, R. W. Chantrell, U. Nowak, A. Lyber-\natos, and H.-J. Richter, Appl. Phys. Lett. 100, 102402\n(2012)."    },    {        "title": "1312.3439v1.Exact_results_for_a_generalized_spin_1_2_Ising_Heisenberg_diamond_chain_with_the_second_neighbor_interaction_between_nodal_spins.pdf",        "content": "arXiv:1312.3439v1  [cond-mat.stat-mech]  12 Dec 2013physica statussolidi\nExact results for a generalized\nspin-1/2 Ising–Heisenberg diamond\nchain with the second-neighbor\ninteractionbetweennodal spins\nBohdanLisnyi1,2, Jozef Stre ˇcka2,*\n1Institute for Condensed Matter Physics,National Academy o fSciences ofUkraine, 1 SvientsitskiiStreet,Lviv79011, U kraine\n2Insitute of Physics,Facultyof Science, P.J. ˇSaf´ arikUniversity, ParkAngelinum 9,040 01Koˇ sice, Slov akRepublic\nReceivedXXXX, revisedXXXX,accepted XXXX\nPublishedonline XXXX\nKeywords: Ising–Heisenberg diamond chain, spin frustration,magnet ization plateaux, exact results\n∗Corresponding author: e-mail jozef.strecka@upjs.sk , Phone: +421-55-2342276, Fax:+421-55-6222124\nThe ground state and thermodynamics of a generalized spin-1 /2 Ising–Heisenberg diamond chain with the second-\nneighbor interaction between nodal spins are calculated ex actly using the mapping method based on the decoration-\niteration transformation. Rigorous results for the magnet ization, susceptibility, and heat capacity are investigat ed in\ndependenceon temperatureandmagneticfield for the frustra teddiamondspin chainwith the antiferromagneticIsing\nand Heisenberginteractions.It is demonstratedthat the se cond-neighborinteractionbetweennodalspinsgivesrise t o\na greater diversity of low-temperaturemagnetization curv es, which may include an intermediate plateau at two-third\nof the saturation magnetization related to the classical fe rrimagnetic (up-up-up-down-up-up-...) ground state with\ntranslationallybrokensymmetrybesidesanintermediateo ne-thirdmagnetizationplateaureflectingthetranslation ally\ninvariantquantumferrimagnetic(monomer-dimer)spinarr angement.\nCopyrightlinewillbe provided by the publisher\n1 Introduction Decorated spin chains, which can\nbe exactly treated by combining the decoration-iteration\ntransformation with the transfer-matrix method [1,2,3,4,\n5,6], are of practical importance for the qualitative inter -\npretation of magnetic phenomena emergent in real solid-\nstate materials. In particular, several exactly solved Isi ng–\nHeisenberg decorated spin chains provide in-depth under-\nstanding of a striking interplay between geometric spin\nfrustration and quantum fluctuations, which may manifest\nitself through various intriguing phenomena such as the\nappearance of intermediate plateaux in low-temperature\nmagnetizationcurves,the formationof additional maxima\nin the temperature dependence of the susceptibility and\nspecific heat, the enhanced magnetocaloric effect during\nthe adiabatic demagnetizationandso on. Despite a certain\noversimplification, some exactly solved Ising–Heisenberg\nspin chains afford a plausible quantitative description ofthe magnetic behavior of real spin-chain materials [7,8,9,\n10,11,12,13].\nDuring the past decade, the natural mineral azurite\nCu3(CO3)2(OH)2has attracted a considerable research\ninterest, because it provides a long sought experimental\nrealization of the spin-1/2diamondchain with spectacular\nmagneticproperties[14,15,16,17].Owingtothisfact,alo t\nofattentionhasbeenpaidtoarigoroustreatmentofvarious\nversions of the Ising–Heisenberg diamond chain [18,19,\n20,21,22,23,24,25,26,27]. Although a correct descrip-\ntion of magnetic properties of the azurite would require\nmodeling based on a more complex spin-1/2 Heisenberg\nmodel [28,29], the simplified but still exactly tractable\nspin-1/2 Ising–Heisenberg diamond chain qualitatively\nreproduces the most prominent experimental features re-\nported for the azurite such as an intermediate one-third\nmagnetizationplateauaswellasthedouble-peaktempera-\nturedependencesofspecificheatandsusceptibility[14,15 ,\nCopyrightlinewillbe provided by the publisher2 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\n16,17]. Inaddition,it has beentheoreticallypredictedth at\nthe asymmetric spin-1/2 Ising–Heisenberg diamond chain\n[21] relevant to the azurite may display much richer mag-\nnetic behavior than its symmetric counterpart [18], since\ntheasymmetryinexchangeinteractionsalongthediamond\nsides may cause new peculiarities such as an existence of\nthe additional magnetization plateau at zero magnetiza-\ntion and/or unusual temperature dependence of zero-field\nspecificheatwiththreedistinctroundpeaks[21].\nTo provide consistent description of all experimental\ndata reported so far for the azurite (i.e. low-temperature\nmagnetization curve, INS and NMR data, magnetic sus-\nceptibility and specific heat [14,15,16,17]), the first-\nprinciplescalculationsbasedonthedensityfunctionalth e-\nory have been recently combined with the state-of-the-art\nnumericalcalculations[28,29].Thispowerfulcombinatio n\nof accurate methods has convincingly evidenced that the\nasymmetricspin-1/2Heisenbergdiamondchainsuggested\nin the early studies as a feasible model of the azurite [14,\n15,16,17] must be inevitably extended in order to include\nthe second-neighbor interaction between the nodal spins\nas well as non-negligible inter-chain interactions [28].\nHowever, a closer inspection of the exchange interactions\nin the azurite allows one to map a rather complex three-\ndimensionalHeisenbergmodeltotheasymmetricspin-1/2\nHeisenberg diamond chain with the second-neighbor in-\nteraction between the nodal spins under additional small\nrefinementofall exchangeconstants[29].\nBearing all this in mind, the main purpose of this\nwork is to examine magnetic properties of the asymmet-\nric spin-1/2 Ising–Heisenberg diamond chain refined by\nthe second-neighbor interaction between the nodal spins.\nIt will be evidenced that the second-neighbor interaction\nbetween the nodal spins may cause an emergence of the\nadditional intermediate plateau at two-thirds of the satu-\nration magnetization, the nature of which is completely\nthe same as recently proposed for the symmetric spin-\n1/2 Ising–Heisenberg diamond chain accounting for an\nextra four-spin interaction [25,26]. It is worthwhile to\nremark, moreover, that an investigation of the general-\nized spin-1/2 Ising–Heisenberg diamond chain with the\nsecond-neighborinteractionbetweenthe nodalspinsis in-\ntriguingbecausethismodelisisomorphicwiththespin-1/2\nIsing–Heisenberg doubly decorated chain [30,31] refined\nby the additional further-neighbor interactions as well\nas the spin-1/2 Ising–Heisenberg tetrahedral chain [32].\nConsequently,the exact results presented hereafter for th e\ngeneralizedspin-1/2Ising–Heisenbergdiamondchainwith\nthe second-neighbor interaction between the nodal spins\nhave some important implications also for the remarkable\nquantumantiferromagneticorderexperimentallyobserved\nin the copper-based tetrahedral chain Cu 3Mo2O9[33,34,\n35,36,37,38].\nThe organization of this paper is as follows. Section\n2 provides an exact solution of the generalized spin-\n1/2 Ising–Heisenberg diamond chain with the second-neighbor Ising interaction between the nodal spins and\nthe XYZ Heisenberg interaction between the interstitial\nspins in the spirit of the decoration-iteration mapping\ntransformation. The most interesting results obtained for\nthe ground-state phase diagrams are discussed together\nwith typical magnetic field and temperature dependences\nof the magnetization, susceptibility, and specific heat in\nSection 3. A few experimental results reported previ-\nously for solid-state representatives of the diamond spin\nchain Cu 3(CO3)2(OH)2and the tetrahedral spin chain\nCu3Mo2O9are also briefly qualitatively interpreted with\nthe help of the studied model in Section 4. Finally, some\nconclusionsandfutureoutlooksare drawninSection5.\n2 Ising–Heisenberg diamond chain Let us con-\nsider the generalized spin-1/2 Ising–Heisenberg diamond\nchain in a presenceofthe externalmagneticfield. A prim-\nitive cell of the diamond spin chain is determined by the\nnodeskandk+1asillustratedinFig.1.Allnodalsitesare\noccupiedbytheIsingspins µk=±1/2,whicharecoupled\nwith their neighbors solely through the Ising interactions .\nOnthecontrary,twointerstitialsites (k,1)and(k,2)from\nthek-th primitive cell are occupied by the Heisenberg\nspinsSk,1andSk,2, which are mutually coupled through\nthe spatially anisotropic XYZ Heisenberg interaction. For\nfurther convenience, the total Hamiltonian of the gener-\nalized spin-1/2 Ising–Heisenberg diamond chain can be\ndefinedasa sumovercell Hamiltonians ˆHk:\nˆH=N/summationdisplay\nk=1ˆHk, (1)\nwhere each cell Hamiltonian ˆHkinvolves all the interac-\ntiontermsbelongingto the k-thprimitivecell\nˆHk=J1ˆSx\nk,1ˆSx\nk,2+J2ˆSy\nk,1ˆSy\nk,2+J3ˆSz\nk,1ˆSz\nk,2+I3µkµk+1\n+µk/parenleftBig\nI1ˆSz\nk,1+I2ˆSz\nk,2/parenrightBig\n+µk+1/parenleftBig\nI2ˆSz\nk,1+I1ˆSz\nk,2/parenrightBig\n−hH/parenleftBig\nˆSz\nk,1+ˆSz\nk,2/parenrightBig\n−hI\n2(µk+µk+1),(2)\nIn above, ˆSα\nk,i(α=x,y,z;i= 1,2) denote spatial com-\nponentsofthespin-1/2operator,theparameters J1,J2and\nJ3determinethespatiallyanisotropicXYZinteractionbe-\ntween the nearest-neighbor Heisenberg spins, I1andI2\nlabel the Ising interactions between the nearest-neighbor\nIsing and Heisenberg spins residing the diamond sides, I3\nrepresents the second-neighbor Ising interaction between\nthe nodal spins, hIandhHare the magnetic fields acting\non the Ising and Heisenberg spins, respectively. It should\nbementionedthatafewparticularcasesoftheHamiltonian\n(2)havebeenextensivelystudiedinthepast.Theparticula r\ncaseI2=I3= 0(orI1=I3= 0)correspondstothespin-\n1/2 Ising–Heisenbergdoubly decorated chain [30,31], the\notherparticularcase I1=I2andI3= 0correspondstothe\nsymmetricspin-1/2Ising–Heisenbergdiamondchain[18],\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nµk µk+1I1 I2\nI2 I1Sk,1\nSk,2hI\nhHhH\nI3{J1, J2, J3}\nFigure 1 A fragment of the generalized spin-1/2 Ising–\nHeisenberg diamond chain. The Ising spins µkandµk+1\nat two nodal sites and the Heisenberg spins Sk,1andSk,2\nattwointerstitialsitesofthe k-thprimitivecellaremarked.\nwhilethemostsymmetricspecialcase I1=I2=I3corre-\nsponds to the spin-1/2 Ising–Heisenberg tetrahedral chain\n[32].\nLet us calculate the partition function of the general-\nized spin-1/2 Ising–Heisenberg diamond chain. With re-\ngard to a validity of commutative relation between differ-\nent cell Hamiltonians [ˆHi,ˆHj] = 0, the partition function\nZcanbepartiallyfactorizedintothefollowingproduct:\nZ=/summationdisplay\n{µk}N/productdisplay\nk=1TrSk,1,Sk,2exp/parenleftBig\n−βˆHk/parenrightBig\n,(3)\nwhereβ= 1/(kBT),kBisthe Boltzmann’sconstant, Tis\ntheabsolutetemperature,thesymbol/summationtext\n{µk}denotessum-\nmationovera completeset ofspinstates ofthe Isingspins\nandthe symbolTr Sk,1,Sk,2marksa traceoverspindegrees\noffreedomoftwoHeisenbergspinsfromthe k-thprimitive\ncell.To proceedfurtherwitha calculation,onenecessaril y\nneedsto evaluatetheeffectiveBoltzmann’sfactor\nZk(µk,µk+1) =TrSk,1,Sk,2exp/parenleftBig\n−βˆHk/parenrightBig\n,(4)\nwhich naturally appears behind the product symbol in the\nfactorized form (3) of the partition function. For this pur-\npose, it is quite advisable to pass to the matrix representa-\ntion of the cell Hamiltonian ˆHkin the basis spanned over\nfouravailablestatesoftwoHeisenbergspins Sz\nk,1andSz\nk,2:\n|↑,↑/an}bracketri}htk=|↑/an}bracketri}htk,1|↑/an}bracketri}htk,2,|↓,↓/an}bracketri}htk=|↓/an}bracketri}htk,1|↓/an}bracketri}htk,2,\n|↑,↓/an}bracketri}htk=|↑/an}bracketri}htk,1|↓/an}bracketri}htk,2,|↓,↑/an}bracketri}htk=|↓/an}bracketri}htk,1|↑/an}bracketri}htk,2,(5)\nwhereas|↑/an}bracketri}htk,iand|↓/an}bracketri}htk,idenote two eigenvectors of the\nspin operator ˆSz\nk,iwith the respective eigenvalues Sz\nk,i=\n1/2and−1/2. After a straightforward diagonalization of\nthe cell Hamiltonian ˆHkone obtains the following four\neigenvalues:\nE1,2=−hI\n2(µk+µk+1)+I3µkµk+1+J3\n4±Q+,\nE3,4=−hI\n2(µk+µk+1)+I3µkµk+1−J3\n4±Q−,(6)\nwhere\nQ±=1\n2/radicalBigg/parenleftbiggJ1∓J2\n2/parenrightbigg2\n+[(I1±I2)(µk±µk+1)−hH∓hH]2.Now, one may simply use the eigenvalues (6) in order to\ncalculate the Boltzmann’s factor (4) according to the re-\nlationZk(µk,µk+1) =4/summationtext\ni=1e−βEi. The explicit form of\nthe relevant Boltzmann’s factor can be subsequently re-\nplaced through the generalized decoration-iteration tran s-\nformation[1,2,3,4]:\nZk(µk,µk+1)=2exp/bracketleftBig\nβhI\n2(µk+µk+1)−βI3µkµk+1/bracketrightBig\n×/bracketleftBig\nexp/parenleftBig\n−βJ3\n4/parenrightBig\ncosh(βQ+)+exp/parenleftBig\nβJ3\n4/parenrightBig\ncosh(βQ−)/bracketrightBig\n=Aexp/bracketleftBig\nβRµkµk+1+βH0\n2(µk+µk+1)/bracketrightBig\n.(7)\nThe mapping parameters A,R, andH0are unambigu-\nously determined by a ’self-consistency’ condition of the\ndecoration-iteration transformation (7), which requires a\nvalidity of the mapping transformation independently of\nthe spin states of two Ising spins µkandµk+1involved\ntherein. The decoration-iteration mapping transformatio n\n(7) satisfies the’self-consistency’conditionif andonlyi f\nA=/bracketleftbigg\nZk/parenleftbigg1\n2,1\n2/parenrightbigg\nZk/parenleftbigg\n−1\n2,−1\n2/parenrightbigg\nZ2\nk/parenleftbigg1\n2,−1\n2/parenrightbigg/bracketrightbigg1/4\n,\nβR= ln/bracketleftBigg\nZk/parenleftbig1\n2,1\n2/parenrightbig\nZk/parenleftbig\n−1\n2,−1\n2/parenrightbig\nZ2\nk/parenleftbig1\n2,−1\n2/parenrightbig/bracketrightBigg\n,\nβH0= ln/bracketleftBigg\nZk/parenleftbig1\n2,1\n2/parenrightbig\nZk/parenleftbig\n−1\n2,−1\n2/parenrightbig/bracketrightBigg\n. (8)\nAn important consequence directly follows from the ex-\nplicit formulasfor the mapping parameters(8), which im-\nply that the effective coupling βRis the only mapping\nparameter that depends on the second-neighbor interac-\ntionI3between the nodal spins through a trivial shift\nβR=βR|I3=0−βI3while the other two mapping pa-\nrametersAandβH0aretotallyindependentof I3.\nByinsertingthedecoration-iterationtransformation(7)\ninto Eq. (3) one readily gets a rigorous mapping rela-\ntion between the partition function Zof the generalized\nspin-1/2Ising–Heisenbergdiamondchainandthepartition\nfunctionZ0of the simple spin-1/2 Ising chain with the\neffective nearest-neighbor interaction Rand the effective\nmagneticfield H0:\nZ=ANZ0(β,R,H 0). (9)\nItisworthytorecallthatthespin-1/2Isingchaininamag-\nnetic field is well-known exactly tractable model, which\ncan be solved for instance through the classical transfer-\nmatrix method [39]. The partition function of the spin-1/2\nIsing chain in a magnetic field is determined by a sum of\ntwo eigenvalues of the transfer matrix Z0=λN\n++λN\n−,\nwhich are for the sake of completeness given by the fol-\nlowingexpressions:\nλ±= eβR\n4/bracketleftBigg\ncosh/parenleftbiggβH0\n2/parenrightbigg\n±/radicalBigg\nsinh2/parenleftbiggβH0\n2/parenrightbigg\n+e−βR/bracketrightBigg\n.\nCopyrightlinewillbe provided by the publisher4 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\nFrom this point of view, the exact calculation of the\npartition function Zof the generalized spin-1/2 Ising–\nHeisenbergdiamondchainwiththesecond-neighborinter-\nactionbetweenthenodalspinsisformallycompleted.\nExact results for other thermodynamic quantities fol-\nlow quite straightforwardly from the mapping relation (9)\nbetween the partition functions. In the thermodynamic\nlimitN→∞, the free energy per unit cell can be evalu-\natedfromtheformula\nf= lim\nN→∞−1\nNβ−1lnZ=−1\nβ(lnA+lnλ+),(10)\nwhich also allows a straightforward calculation of the en-\ntropysandtheheatcapacity cperunitcell:\ns=kBβ2∂f\n∂β, c=−β∂s\n∂β.(11)\nThe single-site magnetization of the Ising spins mI≡\n/an}bracketle{tµk+µk+1/an}bracketri}ht/2and the single-site magnetization of the\nHeisenbergspins mH≡/angbracketleftBig\nˆSz\nk,1+ˆSz\nk,2/angbracketrightBig\n/2canbeobtained\nby a differentiation of the free energy (10) with respect to\ntheparticularmagneticfields:\nmI=−∂f\n∂hI, m H=−1\n2∂f\n∂hH.(12)\nOnce evaluated, the total magnetization can be expressed\nintermsofbothsingle-sitemagnetizationsthroughthefor -\nmulam= (mI+2mH)/3.\n3 ResultsandDiscussion Inthissection,letusdis-\ncuss the most interesting results obtained for the gener-\nalized spin-1/2 Ising–Heisenberg diamond chain with the\nsecond-neighbor interaction between the nodal spins. For\nsimplicity, our further attention will be focused just on\nthe particular case with the antiferromagnetic Ising in-\nteractions I1,I2,I3>0and the antiferromagnetic XXZ\nHeisenberginteraction J1=J2=J∆,J3=J >0. The\nmain motivation for a detailed study of the antiferromag-\nneticspin-1/2Ising–Heisenbergdiamondchainliesin that\nthis special case should exhibit the most obviousmanifes-\ntations of a mutual interplay between the geometric spin\nfrustration and local quantum fluctuations. Note further-\nmorethatthedimensionlessparameter ∆determinesaspa-\ntial anisotropy in the XXZ Heisenberg interaction and the\nspecialcaseof ∆= 1correspondstotheisotropicHeisen-\nberg coupling between the nearest-neighboringinterstiti al\nspins. To further reducethe numberof free interactionpa-\nrameters,we willalsoassumeequalmagneticfieldsacting\non the Ising and Heisenberg spins h=hI=hHwhat\nphysicallycorrespondstosettingg-factorsoftheIsingan d\nHeisenberg spins equal one to each other. Another impor-\ntant observation can be made from the Hamiltonian (2) of\nthe generalized spin-1/2 Ising–Heisenberg diamond chain\nthatisinvariantwithrespecttotheinter-change I1←→I2under simultaneous re-numbering of the interstitial sites\nk,1←→k,2and hence, one may also consider I1≥I2\nwithout loss of generality.This fact allows us to introduce\na difference between two Ising interactions along the dia-\nmondsides δI=I1−I2≥0andtousethestrongeramong\ntwo Isinginteractionsastheenergyunitwhendefiningthe\nfollowingset ofdimensionlessinteractionparameters:\n˜J=J\nI1, δ˜I=δI\nI1,˜I3=I3\nI1,˜h=h\nI1.(13)\nThereducedinteractionparametersgivenbyEq.(13)mea-\nsure a relative strength of the Heisenberg interaction, the\nasymmetry of two Ising interactions along the diamond\nsides, the second-neighbor interaction between the nodal\nspins, and the external magnetic field, all normalizedwith\nrespect to the strongerIsing interaction ( I1) along the dia-\nmondsides. Itisquiteevidentthat theaccessiblevaluesof\nthe parameter δ˜I, whose physical sense lies in the degree\nof asymmetryof two Ising interactionsalong the diamond\nsides, arethenrestrictedto theinterval δ˜I∈[0,1].\nFirst,letusexaminethegroundstateofthegeneralized\nspin-1/2 Ising–Heisenberg diamond chain. The ground\nstatecanbetriviallyconnectedtothelowest-energyeigen -\nstate of the cell Hamiltonian (6) obtained by taking into\naccount all four states of two nodal Ising spins µkand\nµk+1thatenterintotherespectiveeigenvalues.Depending\non a mutual competition between the interaction param-\neters˜J,∆,δ˜I,˜I3and˜hone finds in total five different\nground states: the saturated paramagnetic state SPA, two\nclassical ferrimagneticstates FRI 1and FRI 2, the quantum\nferrimagneticstateQFIandthequantumantiferromagnetic\nstate QAF givenbytheeigenvectors\n|SPA/an}bracketri}ht=N/productdisplay\nk=1|+/an}bracketri}htk|↑,↑/an}bracketri}htk,\n|FRI1/an}bracketri}ht=N/productdisplay\nk=1|−/an}bracketri}htk|↑,↑/an}bracketri}htk,\n|QFI/an}bracketri}ht=N/productdisplay\nk=1|+/an}bracketri}htk1√\n2/bracketleftbig\n|↑,↓/an}bracketri}htk−|↓,↑/an}bracketri}htk/bracketrightbig\n,\n|QAF/an}bracketri}ht=\n\nN/producttext\nk=1/vextendsingle/vextendsingle[−]k/angbracketrightbig\nk/bracketleftbig\nA[−]k|↑,↓/an}bracketri}htk−A[−]k+1|↓,↑/an}bracketri}htk/bracketrightbig\nN/producttext\nk=1/vextendsingle/vextendsingle[−]k+1/angbracketrightbig\nk/bracketleftbig\nA[−]k+1|↑,↓/an}bracketri}htk−A[−]k|↓,↑/an}bracketri}htk/bracketrightbig,\n|FRI2/an}bracketri}ht=\n\nN/producttext\nk=1/vextendsingle/vextendsingle[−]k/angbracketrightbig\nk|↑,↑/an}bracketri}htk\nN/producttext\nk=1/vextendsingle/vextendsingle[−]k+1/angbracketrightbig\nk|↑,↑/an}bracketri}htk. (14)\nIn above, the ket vector |±/an}bracketri}htkdetermines the state of the\nnodal Ising spin µk=±1/2, the symbol [−]k∈{−,+}\nmarks the sign of the number (−1)k, the spin states rele-\nvant to two Heisenberg spins from the kth primitive cell\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5\nare determinedby the notation(5) and the probabilityam-\nplitudesA±areexplicitlygivenbythe expressions:\nA±=1√\n2/radicaltp/radicalvertex/radicalvertex/radicalbt1∓δ˜I/radicalBig\n(δ˜I)2+(˜J∆)2.(15)\nTheeigenenergiesperprimitivecell that correspondto the\nrespectivegroundstates(14) aregivenasfollows:\n˜ESPA=˜J\n4+1−δ˜I\n2+˜I3\n4−3˜h\n2,\n˜EFRI1=˜J\n4−1+δ˜I\n2+˜I3\n4−˜h\n2,\n˜EQFI=−˜J\n4−˜J∆\n2+˜I3\n4−˜h\n2,\n˜EQAF=−˜J\n4−1\n2/radicalBig\n(δ˜I)2+(˜J∆)2−˜I3\n4,\n˜EFRI2=˜J\n4−˜I3\n4−˜h. (16)\nLet us shortly comment on respective spin arrange-\nment inherent to the ground states (14). At high magnetic\nfields,theantiferromagneticspin-1/2Ising–Heisenbergd i-\namond chain naturally ends up at SPA ground state with\nall nodal Ising and interstitial Heisenberg spins fully po-\nlarized by the external magnetic field. Contrary to this,\nthe ground-state spin alignment is much more diverse at\nlower magnetic fields when either one of three ferrimag-\nnetic ground states (FRI 1, FRI2or QFI) or the unique\nquantum antiferromagnetic ground state QAF is realized.\nThegroundstateFRI 1correspondstoaclassicalferrimag-\nneticspinarrangement,inwhichallinterstitialHeisenbe rg\nspinsarefullyalignedwiththemagneticfieldandallnodal\nIsing spins point in an opposite direction due to the anti-\nferromagnetic coupling with their nearest-neighbor inter -\nstitial spins. However, it is energetically more favorable\nfor the Heisenberg spin pairs to form the singlet-dimer\nstateprovidedthattheantiferromagneticcouplingbetwee n\nthe Heisenberg spins is strong enough. Under this condi-\ntion, the antiferromagnetic spin-1/2 Ising–Heisenberg di -\namond chain rests in the quantum ferrimagnetic ground\nstate QFI with a character of the dimer-monomer state,\nbecause the nodal Ising spins tend to align with the mag-\nnetic field on behalf of a spin frustration that effectively\nswitchesoffthecouplingbetweenthenearest-neighborin-\nterstitial and nodal spins. The second-neighbor coupling\nbetween the nodal Ising spins may additionally cause the\nantiferromagneticalignmentofthenodalIsingspinsatlow\nenough magnetic fields, which consequently leads to the\nuniquequantum antiferromagneticgroundstate QAF. The\nmost striking feature of QAF is that the antiferromagnetic\nalignment of the nodal Ising spins is surprisingly trans-\nferred to a quantum superposition of two intrinsically an-\ntiferromagnetic states ( |↑,↓/an}bracketri}htkand|↓,↑/an}bracketri}htk) of the Heisen-\nbergspinpairs,whichfallintoaperfectsinglet-dimersta tejust for the symmetric diamond chain δ˜I= 0while any\nasymmetry δ˜I/ne}ationslash= 0causes according to Eqs. (14)-(15) the\nspin-singlet-like state with a non-zero staggered magneti -\nzation on the Heisenberg spin pairs. The second-neighbor\ninteraction between the nodal Ising spins may be also re-\nsponsible for an appearance of another classical ferrimag-\nnetic ground state FRI 2with translationally broken sym-\nmetry, which cannot be in principle found in the spin-1/2\nIsing–Heisenberg diamond chain without this interaction\nterm [21]. The ground state FRI 2can be characterized by\nafullalignmentofallinterstitialHeisenbergspinswitht he\nmagneticfield,whereastheantiferromagneticarrangement\nof the nodal Ising spins arises from the antiferromagnetic\nsecond-neighborcoupling ˜I3in betweenthem.\nNow, let us proceed to a detailed analysis of the\nground-state phase diagram. The ground-state phase di-\nagram in the δ˜I−˜hplane in an absence of the second-\nneighbor interaction between the nodal spins might have\nthree different topologies depending on a size of the\nparameter ˜J(1 +∆)[21]: the topology of type 1 for\n˜J(1 +∆)≤1shown in Fig. 2(a1), the topology of type\n2 for1<˜J(1 +∆)<2displayed in Fig. 2(b1), and\nthe topology of type 3 for ˜J(1 +∆)≥2illustrated in\nFig. 2(c1). The relevant ground-state boundaries for the\nspecial case of ˜I3= 0are shown in Fig. 2(a1),(b1),(c1)\nby dotted lines for the illustrative case of the isotropic\nHeisenberg interaction ( ∆= 1). In what follows, we\nwill concentrate our attention only to the influence of the\nsecond-neighbor interaction ˜I3on the topology of the re-\nspective ground-state phase diagrams, whereas the reader\ninterested in more details concerned with the special case\n˜I3= 0isreferredtoRef.[21].\nConsider first the changes in the ground-state phase\ndiagram of type 1 invoked by the strengthening of the\nsecond-neighbor interaction ˜I3. It is quite obvious from\nFig. 2(a1) that the direct field-induced transition between\nthe FRI 1and SPA phases observable for the special case\n˜I3= 0along the line ˜h= 2−δ˜Ivanishes on account\nof a presenceof the band-likeregionpertinentto the FRI 2\nphase. A cross-section of the band-like region in parallel\nto the field axis ˜hequals to 2˜I3, which means that the\nfield rangeinherentto the FRI 2phasebecomesthe greater\nthe stronger the second-neighborinteraction ˜I3is. At zero\nmagnetic field, the FRI 1phase is replaced with the QAF\nphaseabovetheboundaryvalue\nδ˜I=2−˜J−˜I3\n2−(˜J∆)2\n2/parenleftBig\n2−˜J−˜I3/parenrightBig, (17)\nwhichmonotonicallydecreaseswithincreasingthesecond-\nneighborinteraction ˜I3untilitreacheszeroatthethreshold\nvalue˜I3= 2−˜J(1 +∆)[see Fig. 2(a2)]. If the strength\nofthesecond-neighborinteraction ˜I3isfromtheinterval\n1−1\n2/parenleftbigg\n˜J+/radicalBig\n(˜J∆)2+1/parenrightbigg\n≤˜I3≤2−1\n2˜J(1+∆),(18)\nCopyrightlinewillbe provided by the publisher6 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\n0    0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 \n0    0.2 0.4 0.6 0.8 1.0 123\n0    0.2 0.4 0.6 0.8 1.0 123\n0    0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 \n0    0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 \n0    0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 \n0    0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 \n0    0.2 0.4 0.6 0.8 1.0 123\n0    0.2 0.4 0.6 0.8 1.0 123 I3/I1 = 0.19098     J/I1 = 0.5 \nFRI 2\n I3/I1 = 0.1 a1 \nQAF FRI 1SPA h / I1  \nδI / I1  I3/I1 = 1.0 J/I1 = 0.5 \nFRI 2a2 \nQAF SPA h / I1  \nδI / I1 \n I3/I1 = 0.5    I3/I1 = 1.5   J/I1 = 0.5 \nFRI 2a3 \nQAF FRI 1SPA h / I1  \nδI / I1 \n I3/I1 = 0.25       \nQFI  I3/I1 = 0.1 FRI 2J/I1 = 0.75 b1 \nQAF FRI 1SPA h / I1 \nδI / I1 QFI     \n I3/I1 = 0.35 J/I1 = 0.75 \nFRI 2b2 \nQAF FRI 1QFI SPA h / I1 \nδI / I1 I3/I1 = 0.55     \nI3/I1 = 0.6743 b3 J/I1 = 0.75 \nFRI 2\nQAF FRI 1SPA h / I1 \nδI / I1 \n  I3/I1 = 0.25 \n  I3/I1 = 0.5 J/I1 = 1.0 c1 h / I1 FRI 2\nδI / I1 QAF QFI SPA  \n QFI  I3/I1 = 0.7929 J/I1 = 1.0 c2 h / I1 FRI 2\nδI / I1 QAF SPA  \n QFI  I3/I1 = 0.9 J/I1 = 1.0 c3 h / I1 FRI 2\nδI / I1 QAF SPA  \n \nFigure2 Ground-statephasediagramsinthe δ˜I−˜hplaneconstructedbyconsideringseveralvaluesoftherela tivestrength\nofthesecond-neighborinteraction ˜I3andthreedifferentrelativestrengthsoftheisotropicHei senberginteraction( ∆= 1):\n(a)˜J= 0.5, (b)˜J= 0.75, and (c) ˜J= 1.0. The dotted lines shown in Fig. 2(a1),(b1),(c1)correspond to the special case\nwithoutthesecond-neighborcoupling ˜I3= 0.\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 7\nthen, the ground-state phase diagram contains a special\ntriplepointwiththe coordinates:\nδ˜I=1\n3/bracketleftBigg\n2/parenleftBig\n4−2˜I3−˜J/parenrightBig\n−/radicalbigg\n3(˜J∆)2+/parenleftBig\n4−2˜I3−˜J/parenrightBig2/bracketrightBigg\n,\n˜h=1\n3/bracketleftBigg\n2/parenleftBig\n˜J−1/parenrightBig\n+˜I3+/radicalbigg\n3(˜J∆)2+/parenleftBig\n4−2˜I3−˜J/parenrightBig2/bracketrightBigg\n,\n(19)\nat which the FRI 1, QAF and FRI 2phases coexist together\n[see Fig. 2(a2)]. The coexistence point of the FRI 1, QAF\nand FRI 2phases graduallymovestowards lower valuesof\nthe asymmetry parameter δ˜Iwith increasing the second-\nneighborinteraction ˜I3(alongtheimaginarypartoftransi-\ntionlinebetweentheQAFandFRI 2phases)untilitreaches\nthe symmetric point δ˜I= 0for˜I3= 2−˜J(1 +∆)/2.\nHerewith the FRI 1phase completely disappears from the\nground-statephasediagramasitisillustratedinFig.2(a3 ),\nwhereasa further increase in the second-neighborinterac-\ntion˜I3onlyextendstheareapertinenttotheFRI 2phasebut\nit does not qualitatively change the topology of the phase\ndiagram.\nThe effect of the second-neighbor interaction ˜I3upon\nthe ground-state phase diagram of type 2 is quite similar\nas in the previous case, but the relevant phase diagram is\nin general much more complicated due to a presence of\nthe QFI phase residing a parameter space with a rather\nhigh asymmetry of two Ising interactions along the dia-\nmondsides.Thesecond-neighborinteraction ˜I3repeatedly\ngivesrisetotheband-likeregioncorrespondingtotheFRI 2\nphase, which emerges instead of the direct field-induced\ntransitionbetweentheFRI 1andSPAphasesunlikethespe-\ncial case ˜I3= 0[see Fig. 2(b1)]. In addition, the param-\neter region inherent to the FRI 2phase wedges in between\nthe SPA and QFI phases, whereas the apex of this wedge\nforms the triple point that determines a coexistence of the\nSPA, QFIandFRI 2phasesat [Fig.2(b1)]\nδ˜I=2−˜J(1+∆)+2˜I3,\n˜h=˜J(1+∆)−˜I3. (20)\nThis triple point is shifted towards higher values of the\nasymmetry parameter δ˜Iwith increasing of the second-\nneighbor interaction ˜I3until it completely vanishes from\nthe phase diagram for ˜I3>[˜J(1 +∆)−1]/2[see\nFig. 2(b2)]. Besides, two phase boundaries between the\nFRI2-QFIandQFI-QAFphasesaregraduallyapproaching\neach other upon further increase of the second-neighbor\ncoupling ˜I3until both transition lines meet at a new triple\npointwhenever\n˜I3≥˜J(1+∆)−1\n2/parenleftBigg\n˜J+/radicalbigg/parenleftBig\n˜J∆/parenrightBig2\n+1/parenrightBigg\n. (21)Apparently,theaforementionedtriplepointdefinesacoex-\nistence ofthe FRI 2, QFIandQAF phasesgivenby\nδ˜I=/radicalbigg/parenleftBig\n˜J+2˜J∆−2˜I3/parenrightBig2\n−(˜J∆)2,\n˜h=˜J(1+∆)−˜I3, (22)\nwhich can be clearly seen in Fig. 2(b2)-(b3).The locus of\nthe last triple point movesto lower values of the asymme-\ntry parameter δ˜Iwith increasing the second-neighbor in-\nteraction ˜I3(alongtheimaginarypartofthetransitionline\nbetween the FRI 2and QAF phases), which consequently\nreducestheparameterregioninherenttotheQFIphase[see\nFig. 2(b3)].Ifthesecond-neighborinteractionequalsto\n˜I3=˜J/parenleftbigg1\n2+∆/parenrightbigg\n−1\n2/radicalbigg/parenleftBig\n˜J∆/parenrightBig2\n+/bracketleftBig\n˜J(1+∆)−2/bracketrightBig2\n,(23)\nthen, all three aforedescribed triple points determining a\ncoexistence of the FRI 1-FRI2-QFI, FRI 1-QFI-QAF, and\nFRI2-QFI-QAF phases merge together owing to a com-\nplete disappearance of the QFI phase from the ground-\nstate phase diagram as displayed in Fig. 2(b3). As a re-\nsult, the phase diagram gains for stronger values of the\nsecond-neighbor interaction ˜I3the same topology as de-\nscribed previously in Fig. 2(a2) with only one triple point\ndetermining a phase coexistence between the FRI 1, FRI2\nand QAF phases [cf. Fig. 2(b2) with Fig. 2(a2)]. If the\nsecond-neighbor interaction exceeds the threshold value\n˜I3>2−˜J(1 +∆)/2, the triple point corresponding to\nthe phase coexistence between the FRI 1, FRI2and QAF\nphases vanishes and one recovers qualitatively the same\nphasediagramasillustratedinFig.2(a3).\nLast, let us comment on changes in the ground-state\nphasediagramoftype3causedbythesecond-neighborin-\nteraction as shown in the lower panel of Fig. 2. The most\nfundamentaldifferenceis that the FRI 2phase does not in-\nstantaneouslyappearintherelevantground-statephasedi -\nagramuponrisingthesecond-neighborinteraction ˜I3from\nzeroincontrasttotheprevioustwocases.Indeed,theFRI 2\nphase emerges first in between the QFI and SPA phases\njust if the second-neighborinteraction is stronger than th e\nboundary value ˜I3≥[˜J(1 +∆)−2]/2. The parameter\nregion inherent to the FRI 2phase is then delimited by the\nsymmetric point δ˜I= 0and the triple point (20) deter-\nmining a coexistence of the SPA, QFI and FRI 2phases.\nThe triple point of the phase coexistence SPA-QFI-FRI 2\nis shifted towards higher values of the asymmetry param-\neterδ˜Iupon strengthening of the second-neighbor inter-\naction˜I3until it completely vanishes from the phase dia-\ngramfor ˜I3>[˜J(1+∆)−1]/2[seeFig.2(c1)].Thenext\ntriplepointdeterminingacoexistenceoftheFRI 2,QFIand\nQAF phases occurs whenever the second-neighbor cou-\npling satisfies the condition(21), whereasthe locusof this\ntriple point given by Eq. (22) gradually moves towards\nlower values of the asymmetry parameter δ˜I(along the\nimaginarypart of the transition line between the FRI 2and\nCopyrightlinewillbe provided by the publisher8 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\nQAF phases) as the second-neighbor interaction further\nstrengthens[seeFig.2(c2)-(c3)].Theaforedescribedtri ple\npoint cannot be found in the ground-state phase diagram\nfor˜I3>˜J(1+∆)/2due to a complete disappearance of\nthe QFI phase and the phase diagram finally recovers the\nsametopologyasdiscussedpreviouslyforFig.2(a3).\nBeforeproceedingtoadiscussionoffinite-temperature\nproperties, it is worth mentioning that the total magneti-\nzation of two ferrimagnetic ground states FRI 1and QFI\nequalsto one-thirdof the saturationmagnetizationin con-\ntrast to the total magnetization of the other ferrimagnetic\nground state FRI 2being equal to two-thirds of the satura-\ntionmagnetization.Forthisreason,threeremarkableferr i-\nmagneticgroundstatesshouldmanifestthemselvesinlow-\ntemperaturemagnetizationcurvesasintermediateplateau x\nat one-third and/or two-thirds of the saturation magneti-\nzation. Let us consider first the field dependence of the\ntotal magnetization normalized with respect to the satura-\ntion magnetization as depicted in Figs. 3(a) and 4(a) for\na few temperatures and the isotropic Heisenberg coupling\n(∆= 1)oftherelativestrength ˜J= 0.75.Thedottedlines\nshow the magnetization curves for the special case with-\nout the second-neighbor interaction ( ˜I3= 0), while the\nsolidlinesdisplaytherelevantchangeinthemagnetizatio n\ncurvesachieveduponswitchingonthesecond-neighborin-\nteractionofmoderatestrength ˜I3= 0.35.Themostcrucial\nchange in the low-temperature magnetization curves due\nto the non-zero second-neighbor interaction ˜I3definitely\nrepresents the novel two-thirds intermediate plateau con-\nnected with the ground state FRI 2. In fact, the two-thirds\nmagnetization plateau may emerge both for low as well\nas high value of the asymmetry parameter as depicted in\nFigs. 3(a) and 4(a) for two specific cases δ˜I= 0.1and\n0.7, respectively, while the two-thirds plateau cannot be\nbasically found in the magnetization process of the spin-\n1/2 Ising–Heisenberg diamond chain without the second-\nneighbor interaction [21]. Moreover, it is quite obvious\nfrom Figs. 3(a) and 4(a) that the asymmetry parameter δ˜I\nplays an essential role whether or not the magnetization\ncurve might have plateau at zero magnetization, because\nthe asymmetry parameter generally favors the QAF phase\nwith a zero total magnetizationbeforeenteringto the one-\nthirdplateauFRI 1phase.Therisingtemperaturegenerally\nsmoothens the stepwise magnetization curves observable\natlowenoughtemperaturesuntiltheintermediateplateaux\ncompletelydisappearfromthe magnetizationcurves.\nNext, let us turn our attention to temperature depen-\ndences of the total magnetization shown in Fig. 3(b) and\n4(b). The following general trends can be deduced from\nthe displayed thermal variations of the total magnetiza-\ntion. The magnetization exhibits the marked temperature-\ninduced changes whenever the external magnetic field is\nsufficiently close to critical fields determining a phase co-\nexistence between two different ground states, whereas\nthe vigorous thermally-induced increase (decrease) of the\ntotal magnetization is observable for the magnetic fieldsslightly below (above) the respective critical fields. Con-\ntrary to this, the magnetization falls down rather steadily\nwiththerisingtemperatureifthemagneticfieldisselected\nfromthemiddlepartofthemagnetizationplateauorabove\nthe saturation field. It is noteworthy that the monotonous\ndecrease of the total magnetization with increasing tem-\nperature can be also found exactly at critical fields rele-\nvant to a phase coexistence of two different groundstates,\nwhich are denoted in Fig. 3(b) and 4(b) by triangle sym-\nbols. Underthis condition,the total magnetizationasymp-\ntotically reachesnon-trivialvaluesas temperaturetends to\nzero, namely, 1/(3√\n5)≈0.1491for a coexistence point\noftheQAF-QFIphases, (2√\n5−1)/(3√\n5)≈0.5176fora\ncoexistence point of the FRI 1-FRI2and QFI-FRI 2phases,\n(2√\n5+1)/(3√\n5)≈0.7157fora coexistencepointofthe\nFRI2-SPA phases.\nThe temperature variation of the zero-field suscepti-\nbility times temperature product is depicted in Fig. 5 for\nthe particular case of the isotropic Heisenberg coupling\n(∆= 1,˜J= 0.75) and the moderate strength of the\nsecond-neighbor interaction ˜I3= 0.35. If the asymme-\ntry parameter δ˜I <0.1375is small enough in order to\nestablish the ferrimagnetic ground-state FRI 1, the suscep-\ntibility times temperature product exhibits a striking non -\nmonotonous dependence upon lowering temperature with\naflatminimumprecedinglow-temperaturedivergencethat\nis quite typical for ferrimagnets [40]. On the other hand,\nthe susceptibility times temperature product shows for\nhigher values of the asymmetry parameter δ˜I >0.1375\na monotonous thermal dependence when it asymptoti-\ncally tends to zero with decreasing temperature owing\nto the quantum antiferromagnetic ground state QAF. The\nstronger the antiferromagnetic second-neighbor interac-\ntion˜I3is, the less pronounced the temperature-induced\nincreaseof χTproductcanbeobserved.\nFinally, let us examine in detail temperaturevariations\nof the zero-field specific heat. For this purpose, typical\ntemperaturedependencesofthezero-fieldspecificheatare\nplotted in Fig. 6 for the particular case of the isotropic\nHeisenberg coupling ( ∆= 1,˜J= 0.75), the second-\nneighbor interaction ˜I3= 0.35and several values of\nthe asymmetry parameter δ˜I. It can be clearly seen from\nFig. 6(a) that the round maximum observable at higher\ntemperatures gradually decreases in height with increas-\ning the asymmetry parameter δ˜Ias far as the FRI 1phase\nconstitutes the ground state. Moreover, there also appears\nthe additional Schottky-type peak at lower temperatures,\nwhichisshiftedtowardslowertemperaturesuponstrength-\nening of the asymmetry parameter δ˜I. The special case\nδ˜I= 0.1375corresponds to a phase coexistence between\nthe FRI 1and QAF phases, which is characterized through\nthe notable thermal dependenceof the heat capacity with-\nout the low-temperature peak but with a shoulder super-\nimposed on ascending part of the round high-temperature\nmaximum [see thick lines displayed in Fig. 6(a) and (b)].\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 9\n0    0.5 1.0 1.5 2.0 2.5 0.2 0.4 0.6 0.8 1.0 \n0    0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 \nFRI 1 → FRI 2 → SPA 0.05 a\n0.17 0.1 kBT/I1 = 0 \nm / ms \n \n \nh / I1 kBT / I1 m / ms  \n 2.7 \n2.4 \n1.6 2.25 \n1.55 b\n0.05 0.3 0.9 1.5 1.9 2.2 h/I1 = 3.5 \nδI/I1 = 0.1 δI/I1 = 0.1 \n \n \nFigure 3 (a) The total magnetizationas a function of the magnetic fiel d at a few different temperaturesfor the particular\ncase of the Heisenberg interaction ∆= 1and˜J= 0.75by assuming one fixed value of the asymmetry parameter\nδ˜I= 0.1. The dotted lines correspondto the special case without the second-neighborinteraction ˜I3= 0, the solid lines\ntotheparticularcasewiththesecond-neighborinteractio n˜I3= 0.35ofamoderatestrength.(b)Thermalvariationsofthe\ntotal magnetization for the particular case of the Heisenbe rg interaction ∆= 1and˜J= 0.75, the asymmetry parameter\nδ˜I= 0.1and the second-neighbor interaction ˜I3= 0.35at several values of the magnetic field. The triangle symbols\ndenotecriticalfieldsatwhichtwodifferentgroundstatesc oexisttogether.\nIt worthwhile to remark that the double-peak temperature\ndependence of the specific heat is recovered if the asym-\nmetry parameter δ˜Iis strong enoughto establish the QAF\nground state [see Fig. 6(b)]. In opposite to the previous\ncase, the round high-temperature maximum increases in\nheight with increasing the asymmetry parameter δ˜Iand\nthe low-temperature Schottky-type peak shifts towards\nhigher temperatures until a complete coalescence of the\nlow- and high-temperature peaks is achieved. The most\nspectacular thermal dependence of the heat capacity with\nthreedistinctroundpeakscanbedetectedfortheasymme-\ntry parameter close to δ˜I≈0.2when a mutual overlapof\nthe low- and high-temperaturepeaks gives rise to a subtle\nintermediate (third) maximum significantly supported by\nthe shoulder superimposed ascending part of the round\nhigh-temperature maximum [see the curve δ˜I= 0.2in\nFig. 6(b)]. It is quite apparent that the sharpest peak ob-\nservableatthelowesttemperaturecanbealwaysattributed\nto thermal excitations from the FRI 1phase towards the\nQAFphaseorviceversa.\n4 Experimental implications In this section, let us\ndraw a few implications for experimental representatives\nof the diamond spin chain Cu 3(CO3)2(OH)2(azurite)and the tetrahedral spin chain Cu 3Mo2O9on the basis\nof the exactly solved spin-1/2 Ising–Heisenberg diamond\nchain with the second-neighbor interaction between the\nnodal spins. It has been argued in Refs. [28,29] that the\nasymmetric spin-1/2 Heisenberg diamond chain with the\nsecond-neighbor interaction between the nodal spins pro-\nvidesacomprehensivedescriptionofallexperimentaldata\nreportedyetfortheazurite.Althoughthegeneralizedspin -\n1/2 Ising–Heisenberg diamond chain surely represents\na considerable simplification of the analogous spin-1/2\nHeisenbergdiamondchain,it might be quiteinterestingto\nascertain to what extent it explains the most pronounced\nfeatures of the azurite because this simplified model still\ncorrectly reproduces the strongest Heisenberg interactio n\nbetween the nearest-neighbor interstitial spins. Accordi ng\nto Refs. [28,29], the asymmetric spin-1/2 Heisenberg dia-\nmond chain with the second-neighborinteractionbetween\nthe nodalspins quantitativelyreproducesthe experimenta l\ndata of the azurite by assuming the following specific val-\nues of the exchange constants (see Fig. 1 for the notation\nused):J/kB= 33K,I1/kB= 15.5K,I2/kB= 6.9K,\nI3/kB= 4.6K and the gyromagneticratio g = 2.06. This\nresult actually implies that the Heisenberg coupling be-\ntween the nearest-neighbor interstitial spins is by far the\nmostdominantexchangeinteraction.\nCopyrightlinewillbe provided by the publisher10 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\n0    0.5 1.0 1.5 2.0 2.5 0.2 0.4 0.6 0.8 1.0 \n0    0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 \nQAF → QFI → FRI 2 → SPA 0.02 \n0.1 0.05 kBT/I1 = 0 \nδI/I1 = 0.7 am / ms \nm / ms \nkBT / I1 h / I1  \n 2.5 \n1.6 \n1.2 \n0.9 \n0.626 1.15 1.65 \n0.2 0.5 0.7 1.1 1.4 1.8 h/I1 = 3.5 \nδI/I1 = 0.7 b \n \nFigure 4 (a) The total magnetizationas a function of the magnetic fiel d at a few different temperaturesfor the particular\ncase of the Heisenberg interaction ∆= 1and˜J= 0.75by assuming one fixed value of the asymmetry parameter\nδ˜I= 0.7. The dotted lines correspondto the special case without the second-neighborinteraction ˜I3= 0, the solid lines\ntotheparticularcasewiththesecond-neighborinteractio n˜I3= 0.35ofamoderatestrength.(b)Thermalvariationsofthe\ntotal magnetization for the particular case of the Heisenbe rg interaction ∆= 1and˜J= 0.75, the asymmetry parameter\nδ˜I= 0.7and the second-neighbor interaction ˜I3= 0.35at several values of the magnetic field. The triangle symbols\ndenotecriticalfieldsatwhichtwodifferentgroundstatesc oexisttogether.\n0    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 \n0.1375 \n10.2 \n0.5 0.1 \nkBT / I1 δI/I1 = 0 \nχ kBT  \n \nFigure 5 The zero-field susceptibility times temperature\nproductasafunctionoftemperaturefortheparticularcase\nof the Heisenberg interaction ∆= 1and˜J= 0.75, the\nsecond-neighborinteraction ˜I3= 0.35and several values\noftheasymmetryparameter.\nThe ground-state phase diagram of the generalized\nspin-1/2 Ising–Heisenberg diamond chain is depicted in\nFig. 7 in the form of dependence magnetic field versusthe second-neighbor interaction for the fixed values of\nexchange constants relevant for the azurite. The magneti-\nzation curve of the azurite should exhibit according to the\ngeneralized Ising-Heisenberg diamond chain an interme-\ndiate plateau at one-third of the saturation magnetization\nin the field range from 4.13 T to 31.96 T, which corre-\nsponds to the quantum ferrimagnetic phase QFI with a\ncharacterofthedimer-monomerstate. Theseresultsmight\nbe contrasted with the state-of-the-art DMRG data for the\nanalogousspin-1/2Heisenbergdiamondchain,whichpre-\ndict the intermediate one-third plateau associated with th e\ndimer-monomer state in the field range from ≃9.5 T to\n≃31 T in accordancewith the experimentalmagnetization\ndata[28,29].Whiletheloweredgeoftheone-thirdplateau\nis under-estimated within the generalized spin-1/2 Ising–\nHeisenbergdiamondchainapproximativelytwo times, the\nupper edge of the one-third plateau quantitatively coin-\ncides almost exactly with the experimental data and the\nrelevant results of the generalized spin-1/2 Heisenberg\ndiamond chain (the estimated error is around 3 %). Alto-\ngether, it could be concludedthat the generalized spin-1/2\nIsing–Heisenbergdiamondchain notonlyqualitativelyre-\nproduces a character of the ferrimagnetic dimer-monomer\nstate within the one-third plateau region, but it quantita-\ntively reproducesthe upperedgeof the one-thirdmagneti-\nzation plateau. The main reason for this surprisingly good\nquantitative concordance is the fact that the quantum ( xy)\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 11\n0     0.1 0.2 0.3 0.4 0.5 0.05 0.10 0.15 0.20 0.25 \n0     0.1 0.2 0.3 0.4 0.5 0.6 0.05 0.10 0.15 0.20 0.25 0.30 0.1375 δI/I1 = 0 \n0.05 \n0.1 c / 3 kB kBT / I1 a\n \n \nkBT / I1 c / 3 kB  \n \n0.17 \n0.1375 0.2 0.3 0.4 0.5 bδI/I1 = 0.6  \n \nFigure 6 The temperature dependences of the zero-field\nspecific heat for the Heisenberg interaction ∆= 1and\n˜J= 0.75, the second-neighborinteraction ˜I3= 0.35and\nseveral valuesof the asymmetry parameter δ˜I. The partic-\nular values of the asymmetry parameter depicted in Fig.\n6(a) coincide with the ground state FRI 1, while the ones\ndisplayedinFig. 6(b)correspondtothegroundstateQAF.\npart of the exchange interactions I1,I2andI3becomes\nirrelevant once the nodal spins are fully polarized by the\nmagnetic field within the one-third magnetization plateau\ncorresponding to the quantum dimer-monomer state. It is\nquitetemptingto conjecture,moreover,thatthe two-third s\nplateaucouldbedetectedinthemagnetizationcurveofthe\nazurite just if the second-neighborinteractionbetween th e\nnodalspins would be greaterthan I3/kB≥21.8K, i.e., if\nit wouldberoughlyfivetimesstrongerthanit isin reality.\nLast but not least, let us employ the exact solution for\nthe generalized spin-1/2 Ising–Heisenberg diamond chain\nto gain some insight into the magnetism of the copper-\nbased chain of corner-sharing tetrahedra Cu 3Mo2O9[33,\n34,35,36,37,38] to be further referred to as the distorted\ntetrahedral chain. First, it is worth mentioning that the0    5 10 15 20 25 30 35 510 15 20 25 30 35 40 \nFRI 2\n \nQAF QFI SPA B [T]  \nI3/ kB [K] \nFigure 7 The ground-state phase diagram of the gen-\neralized spin-1/2 Ising–Heisenberg diamond chain in the\nI3−Bplane for the fixed values of exchange constants:\nJ/kB= 33K (∆= 1),I1/kB= 15.5K,I2/kB= 6.9K\nand the gyromagnetic ratio g = 2.06. The vertical broken\nline atI3/kB= 4.6K shows the magnetization process\nrelevantfortheazurite.\nasymmetric spin-1/2 Ising–Heisenberg diamond chain ac-\ncounting for the additional second-neighbor interaction\nbetween the nodal spins is isomorphous with the dis-\ntorted spin-1/2 Ising–Heisenberg tetrahedral chain with\nfourdifferentexchangeinteractionswithinthe tetrahedr on\nunit. Even though the magnetic compound Cu 3Mo2O9is\nagaintheexperimentalrealizationofthedistortedspin-1 /2\nHeisenberg tetrahedral chain, it is our hope that the sim-\nplified spin-1/2 Ising–Heisenberg tetrahedral chain may\ncapture some important vestiges of its magnetic behav-\nior. Recent experimental measurements performed on the\ndistorted tetrahedral chain Cu 3Mo2O9serve in evidence\nof the spectacular quantum antiferromagnetic order, in\nwhich the uniform N´ eel order of the nodal spins along\nthe chain direction is accompanied with the spin-singlet-\nlike state of the interstitial spins [33,34,35,36,37,38].\nAccording to our notation (see Fig. 1), the following ex-\nchange constants have been extracted from the inelastic\nneutron scattering data for two strongest exchange inter-\nactionsJ/kB= 67K,I3/kB= 75K, and the respec-\ntive difference between two weaker exchange interactions\nδI/kB= (I1−I2)/kB= 35K [37]. Despite the fact that\nthe absolute values of two weaker interactions I1andI2\ncannotbesimplyfiguredoutfromtheavailableexperimen-\ntal data and they are still under debate, the rough estimate\nof the weakest interaction is around I2/kB≈12K [35].\nRegardlessof the aforementionedambiguity,the strongest\nexchange interaction I3/kB= 75K definitely drives the\nzero-field ground state into the unique quantum antiferro-\nmagnetic phase (14)-(15) with the N´ eel order of the nodal\nspins and the spin-singlet-likestate of the interstitial s pins\nCopyrightlinewillbe provided by the publisher12 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\ncharacterizedbythestaggeredmagnetization:\nmstag=/an}bracketle{tQAF|1\n2(ˆSz\nk,1−ˆSz\nk,2)|QAF/an}bracketri}ht\n=1\n2δI/radicalbig\n(δI)2+(J∆)2≃0.23, (24)\nwhich implies a quantum reduction of the magnetic mo-\n20 25 30 35 40 45 50 55 30 40 50 60 70 80 90 100 \nQAF FRI 2SPA B [T]  \nI1 / kB [K] \nFigure 8 The ground-state phase diagram of the gen-\neralized spin-1/2 Ising–Heisenberg diamond chain in the\nI1−Bplane for the fixed values of exchange constants:\nJ/kB= 67K (∆= 1),I3/kB= 75K,δI/kB=\n(I1−I2)/kB= 35K and the gyromagnetic ratio g =\n2.154 that are relevant for the distorted tetrahedral chain\nCu3Mo2O9. The vertical broken line at I1/kB= 47K\nshowstheestimatedmagnetizationprocess.\nment of interstitial spins roughly to 47 % of its satura-\ntion value. It is worth noticing that the quantum reduc-\ntion of staggered magnetization depends just on a relative\nstrength of the coupling Jbetween the nearest-neighbor\ninterstitial spins and the difference of exchange interac-\ntionsδI=I1−I2, which are known quite accurately\nfrom the experimental data unlike the absolute values of\ntheexchangeinteractions I1andI2.Withthisbackground,\nwe have constructed for the distorted tetrahedral chain\nCu3Mo2O9the ground-state phase diagram in the I1−B\nplane displayed in Fig. 8. The interaction constants esti-\nmated for the distorted tetrahedral chain Cu 3Mo2O9evi-\ndently fall into the parameter region, where the interme-\ndiate one-third magnetization plateau is absent but there\nexists the two-thirds magnetization plateau connected to\nthe classical FRI 2ground state emerging at sufficiently\nhigh magnetic fields. This theoretical prediction is consis -\ntent with recent high-field measuremensperformedon the\nsingle-crystalsampleofCu 3Mo2O9,whichgiveaclearev-\nidenceforthe two-thirdsmagnetizationplateauthe micro-\nscopicoriginofwhichiscurrentlyunderinvestigation[38 ].\nIt is worthwhile to remark that the lower edge of inder-mediate two-thirds plateau is independent of the absolute\nvalue of the exchange constant I1(it depends only on the\nexchange constant Jand the difference δI), which allows\nus to fix the lower edge of two-thirds plateau quite accu-\nrately to the value B= 49.3T that is in a relatively good\nquantitative accord with the values B= 52.3,60.3and\n47.5Treportedforthemagnetizationdatameasuredalong\nthreecrystalographicaxesofthedistortedtetrahedralch ain\nCu3Mo2O9[38].Fromthisperspective,onemayinferthat\nthe two-thirdsplateauactuallybearsa connectionwith the\nclassical ferrimagneticgroundstate FRI 2.\n5 Conclusion In the present article, the ground state\nand thermodynamics of the asymmetric spin-1/2 Ising–\nHeisenberg diamond chain generalized by the second-\nneighborinteractionbetweenthenodalspinsareexamined\nbyarigorouscalculation.Exactresultsforthefreeenergy ,\nmagnetization,susceptibility,entropyand heat capacity of\nthe generalized spin-1/2 Ising–Heisenberg diamond chain\nhave been derived by applying the method of decoration-\niteration transformation. In particular, our attention ha s\nbeen focused on exploring the magnetic behavior of the\ngeneralizedspin-1/2Ising–Heisenbergdiamondchainwith\nthe antiferromagnetic interactions, which should exhibit\nthe most intriguing magnetic features in relation with a\nstrong interplay between the geometric frustration and\nlocalquantumfluctuations.\nAmong the most interesting results one could men-\ntion a considerable diversity of ground-state phase dia-\ngrams, which may include in total five different ground\nstates: the saturated paramagnetic ground state SPA, two\nclassical ferrimagnetic ground states FRI 1and FRI 2, one\nquantum ferrimagnetic ground state QFI and the unique\nquantum antiferromagneticground state QAF. Notably all\nferrimagnetic ground states should manifest themselves\nin low-temperature magnetization curves as intermediate\nplateaux at fractional values of the saturation magnetiza-\ntion. While the total magnetization of two translationally\ninvariantclassicalandquantumferrimagneticphasesFRI 1\nand QFI equals to one-third of the saturation magnetiza-\ntion,thetotalmagnetizationoftheotherclassicalferrim ag-\nneticphaseFRI 2(up-up-up-down-up-up-...)withatransla-\ntionally brokensymmetryequalsto two-thirdsof the satu-\nration magnetization. It is worthy of notice that the pecu-\nliar two-thirds magnetization plateau related to the clas-\nsical ferrimagnetic phase FRI 2cannot be definitely found\nin the spin-1/2 Ising–Heisenberg diamond chain without\nthe second-neighbor interaction between the nodal spins\n[21] but the four-spincouplingmight representanalterna-\ntivemechanismforastabilizationofthetwo-thirdsplatea u\n[25,26].Besides,we havealsodemonstratedarichvariety\noftemperaturedependencesofthezero-fieldsusceptibilit y\nand zero-field specific heat, whereas thermal dependences\nof zero-field specific heat may display one or two anoma-\nlous low-temperaturepeaksin additionto the roundmaxi-\nmumobservableat highertemperatures.\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 13\nThe exact solution presented for the generalized spin-\n1/2 Ising–Heisenberg diamond chain has also proved\nits usefulness in elucidating magnetic properties of two\ncopper-based chains Cu 3(CO3)2(OH)2and Cu 3Mo2O9,\nwhich provide outstanding experimental realizations of\nthe diamond spin chain and the distorted tetrahedral spin\nchain, respectively. As a matter of fact, the generalized\nspin-1/2 Ising–Heisenbergdiamond chain correctly repro-\nduces the intermediate one-third magnetization plateau of\nthe azurite as macroscopic manifestation of the quantum\nferrimagnetic (dimer-monomer) phase, whereas an upper\nedge of the intermediate plateau coincides almost exactly\nwith the experimental results and the state-of-the-art nu-\nmerical calculations for the analogous but more sophis-\nticated Heisenberg model [28,29]. Moreover, the exactly\nsolved spin-1/2 Ising–Heisenberg diamond chain sheds\nlight on the spectacular quantum antiferromagnetic state\nQAF of the distorted tetrahedral chain Cu 3Mo2O9, which\nis characterized by the N´ eel order of the nodal spins and\nthe spin-singlet-like state of the interstitial spins. Our rig-\norous results have enabled us to conjecture to what extent\nthe staggered magnetizationof interstitial spins is reduc ed\nbyquantumfluctuationswithintheQAF,aswellas,topro-\npose the microscopic nature of two-thirds magnetization\nplateauverifiedbyrecenthigh-fieldmeasurements[38].\nAcknowledgements B.L. acknowledges the financial sup-\nport provided by the National Scholarship Programme of the\nSlovak Republic for the Support of Mobility of Students, PhD\nStudents, University Teachers, Researchers and Artists. J .S. ac-\nknowledges the financial support provided by the grant of The\nMinistry of Education, Science, Research and Sport of the Sl o-\nvakRepublicunderthecontractNo.VEGA1/0234/12andbythe\ngrants of the Slovak Research and Development Agency under\nthe contracts Nos. APVV-0132-11and APVV-0097-12.\nReferences\n[1] I.Syozi, Prog.Theor. Phys. 6, 341 (1951).\n[2] M. 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Ito,H.Eisaki, arxiv: 1311.2706.\nCopyrightlinewillbe provided by the publisher14 B.Lisnyi and J.Stre ˇcka: Generalized spin-1/2 Ising–Heisenberg diamond chain\n[39] R.J.Baxter,ExactlySolvedModelsinStatisticalMech anics,\n(Academic Press,London, 1982), chap. 2.\n[40] S.Yamamoto, Phys.Rev. B 59, 1024 (1999).\nCopyrightlinewillbe provided by the publisher"    },    {        "title": "2105.06102v2.Thermal_instability_in_a_ferrimagnetic_resonator_strongly_coupled_to_a_loop_gap_microwave_cavity.pdf",        "content": "arXiv:2105.06102v2  [cond-mat.str-el]  6 Aug 2021Thermal instability in a ferrimagnetic resonator strongly coupled to a loop-gap\nmicrowave cavity\nCijy Mathai,1Oleg Shtempluck,1and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: August 9, 2021)\nWe study nonlinear response of a ferrimagnetic sphere reson ator (FSR) strongly coupled to a\nmicrowave loop gap resonator (LGR). The measured response i n the regime of weak nonlinearity\nallows the extraction of the FSR Kerr coefficient and its cubic damping rate. We find that there is\na certain range of driving parameters in which the system exh ibits instability. In that range, self-\nsustained modulation of the reflected power off the system is g enerated. The instability is attributed\nto absorption-induced heating of the FSR above its Curie tem perature.\nI. INTRODUCTION\nFerromagnetic and ferrimagnetic resonators [1–3] are\nwidely employed in a variety of microwave (MW) de-\nvices, including narrow band oscillators [4], filters [5],\nand parametric amplifiers [6]. These resonators exhibit a\nvariety of intriguing physical effects [7], including Bose-\nEinstein condensation [8] and magneto-optical coupling\n[9–12]. Here we study a strongly coupled hybrid system\ncomposed of a loop gap resonator (LGR) integrated with\na ferrimagnetic sphere resonator (FSR) made of yttrium\niron garnet (YIG) [13, 14]. We focus on the regime of\nnonlinearresponse. InsectionIIIbelowweexploretheef-\nfect onnonlineardampingin the regionofrelativelyweak\nmicrowavedriving. Aninstability, whichisobservedwith\na much stronger driving, is reported in section IV below,\nand a theoretical model, which attributes the instability\nto a driving-induced heating, is presented.\nMany nonlinear dynamical effects have been observed\nbefore in FSRs, including auto-oscillations [15, 16], opti-\ncal cooling [17], frequency mixing [18, 19] and bistabil-\nity [20–24]. The Suhl instability (of both first and sec-\nond orders)hasbeen observedwith transversemicrowave\ndriving, whereas parallel pumping instability has been\nobserved with longitudinal driving [25]. Applications of\nnonlinearity for quantum data processing have been ex-\nplored in [26–33].\nHeating a YIG sphere from room temperature to 400K\nby microwave driving having power of 450mW has been\nreported in [34]. At a Curie temperature given by Tc=\n560 K, YIG undergoes a phase transition between an\nordered ferrimagnetic state (FS) and a disordered para-\nmagnetic state (PS). Thermal instability was observed\nin a cavity magneto-mechanical system [35]. Microwave\noscillations induced by injecting spin-polarized current\n[36] into a magnetic-multilayer structure have been re-\nported in [37]. Self-excited oscillations induced by ohmic\nheating in a Y 3Fe5O12/Pt bilayer nanowire have been\ninvestigated in [38]. Imaging of heating induced by the\nspin Peltier effect has been demonstrated in [39].\nFIG. 1: FSR-LGR coupling: (a) A sketch of the FSR made\nof YIG having radius of Rs= 1mm that is integrated inside\nthe aluminum cylindrical LGR having gap width of 0 .3mm.\nThe sphere is held by ceramic ferrules (CFs). A sapphire\nwafer (labeled as S) is inserted into the gap to increase the\ncapacitance. (b) The numerically calculated magnetic field\nenergy density distribution (normalized with respect to th e\nmaximum value) corresponding to driving at the resonance\nfrequency ωe/(2π) = 3.3GHz. (c) A VNA reflectivity |S11|2\nmeasurement as a function of magnon frequency ωs(propor-\ntional to the externally applied magnetic field). The cou-\nplingcoefficient geffis extractedfrom thetheoretical fit(white\ndashed lines) following Eq. (2).\nII. LOOP GAP RESONATOR\nWith relatively low input power, the main mecha-\nnisms responsible for FSR nonlinear response are mag-\nnetic anisotropy [40] and exchange interaction[13]. Con-\nsider a MW cavity mode having angular frequency ωe\nand an integrated FSR having radius Rs. It is assumed\nthat the applied static magnetic field Hsis parallelto the\neasy axis. In the Holstein-Primakoff approximation [41]\n(which assumes that magnetization is nearly saturated),\nthe Hamiltonian of the system HDis expressedas [21, 42]\n/planckover2pi1−1HD=ωeNe+ωsNs+KMN2\ns\n+geff/parenleftbig\nA†\neAs+AeA†\ns/parenrightbig\n,\n(1)\nwhereNe=A†\neAe(Ns=A†\nsAs) is a cavity mode (FSR\nKittel mode) number operator, ωs=γgHsis the Kit-\ntel mode angular frequency, γg/2π= 27.98 GHz T−1is2\nthe gyromagnetic ratio, KM=/planckover2pi1γ2\ngKc1//parenleftbig\nVsM2\ns/parenrightbig\nis the\nanisotropy-induced Kerr frequency, Kc1is the first-order\nanisotropy constant, Vs= 4πR3\ns/3 is the volume of the\nsphere,Msis the saturation magnetization, and geffis\nthe cavity-FSR coupling coefficient. For YIG at room\ntemperature, Ms= 140 kA /m andKc1=−610 J/m3,\nhenceKM=−2.4×10−8Hz×(Rs/(100µm))−3.\nIn the linear regime, where the Kerr nonlinearity can\nbe disregarded, the Hamiltonian HD(1) can be diago-\nnalized. The angular frequencies ω±of the two hybrid\nphoton-magnon eigen modes are given by [43]\nω±=ωe+ωs\n2±/radicalBigg/parenleftbiggωe−ωs\n2/parenrightbigg2\n+g2\neff.(2)\nBoth angular frequencies ω±are positive provided that\ngeff<√ωsωe. Note that the super-radiance Dicke in-\nstability occurs in the ultra-strong coupling region where\ngeff>√ωsωe[44]. In the rotating wave approximation\n(RWA) the Kerrcoefficients K±ofthe hybridmodes hav-\ning angular frequencies ω±are given by Eqs. ( A9) and\n(A10) of appendix A [see Eq. ( A8)].\nIn the current experiment, we explore the response for\na wide range of the MW input powers Pp. We find that\nthe response is well described by the Hamiltonian HD\nprovided that Ppis sufficiently small. However, with suf-\nficiently high Pp, the FSR temperature Tmay exceed\nthe Curietemperature Tcdueto MWabsorption-induced\nheating. We study the response of the FSR-LGR system\nto an injected monochromatic pump tone having a fre-\nquency close to resonance. The off reflected power is\nmeasured using a spectrum analyzer (SA). We find that\nthere is a certain zone in the pump frequency - pump am-\nplitude plane, in which the resonator exhibits limit-cycle\n(LC) response resulting in self-sustained modulation of\nthe reflected power. The observed LC is attributed to\nthermal instability (TI) [45].\nA MW cavity made of an LGR allows achieving a\nrelatively large coupling coefficient geff[46, 47]. The\nMW LGR schematically shown in Fig. 1(a), is made\nof a hollow concentric aluminium tube having an in-\nner and outer radii of RLGR= 1.7mm and 3mm, re-\nspectively, and a height of HLGR= 12mm. A sapphire\nstrip of 260 µm thickness has been inserted into the gap\nin order to increase its capacitance, which in turn re-\nduces the frequency feof the LGR fundamental mode\n[fe=ωe/(2π) = 3.3GHz with sapphire] [48]. An FSR\nmade of YIG having radius of Rs= 1mm is held by two\nferrules inside the LGR. The static magnetic field Hsis\napplied perpendicularly to the LGR axis. The LGR-FSR\ncoupled system has been encapsulated in a metallic rect-\nangular shield made of aluminum. The cavity is weakly\ncoupled to a loop antenna (LA).\nThe numerically calculated magnetic energy density\ndistribution corresponding to the LGR fundamental\nmode is shown in Fig. 1(b). The calculated density is\nhomogeneous ( ≃95%) over the FSR volume, and it is\nwell confined inside the LGR inner volume. Note thatfor our device, the LGR inner volume, which is given\nbyπR2\nLGRHLGR, is 4 orders of magnitude smaller than\nthe volume λ3\ne, whereλe=c/feis the free space wave-\nlength corresponding to the LGR frequency fe, andcis\nthe speed of light in vacuum. Consequently, the coupling\ncoefficient geffcan be made much larger than typical val-\nues obtained with the commonly employed rectangular\ncavities [28], for which the mode volume commonly has\nthe same order of magnitude as λ3\ne.\nBased on Eq. (2) of Ref. [28], together with the eval-\nuated energy density shown in Fig. 1(b), the calcu-\nlated value of the coupling coefficient is found to be\ngeff= 176MHz for the LGR fundamental mode of fre-\nquencyfe= 3.3GHz. Alternatively, geffcan be extracted\nfrom measurements of MW reflection coefficient |S11|2\nas a function of the Kittel mode frequency ωs/(2π) and\ndriving frequency ωNA/(2π). Fitting |S11|2, which is\nmeasured at temperature of 3K using a vector network\nanalyzer (VNA), with Eq. ( 2) [see Fig. 1(c)] yields the\nvaluegeff= 200MHz, which is pretty much close to the\nvalueobtainedfromsimulation. Note that geffisonlyone\norder of magnitude smaller than the threshold value cor-\nresponding to the super-radiance Dicke instability [44].\nIII. KERR COEFFICIENT AND NONLINEAR\nDAMPING\nCavity driving having amplitude Ω pand angular fre-\nquencyωpis taken into account by adding a term given\nby/planckover2pi1Ωp/parenleftbig\nA†\nee−iωpt+Aee−iωpt/parenrightbig\nto the Hamiltonian HD\n(1). Steady state solution of the driven system was cal-\nculated in Ref. [40] for the case where damping is taken\nintoaccountto firstorderonly. Forthat casethe solution\nis found by solving a cubic equation for the FSR dimen-\nsionless energy Es=/angbracketleftNs/angbracketright[given by Eq. (36) of [40]]. We\nfind, however,thatthe calculatedsteadystate yieldsonly\na moderate agreement with experimental data. Better\nagreement can be obtained by taking into account non-\nlinear damping to cubic order [49]. In this approach the\ncubic equation for Esbecomes\n/parenleftbig\nδ′2\ns+γ′2\ns/parenrightbig\nEs=η|Ωp|2, (3)\nwhereδ′\ns=δs−ηδe+ 2KMEs,δs=ωs−ωpand\nδe=ωe−ωpare driving detuning angular frequencies,\nη=g2\neff//parenleftbig\nδ2\ne+γ2\ne/parenrightbig\n,γe=γ1e+γ2ewithγ1e(γ2e) be-\ning the external (intrinsic) cavity damping rate, γ′\ns=\nγs+ηγe+γ3sEs,γsis the FSR linear damping rate and\nγ3sis the FSR cubic nonlinear damping coefficient. Note\nthat|Ωp|2is proportional to the driving power Ppin-\njected into the LA. Note also that when nonlinear damp-\ning is disregarded (i.e. when γ3s= 0) Eq. ( 3) becomes\nidentical to Eq. (36) of [40].\nVNA measurements of the reflection coefficient |S11|2\nfor three different values of Ppare shown in Fig. 2(a-c).\nFor the data presented in both Fig. 2and Fig. 3, the\nradius of the FSR is Rs= 0.1mm. The theoretical fit3\nFIG. 2: Reflection coefficient |S11|2in dB units for three values of MW input power Pp. Panels (a), (b), and (c) present the\nexperimental data corresponding to MW input powers Ppof -20 dBm, -5 dBm, and +10 dBm, respectively. The second row\n[panels (c), (d), and (e)] shows the corresponding theoreti cal fits that are obtained from Eq. (3). The theoretical fit par ameters\nareγ2e= 1.5 MHz,γe= 4 MHz, γs= 1 MHz, KM= 6.325 nHz, δe= 35MHz , and γ3s= 0.001 nHz. To obtain a proper fit,\nNsandgeffare taken as variable values varying as a function of Pp. ForPp=−20 dBm, −5 dBm, and 10 dBm, Nsvalues are\ntaken as 1 ×1019m−3, 5×1019m−3and 8×1019m−3, andgeffvalues are taken as 14MHz , 14 MHz and 12MHz, respectively.\nshown in Fig. 2(d-f) is based on the cubic equation ( 3),\nwhich allows the calculation of the dimensionless energy\nEs, and on Eq. (3) of Ref. [28], which evaluates the\nreflectioncoefficient |S11|2asafunctionof Es. Thevalues\nof parameters assumed for the calculations are listed in\nthe caption of Fig. 2. Note the driving-induced blue\nshift observed in the magnetic resonance frequency [see\nFig.2(a-c)]. This shift cannot be accurately reproduced\ntheoretically when nonlinear damping is disregarded.\nIV. THERMAL INSTABILITY\nFurther insight can be gained by measuring the spec-\ntral density ISAof the signal reflected off the LA using a\nSA (see Fig. 3). We find that for Pp> Pc= 42.5 dBm,\nand for sufficiently small detuning from resonance, the\nmeasured spectral density ISAcontains equally-spaced\nside-bands (SB) on both sides of the driving frequency\nfp=ωp/(2π) [see Fig. 3(a)]. We measure the SB spac-\ning frequency ωSM/(2π) as a function of the driving fre-\nquencyfpand driving power Pp[see Fig. 3(c)].\nThe observed equally spaced SBs are attributed to\na thermal instability mechanism that is discussed in\nRef. [45]. The phase transition occurring at the Curie\ntemperature Tcbetween the FS and the PS gives rise\nto a sharp change in the resonance modes of the hy-\nbrid cavity-FSR system. Consider the case where the\nfrequency of the externally applied driving is tuned very\nclose to the frequency of one the hybrid system modes.\nWith sufficiently high driving amplitude the temperatureTof the FSR may exceeds the Curie temperature Tcdue\nto driving-induced heating. For that case no steady state\nwithT < T c(i.e. FS) exists. The transition from the FS\ntothePSoccurringat Tcisexpectedtogiverisetoareso-\nnance frequency shift. Consequently the driving-induced\nheating is expected to abruptly drop down, since above\nTcthe frequency detuning between the continuous wave\nexternal driving and the resonance frequency becomes\nlarger (in absolute value). Consider the case where the\nreduced heating gives rise to a temperature drop below\nT < T c. For this case, a steady state with T > T c(i.e.\nPS) also becomes impossible. In the region where no\nsteady state is possible, the temperature is expected to\noscillate around Tc. The frequency of temperature oscil-\nlation can be determined from the spacing between the\nmeasured SBs.\nFor the measurements presented in Fig. 3, the driving\nangularfrequency ωpistunedcloseto ω+. Theanalysisis\ngreatly simplified by disregarding the other hybrid eigen\nmode having angular frequency ω−. This approximation\nis applicable in the strong coupling regime, for which the\nresonances having angular frequencies ω±do not overlap\n[see Eq. ( 2)]. In this approach the FSR-cavity system is\ntreated as a single mode having angular frequency ω+=\n2π×3.32GHz, and Kerr coefficient K+=KMsin4(θg/2)\n[see Eq. ( A9)]. The mode damping rate γ+= 30 MHz is\nexpressed as γ+=γ1++γ2+, whereγ1+is the coupling\ncoefficient between the driven mode and the LA, and γ2+\nis the mode intrinsic damping rate (note that γ1+=γ2+\nfor critical coupling).\nTo account for the observed SB, we consider the ef-4\nFIG. 3: Thermal instability. (a) Spectral density ISAof the\nsignal reflected off the LA, as a function of the detuning fre-\nquencyfd, for the driving frequency fp= 3.2224 GHz and\nnormalized driving power Pp/Pc= 1.288 specified by the\nblack cross overlaid in (c). (b) Spectral density ISAin dB as a\nfunction of the driving frequency fpand detuning frequency\nfdforPp/Pc= 1.7 [indicated by the overlaid horizontal\ndashed line in (c)]. (c) The SB spacing frequency ωSM/(2π)\nin MHz as a function of driving frequency fpand normal-\nized driving power Pp/Pc. The overlaid blue (red) dashed\nline represents the threshold condition EF=EcF(EP=\nEcP). The following values are assumed for the calculations\nω+F/2π= 3.317GHz, ω+P/2π= 3.314GHz, γ+F= 1.3×γ+P,\nσF/wTF= 2.6×σP/wTP, (K+F/γ+F)(wTF/σF) = 0.5 and\nK+P= 0.\nfect of driving-induced heating on the FSR magnetic or-\ndering. The externally applied driving gives rise to a\nheating power Qgiven by Q= 2ℏω+γ2+|B|2, whereB\nis the complex amplitude of the driven mode (note that\nnonlinear damping is disregarded here). It is assumed\nthat the FSR temperature Tis uniform, and that the\ncooling power due to the coupling between the FSR and\nits environment at a base temperature of T0is given by\nH(T−T0), where His the heat transfer coefficient. The\nthermal heat capacity of the FSR is denoted by C. It is\nassumed that all the parameters characterizing the mode\nabruptly change at a critical temperature given by Tc. In\nthe adiabatic (diabatic) region, the mode linear damping\nrateγ+is much smaller (larger) than the thermal decay\nrateH/C.\nIn dimensionless form, system’s time evolution is gov-\nFIG. 4: Limit cycle. (a) Numerical integration of the equa-\ntions of motion (4) and (5) is performed with the following\nparameters Im( wF−wP) =−0.1, Re(wF) =−1, Re(wP) =\n−1.5,σF= 0.01,σP= 0.02, and wTF=wTP= 0.01. The\nvalues of driving detuning frequency Im( wF) and driving am-\nplitudew1=w1F=w1Pare indicated by the black cross in\n(b). The LC is shown in (a) as a closed curve in the complex\nBplane, in (c) as a periodic function of Θ −1 vs. the normal-\nized time τ, and in (d) as a periodic function |B|2vs.τ. The\nplane of driving frequency and driving amplitude is shown in\n(b). No steady state solution exists in the region between th e\nblue and red curves (labeled as A).\nerned by [45]\n˙B=wB−w1, (4)\n˙Θ =σ|B|2−wTΘ. (5)\nOverdot denotes a derivative with respect to a di-\nmensionless time τ, which is related to the time t\nbyτ=γ0t, where γ0is a constant rate. The di-\nmensionless complex frequency wis given by w=/parenleftBig\ni/parenleftBig\nωp−ω+−K+|B|2/parenrightBig\n−γ+/parenrightBig\n/γ0, the dimensionless\ndriving amplitude w1is given by w1=iγ−1\n0√2γ1+Ωp,\nthe dimensionless temperature Θ is given by Θ =\n(T−T0)/(Tc−T0), the dimensionless heating coeffi-\ncientσis given by σ= 2ℏω+γ2+γ−1\n0C−1(Tc−T0)−1,\nand the dimensionless thermal rate wTis given by wT=\n(H/C)/γ0.\nThe normalized parameters w,w1,σandwTare as-\nsumed to have a step function dependence on the tem-\nperature. Below (above ) the critical temperature Tc, i.e.\nfor Θ<1 (Θ>1), they take the values wF,w1F,σF\nandwTF(wP,w1P,σPandwTP), respectively. A steady\nstate (i.e. time independent) solution below (above) the\ncritical temperature Tc, i.e. in the region Θ <1 (Θ>1),\nis possible provided that EF< EcF(EP> EcP), where\nEF=|w1F/wF|2andEcF=wTF/σF(EP=|w1P/wP|2\nandEcP=wTP/σP) [see Eqs. ( 4) and (5) and Fig. 4(b)].\nNote that both EFandEPrepresent steady state values5\nof Eq. (4) for|B|2, whereas both EcFandEcPrepresent\nvalues of |B|2, for which Θ = 1 is a steady state value of\nEq. (5).\nHeat can be removed from the FSR by radiation, ex-\nchange with the surrounding air, and exchange with\nthe supporting ferrules, which hold the FSR inside the\nLGR. The contributions to the total heat transfer co-\nefficient Hdue to radiation, air and the ferrules are\ndenoted by hradSs,hairSsandHfer, respectively, where\nSs= 4πR2\nsistheFSRsurfacearea. Thecoefficient hradis\nroughly given by hrad≃αYIGσSB/parenleftbig\nT4\nc−T4\n0/parenrightbig\n/(Tc−T0),\nwhereαYIGis the averaged FSR absorption coefficient\nin the spectral band corresponding to room tempera-\ntureT0≃300K radiation (wavelength λ≃10µm),\nσSB=π2k4\nB//parenleftbig\n60/planckover2pi13c2/parenrightbig\nis the Stefan-Boltzmann constant,\nkBis the Boltzmann’s constant, /planckover2pi1is Plank’s constant,\nandTc= 560K is the YIG Curie temperature. The ab-\nsorption coefficient value αYIG≃10−1[50] yields hrad≃\n2Wm−2K−1. For ambient temperature and pressure\nhair≃15Wm−2K−1, hence ( hrad+hair)Ss(Tc−T0)≃\n0.6mW for a FSR having radius Rs= 0.1mm. In the\nregion where SB are observed the induced heating power\nappliedtotheFSRisabout3ordersofmagnitudeslarger,\nhenceH≃Hfer, i.e. both radiation and air have negligi-\nbly small contributions, and thus heat is mainly removed\nby the ferrules.\nThe thermal heat capacity of a FSR having radius\nRs= 0.1mm and volume Vs= 4πR3\ns/3 is given by\nC= 2.9×106JK−1m−3×Vs= 1.2×10−5JK−1[51],\nhence the thermal decay rate is roughly given by H/C≃\n320Hz×(Qc/W)((Tc−T0)/(260 K))−1, whereQcisthe\nheating power applied to the FSR, for which the steady\nstate temperature is Tc. Hence for the current device\n(H/C)/γ+≃10−5, and thus the diabatic approximation\nis applicable.\nA typical limit cycle (LC) in the diabatic regime is\nshown in Fig. 4. The LC is calculated by numerically\nintegrating the equations of motion ( 4) and (5). The\nblue (red) cross shown in Fig. 4(a) indicates the steady\nstate value w1/wofBcorresponding to the FS (PS), i.e.\nfor Θ<1 (Θ>1), and the blue (red) circle represents\nthe relation |B|2=EcF(|B|2=EcP). In the plane of\ndriving frequency and driving amplitude, which is shown\nin Fig.4(b), the blue and red curves are derived from the\nrelations EF=EcFandEP=EcP, respectively. In the\nregion labeled as A, no steady state solution to Eqs. ( 4)\nand (5) exists. The LC period time τLCcan be calculated\nby integrating Eqs. ( 4) and (5) over a single period. In\nthe diabatic limit, one finds that τP≃ |wP|−1+|wF|−1.\nThe measured value of LC frequency roughly agrees with\nthis theoretical estimation.\nV. SUMMARY\nIn summary, we demonstrate that relatively large cou-\npling coefficient geffcan be obtained by employing anLGR having mode volume much smaller than λ3\ne. The\nresponse of the system in the weak nonlinear regime al-\nlows the extraction of the Kerr coefficient KMand the\ncubic nonlinear damping rate γ3s. An instability is re-\nvealed by driving the system with a relatively high input\npower. Above the instability threshold the response of\nthe system to an externally applied monochromatic driv-\ning exhibits self-modulation. The instability, which is\nattributed to driving-induced heating, occurs in a region\nwhere the response has no steady state value. Further\nstudy will be devoted to developing sensors that exploit\nthis instability for performance enhancement.\nVI. ACKNOWLEDGMENTS\nThis work was supported by the Israeli science founda-\ntion, the Israeli ministry of science, and by the Technion\nsecurity research foundation.\nAppendix A: Rotating wave approximation\nThe Hamiltonian ( 1) can be expressed as\n/planckover2pi1−1HD=/parenleftbig\nA†\neA†\ns/parenrightbig\nM/parenleftbigg\nAe\nAs/parenrightbigg\n+KMN2\ns,(A1)\nwhere the 2 ×2 matrix Mis given by\nM=/parenleftbigg\nωegeff\ngeffωs/parenrightbigg\n. (A2)\nThe eigenvalues ω±of the matrix Mare given by ω±=\nωm±/radicalbig\nω2\nd+g2\neff[see Eq. ( 2)], where ωm= (ωe+ωs)/2\nandωd= (ωe−ωs)/2. The matrix Mcan be expressed\nas\nM=ωm/parenleftbigg\n1 0\n0 1/parenrightbigg\n+/radicalBig\nω2\nd+g2\neff/parenleftbigg\ncosθsinθ\nsinθ−cosθ/parenrightbigg\n,\n(A3)\nwhere\ntanθ=geff\nωd. 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Recent years have \nwitnessed revival of interest in these compounds following the  \ndiscovery of type-II multiferroicity in the Y - and Z -type \nhexaferrites with strong magnetoelectric coupling around roo m \ntemperature2–4. Further, t he M -type hexaferrites  have also \nevinced a lot of attention  due to the discovery of  several exotic \nquantum critical phenomena such as quantum paraelectricity \n(QPE)5,6, quantum electric dipole liquid state (QEDL)7,8, \nquantum tunneling of magnetization9, quantum electric dipole \nglass8, and magnetic quantum critical point10. Based on low -\ntemperature dc magnetization (M(T)), ac susceptibility ( (ω, T)) \nand x -ray absorption spectra (XAS)/x -ray magnetic  circular  \ndichroism (XMCD) studies on single crystal s of BaFe 12O19 \n(BFO) , we report here another novel phenomenon resulting from \nfreezing of the transverse (basal plane ) component of the spins \ninto a spin glass state at low temperatures. Our results show that \nBaFe 12O19 falls in the category of the geometrically frustrated \nordered compounds11–16 showing exotic spin liquid, spin ice, and \nspin glass transitions even in the absence of any apparent  \nsubstitutional disorde r, with one very significant difference. \nUnlike the spin -glass phases in other geometrically frustrated \ncompounds where it emerges from the high -temperature \nparamagnetic phase, the spin -glass phase of BFO  emerge s from \nthe long -range ordered (LRO) ferrimagn etic (FIM)  phase which  \ncontinue s to coexist with the spin -glass phase.   We have used flux -grown crystals of B FO in the present \ninvestigation. The details of crystal growth, characterization and \nphysical property measurements are given in electronic \nsupplementary information ( ESI). The as -grown crystals are \nhexagonal platelet -shaped with well -developed facets as sho wn \nin Fig. 1(a). The crystallinity and symmetry of the crystals were \nchecked using Laue diffraction pattern collected in the reflection \ngeometry using a polychromatic beam incident along the c -axis \nof the hexagonal unit cell (see Fig. 1(b)). The presence o f closely \nspaced diffraction spots along six symmetry -related directions \nnot only confirms the crystallinity but also confirms the \nhexagonal symmetry of the as -grown crystals. The \na. School of Materials Science and Technology, Indian Institute of Technology \n(Banaras Hindu University), Varanasi, India -221005.  \nb. Universität Hamburg,  Institut fü r Experimentalphysik Luruper  Chaussee 149, D -\n22761 Hamburg, Germany . \nElectronic s upplementary information  (ESI) available : Powder synthesis, crystal \ngrowth and characterization detail, see DOI: 10.1039/x0xx00000x  \nFig. 1: (a) Photograph of as -grown crystal, (b) Laue pattern of BaFe 12O19 \nsingle crystal with x -ray beam along [00l] direction, and (c) kagome bilayer \nconfiguration linked via pyrochlore slabs with different nearest neighbour \nbond lengths r 1, r2, and r 3.COMMUNICATION  Journal Name  \n2 | J. Name ., 2020 , 00, 1-3 This journal is © The Royal Society of Chemistry 20 xx Please do not adjust margins  \nPlease do not adjust margins  magnetoplumbite structure of the BFO in the P63/mmc space \ngroup was confirmed by Rietveld technique using x -ray powder \ndiffraction pattern collected on calcined powder samples  (see th e \nESI).  The refined structural parameters and selected bond \nlengths are given in table S2 & S3 of  ESI.  \n As per the classical Gorter model17, the magnetic structure of \nbarium hexaferrite comprises 3d5Fe3+ spins at the 2a, 2b, and 12k \nWyckoff sites of the P6 3/mmc space group with spin up  \nconfiguration and the spins at the 4f iv and 4f vi Wyckoff sites wit h \nspin down configuration, giving rise to an overall ferrimagnetic \nstructure with a net magnetic moment of 20μ B per formula unit17.  \nThis Ising like picture for the 3d5Fe3+ spins is, however, \nquest ionable, since a magnetic transition has been reported in the \nab-plane with strong spin -phonon coupling18. The variation of \nM⊥C(T) and M//C(T), measured  during warming cycle on a zero-\nfield  cooled  (ZFC)  crystal,  with a magnetic field of 100 Oe \napplied perpendicular ( ⊥) and parallel (//) to the c -axis of the unit \ncell, respectively , shown in Fig. 2  reveals that M⊥C(T) increases \nsteadily with decreasing temperature upto ~4 0K and then shows \na peak ~40K,  whereas  M//C(T) decreases continuously with \ndecreasing temperature. This confirms a magnetic transition at \n~40K . Our results suggest that the spins are  not fully aligned \nalong the c -axis of BFO at low temperature s but have a \nsignificant component transverse to the c -axis in the basal plane \n(00l) due to the canting of the spins away from the c -axis.  \n In order to confirm the canting  of the 3d5Fe3+ spins  away \nfrom the c -axis, we investigated the angle -dependent XMCD \nsignals using XAS spectra recorded  on a single crystal of B FO at \n30K in the normal and grazing incidence (GI) geometries where  \nthe angle (θ) between the direction of propagation vector of the \ncircularly polarized  soft x -ray beam and c-axis is 00 and 150, \nrespectively. The incident -flux-normalized x -ray absorption \nspectra (XAS) obtained in the normal and GI geometries using \nleft circularly and right circularly polarized x -ray photons \ncorresponding to the Fe L 2, 3 edges, labeled  as σ + and σ -, are \nshown in Figs.  3(a) and (b), respectively. The XMCD spectra ( σ \n= σ + - σ-) at iron L 2, 3 edges for the normal and grazing angle \nincidence of polarized x -ray photon  are shown in the same figure \nbelow the XAS spectra. From the angle -dependent  XAS and \nXMCD spectra, spin magnetic moment c an be calculated using \nthe following  spin sum rule equation19,20. \n                   mspin + 7mTθ = - (6P−4Q) nh\nR              ………...(1)  \nin which P = ∫(σ+−σ−)dω \nL3, Q = ∫ (σ++σ−)dω] \nL3+L2, R = \n∫ (σ++σ−)dω \nL3+L2, mspin is the total spin magnetic moment in \nunits  of μB/formula unit, nh is the number of Fe 3d holes, mTθ = \n<Tθ>μB/ħ with < Tθ> being the expectation  value of magnetic \ndipole operator, and L 3 and L 2 represent the integration range \nover energies of the two absorption edges. Using equation (1), \nwe obtained ( mspin + 7mT00)≈0.134 μB/ion for the  magnetic  \nmoment parallel to  the c -axis and ( mspin + 7mT150)≈0.06 μB/ion for \nthe transverse  component of the moment. The observation of \nsignificant XMCD signal for the GI geometry clearly suggests \nthat the 3d5Fe3+ spins are canted away from the c -axis. In order \nto further confirm the spin canting, we also analysed the XAS \nspectra and XMCD signals in the GI geometry recorded with dc \nfield ( H=100Oe ) applied parallel to the beam direction (see Fig.3(c)). The significant enhancement of the XMCD signal s in \nthe presence of dc field further confirms that the 3d5Fe3+ spins \nare indeed  canted away from the c -axis of BFO . Thus, both the \ndc magnetization M(T) and XMCD studies reveal that a finite \ncomponent of the magnetic moments, which are primarily \naligned  parallel to  the c -axis, lies in the ab -plane (i.e., basal plane \n(00l)) perpendicular to the c -axis.   \n In an isostructural compound SrCr 9xGa12-9xO19 (SCGO) with \nmagnetoplumbite structure, it has been shown that the spins in \nthe ab -plane undergo exotic spin liquid21 and spin glass22,23 \ntransitions for Ga content 0<x 1 and 0.6 x124, respectively, \ndue to the formation of kagome bi -layer type block of spins  in \nthe basal plane . This geometry is fully frustrated22 and leads to \ncontinuous macroscopic degeneracy25,26. The observation of \nsignificant component of spins in the ab -plane of BFO  in M(T) \nand XMC D studies and occurrence of the magnetic transition \naround 4 0K in M ⊥C(T) plots (see Fig.2) suggests that this \ntransition may also have a spin glass character similar to SCGO.  \nWe therefore measured M⊥C(T) during warming on  zero-field \ncooled ( ZFC) and field cooled (FC) single crystals  and the \nFig. 2: Temperature evolution  of the dc magnetization M ⊥C and M //C of \nBaFe 12O19 measured with dc bias field H =100 Oe applied perpendicular \nand parallel to the c -axis of unit cell, respectively.  \n \nFig. 3 : X-ray absorption spectra (XAS) and x -ray magnetic circular dichroism \n(XMCD) signal at Fe L 2,3-edges of BaFe 12O19 recorded in (a) normal incidence \ngeometry, (b) and (c) grazing angle geometry.Journal Name   COMMUNICATION  \nThis journal is © The Royal Society of Chemistry 20xx  J. Name ., 2020 , 00, 1-3 | 3 Please do not adjust margins  \nPlease do not adjust margins  typical results are depicted in Fig.4(a). The bifurcation of the  \nZFC and FC M(T) curves just above the peak temperature \ndemonstrates the presence of  history -dependent irreversibility \ncharacteristic of spin glass systems27,28. Since such  an \nirreversibility may also arise due to blocking transition involving \nsuperparamagnetic (SPM) c lusters29, we analysed the  variation \nof the real part of the ac susceptibility 𝜒⊥c′(ω,T) with \ntemperature  (T), measured at various f requencies  (ω=2πf) , \nshown in Fig.4(b). It is evident from this figure that the \ntemperature  Tf(ω) corresponding to the peak in the 𝜒⊥c′(ω,T) \ncurve shifts towards the higher side with increasing measuring \nfrequency  as observed in both the spin glas s27,28 and SPM \nsystem29. For SPM blocking  transition , the temperature \ndependence of the spin relaxation time  τ (≃1/ω) follows \nArrhenius behaviour: τ= τ 0exp(-E/k BT), where τ 0 is the attempt \nrelaxation time, E is the activation energy barrier and k B is the \nBoltzmann constant. For spin -glass systems, the spi n dynamics , \non the o ther hand , follow s a power law  bahaviour27,28: τ=τ 0((T f-\nTSG)/T SG)-zν, where z is the exponent for power -law dependence \nof the c orrelation length ξ on τ  (i.e., ξ~τ-z) while ν is the exponent \nfor the temperature dependence of ξ (i.e., ξ~((T f-TSG)/T SG)ν). For \nthe SPM blocking transition, the ln(τ) versus 1/T plot should be \nlinear.  The non -linear nature of the ln(τ) versus 1/T plot shown \nin the inset of  Fig.4 (c) rules out the possibility of SPM  blocking . \nOn the other hand, least -squares fit to the ln(τ) versus ln((T f-\nTSG)/T SG) plot, as per the power -law behaviour  using the \nprocedure given in ref.30, depicted in Fig.4(c) by continuous line, \nshows excellent fit. This confirms that the transition in the ab -\nplane of BFO is due to the critical slowing down of the spin \ndynamics expected for a spin glass transition. The best fit \nobtained between the observed (red circle ) and calculated (black \nsolid line) relaxation time ( τ) in Fig.4(c)  using the power -law \ndynamics corresponds to τ0=(8.20.8)x10-5 s, zν=( 0.500.02) \nand TSG=(46.0 350.005)K.  The large value of τ 0 (τ0 ~10-5 s) \nsuggests cluster spin glass state in BFO, since the characteristic \ntime (τ 0) required to flip a single magnetic spin in atomic glasses \nis ~10-13s28,31. The calculated zν value  for BFO  is anomalously \nsmall as compared to conventional spin-glass systems  where it \nlies in the range  range 4 to 1228. However, several  oxide and \nalloy -based systems have been reported  to exhibit low zν value s \nin the range  2<zν<432, 1zν<232,33 and zν<1 (zν=0.9, & 0.55)34,35.    \nSuch spin -glass systems have been termed as unconventional \nspin-glasses34. The exi stence of  spin-glass state  at low \ntemperature s in the ab -plane  was further confirmed by studying \nthe relaxation of thermoremanent magnetization (TRM) M⊥C(t) \nas a function of time13,36,37. For this , the sample was cooled from \n300K to 40K and then held at this temperature for 600 sec in the \npresence of a dc magnetic field of 1000 Oe applied perpendicular \nto the c -axis. After the elapse of 600 seconds , the magnetic field \nwas switched off and the decay of magnetization M⊥C(t) as a \nfunctio n of time was recorded for 8 hours. Fig .4(d) depicts the \ntime evolution of M⊥C(t). The decay of TRM in spin glass \nsystems is known to follow Kohlrausch -Williams -Watt (KWW) \ntype stretched exponential behaviour13,36,37: M(t)=M 0+M gexp{ -\n(t/τ r)β}, where M 0 is the ferromagnetic component due to the \ncoexistence of ferrimagnetic phase, M g is the glassy component  \nand τr is the characteristic relaxation time . For β=1 , the system follows exponential decay: M(t)=M 0+M gexp{ -t/τr} with a single \nrelaxation time (τ) corresponding to  the potential energy barriers \nof equal height. Since the energy landscape for spin glass \nsystems is multivalleyed with different barrier height s, the \nrelaxation time (τ) has a distribution27,28 and β lies in the range \n0<β<113,27, 28,36 –38. A least -square fit using the KWW function \ngives an excellent match between  the observed and fitted M ⊥C(t) \nfor M 0=0.115±0.0001,  Mg=0.015±0.00018, and β= 0.63±0.01 . \nThe value of β= 0.63 rules out the po ssibility of relaxation of \nTRM due to the pure long -range ordered ferrimagnetic phase of \nBFO  and confirms the presence of metastable states due to the \nglassy phase.  \n The spin -glass transition has conventionally been linked with \nsubstitutional site -disorder that is re sponsible for frustration and \nrandomness of the interactions27,28,39. In this conte xt, the \nobservation of spin -glass transition in an ordered compound like \nBFO without any substitutional disorder may appear  somewhat \nintriguing.  However,  spin-glass phase  has been  reported in \nseveral  geometrically frustrated ordered compounds like \npyrochlores ( e.g., Tb2Mo 2O714), hydronium jarosite s (e.g.,  \n(H3O)Fe 3(SO 4)2(OH) 612) and spinels (e.g.,  MAl 2O440 with \nM=Co, Fe, and Mn ). Recent theoretical studies on  spin-glass \ntransition in geometrically frustrated  ordered compounds suggest \nthat the presence of an infinitesimal disorder due to anisotropic \nexchange  interactions  or magnetoelastic strains is enough to \nstabilize the spin -glass state41,42. The spin -glass phase reported \nin such geometrically frustrated compounds  including BFO  is \nFig.4: (a) Temperature dependence of ZFC & FC magnetization, (b) \nEvolution of  real part of ac susceptibility measured at various frequencies (c)  \nshows the power law fittings; inset shows the  non-linear behaviour of the \nrelaxation time and (d) Evolution of thermoremanent magnetization with \ntime at 40K . COMMUNICATION  Journal Name  \n4 | J. Name ., 2020 , 00, 1-3 This journal is © The Royal Society of Chemistry 20 xx Please do not adjust margins  \nPlease do not adjust margins  different from those reported in CaBaFe 4O7+43, GaFeO 3, Mn3In \nand Na2Mn 3(SO 4)3(μ3-OH) 2(μ2-OH 2)2 like compounds where \nthere is inherent cation disorder due either to mixed -valence \nstates or anti -site disorder  or randomness in the partial occupancy \nof the Wyckoff position . BFO  is isostructural with SCGO where \na kagome bi -layer configuration of the spins has been reported \nfor the basal plane spins.  We depict in Fig.1( c) the possible \nkagome bi -layer configuration  of spins with nearest neighbour \nbond -lengths for BFO in an alogy with SCGO. We believe that \nthe formation of kagome bi -layer configuration for the basal \nplane component of the spins in BFO  along with the anisotropic \nexchange interactions due to variation in the nearest  neighbour \ndistances between 3d5Fe3+ spins parallel and perpend icular to the \nc-axis is responsible for the stabilization of the spin -glass phase  \nobser ved by us  in BFO . However, the an alogy with SCGO  and \nother geometrically f rustrated order ed com pounds like \nTb2Mo 2O714 and  (H3O)Fe 3(SO 4)2(OH) 612 ends  here. The spin -\nglass phase  in all other geometrically fr ustrated ordered \ncompounds emanates from the paramagnetic phase whereas the \nspin-glass phase of BFO results from freezing of the basal plane \ncomponent of the 3d5Fe3+ spins in the LRO ferrimagnetic phase \nwhich con tinues to coexist with the spin -glass phase . In this  \ncontext, this is the first report of coexist ence of spin glass and \nLRO phases in an ord ered compound in the absence of  any \nsubstitutional disorder . We believe that the present results would \nstimu late further theoretical and experimental studies to  \nunderstand the origin of spin glass transition  due to emergence \nof geometrical frustr ation in LRO magnetic system s.  \nAcknowledgment s: Portions of this research  were carried out at \nthe light source PETRA III of DESY, a member of the Helmholtz \nAssociation (HGF). Financial support from the Department of \nScience and Technology (Government of India) within the \nframework of the India@DESY is gratefully acknowledged.  \nAuthor K . Kumar is thankful to Dr. Arun Kumar for his help.  \nConflict s of interest : \nThere are no conflicts to declare.  \nReferences : \n1 R. C. Pullar, Progress in Materials Science , 2012, 57, 1191 –1334.  \n2 Y. Kitagawa, Y. Hiraoka, T. Honda, T. Ishikura, H. Nakamura and \nT. 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"    },    {        "title": "1807.02445v1.Spin_torque_induced_magnetization_dynamics_in_ferrimagnets_based_on_Landau_Lifshitz_Bloch_Equation.pdf",        "content": "Spin-torque-induced magnetization dynamics in ferrimagnets based on\nLandau-Lifshitz-Bloch Equation\nZhifeng Zhu,1,\u0003Xuanyao Fong,1and Gengchiau Liang1,y\n1Department of Electrical and Computer Engineering,\nNational University of Singapore, Singapore 117576\n(Dated: July 9, 2018)\nA theoretical model based on the Landau-Lifshitz-Bloch equation is developed to study the spin-\ntorque e\u000bect in ferrimagnets. Experimental \fndings, such as the temperature dependence, the peak\nin spin torque, and the angular-momentum compensation, can be well captured. In contrast to\nthe ferromagnet system, the switching trajectory in ferrimagnets is found to be precession free.\nThe two sublattices are not always collinear, which produces large exchange \feld a\u000becting the\nmagnetization dynamics. The study of material composition shows the existence of an oscillation\nregion at intermediate current density, induced by the nondeterministic switching. Compared to the\nLandau-Lifshitz-Gilbert model, our developed model based on the Landau-Lifshitz-Bloch equation\nenables the systematic study of spin-torque e\u000bect and the evaluation of ferrimagnet-based devices.\nI. INTRODUCTION\nFerrimagnets (FiMs) with antiferromagnetic exchange\ncoupled transition-metal (TM) and rare-earth (RE) al-\nloys have attracted considerable attention due to the rich\nphysics1{7and their promise in device applications8{10.\nThe FiMs are expected to have fast spin dynamics like\nantiferromagnets (AFMs), but their magnetic states can\nbe electrically sensed using the tunnel magnetoresistance\n(TMR) e\u000bect due to the \fnite net magnetization ( mnet),\nwhich can be tuned by temperature ( T) or material com-\nposition (X). In addition, the FiMs have large bulk\nperpendicular anisotropy, which o\u000bers an alternative to\nthe ferromagnets (FMs) and enables the scaling down of\nMRAM down to 20 nm8. Furthermore, di\u000berent g factors\nbetween sublattices induce an angular-momentum com-\npensation point, which enables fast domain-wall motion9.\nThe FiMs can be manipulated by magnetic \feld or\nlaser heating2,3,11{15, but an electrical method, such as\nthe spin-transfer torque (STT)1or the spin-orbit torque\n(SOT)4,16, is preferred for electrical characterizations\nand applications. Therefore, it is important to study the\nmagnetization dynamics under spin torque using a model\nwhich can incorporate the e\u000bects of TandX. How-\never, the commonly used theoretical model based on the\nLandau-Lifshitz-Gilbert (LLG) equation1,17{19is limited\nat \fxedTdue to the assumption of a \fxed magnetization\nlength (see Appendix A for detailed analysis of the LLG\nmodel). In contrast, the Landau-Lifshitz-Bloch (LLB)\nmodel has been widely used to describe the magneti-\nzation dynamics at elevated T20, where the T-induced\nmagnetization-length change is taken into account by in-\ncluding a longitudinal relaxation term. To date, the LLB\nequations have been implemented for both FMs20and\nFiMs21, and recently the e\u000bect of spin torque in FMs\nhas also been included22. Starting from the atomistic\nLandau-Lifshitz equation, in this work, we extend the\nLLB model to capture the spin-torque e\u000bect in FiMs.\nThe numerical simulation of current-induced switching in\na FiM/heavy-metal (HM) bilayer is then performed, andwe \fnd the modi\fed LLB model can reproduce salient\nexperimental \fndings, such as the magnetization com-\npensation, the reversal of switching direction across the\nmagnetization-compensation temperature ( TMC)23, and\nthe peak in spin torque at TMC24. In addition, the spin-\ntorque-induced sublattice dynamics in FiMs is studied\nand compared to that in AFMs and FMs. The switching\ntrajectory is found to be precession free, and the sublat-\ntices are not always collinear. Finally, the e\u000bect of Xon\nFiM properties is studied.\nII. THE MODIFIED\nLANDAU-LIFSHITZ-BLOCH EQUATION\nAs shown in Fig. 1(a), the device structure we stud-\nied consists of a FiM (Gd X(FeCo) 1X) deposited on top\nof a HM layer. The magnetizations of sublattices are\nmanipulated by the SOT generated by in-plane electrical\ncurrent. The FiM is treated as a two-sublattice model,\ni.e., Gd and FeCo, which is justi\fed by the experimen-\ntal observation that the magnetizations of Fe and Co are\nparallel up to the Curie temperature ( TC)25. The mag-\nnetization dynamics of each sublattice is captured by the\nmodi\fed LLB equation2,3,22,25{30(see Appendix B for the\nderivation)\n_mv=\rv(mv\u0002HMFA\nv )\u0000\u0000v;k(1\u0000(mv\u0001m0;v)\nm2v)mv\n\u0000\u0000v?mv\u0002(mv\u0002m0;v)\nm2v;(1)\nwhere \u0000 v;k= \u0003 v;NB(\u00180;v)=(\u00180;vB0(\u00180;v)) and \u0000 v;?=\n\u0003v;N[\u00180;v=B(\u00180;v)\u00001]=2 are the coe\u000ecients of longitudi-\nnal and transverse relaxation, respectively. The dimen-\nsionless \feld is given by\n\u00180;v=\f\u0016v(HMFA\nv +HI=\u0015v): (2)\nEq. (1) contains two coupled equations for FeCo and Gd\nidenti\fed by the subscript v, which need to be solvedarXiv:1807.02445v1  [cond-mat.mtrl-sci]  6 Jul 20182\nsimultaneously10. The \frst term on the right hand side\ndescribes the magnetization precession around the mean-\nfree \feld\nHMFA\nv =Hext+HA;v+ (J0;v=\u0016v)mv+ (J0;vk=\u0016v)mk;\n(3)\nwhich consists of the external magnetic \feld Hext, the\ncrystalline anisotropy \feld HA;v= (2Dv=\u0016v)mv;zezwith\ncoe\u000ecientDv, and the exchange coupling between sub-\nlattices with coe\u000ecients J0;vandJ0;vk. The damping\ncoe\u000ecient is given by\n\u0003v;N= 2\rv\u0015v=(\f\u0016v); (4)\n\f= 1=(kBT); (5)\nwhere\u0015vis the damping constant, \rvis the gyromag-\nnetic ratio, \u0016vis the magnetic moment, and kBis the\nBoltzmann constant. As previously mentioned, the T-\ninduced magnetization-length change is described by the\nlongitudinal relaxation term using the Brillouin function\nB(\u0018) =coth(\u0018)\u00001=\u0018; (6)\nand the spin-torque e\u000bective \feld HIis given by\nHI=JS\u0016h=(2etFiMjXM S;v\u0000qMS;kj); (7)\nwhere \u0016his the reduced Planck constant, eis the electron\ncharge,tFiMis the thickness of FiM layer, and MSis the\nsaturation magnetization. The spin current density JSis\nformulated as\nJS=\u0012SH\u001b\u0002JC; (8)\nwhere\u0012SHis the spin-Hall angle, \u001bis the polarization\nof spin current, and JCis the charge current. The equi-\nlibrium magnetization m0;vis calculated via the coupled\nCurie-Weiss equation\nm0;v=B(\u00180;v)\u00180;v=\u00180;v: (9)\nSimilar to the LLB in FM22, the e\u000bect of spin torque only\nenters the two relaxation terms. Furthermore, Eq. (1) re-\nduces to Eq. (4) in Ref.21when the spin torque vanishes,\nor to Eq. (7) in Ref.22when FeCo and Gd are not distin-\nguished. The numerical integration of Eq. (1) proceeds\nusing a fourth-order predictor-corrector method10.\nThe parameters used in the simulation are determined\nas follows [see Fig. 1(c)]: First, the Curie-Weiss equation\n[Eq. (9)] for pure Gd and FeCo is solved independently\nand \ft to the experimental M-Tcurves31to determine\nthe exchange coupling coe\u000ecients JGd= 0:98\u000210\u000021J\nandJFeCo = 1:5\u000210\u000021J. Then, the JGdFeCo =\u00007:63\u0002\n10\u000021J is obtained by solving the coupled Curie-Weiss\nequations of FiM. In addition, a su\u000ecient anisotropy is\nused to ensure the perpendicular magnetization. The\n\u0012SHand\u0015are swept with Dto \ft the experimental\nM-HandM-Jloops16, and a good agreement10with\nthe experimental data is obtained with \u0012SH= 0:003732,\n\u0015=0.07, and D= 3:2\u000210\u000026J.\nFiM\nHMt = 3 nmxyz(a)\nSolve Curie -Weiss equation for \nFeCo and Gdindependently\nfit with experimental \n“M vs T”\nJFeCo, JGdfit experimental \n“TMCvs X”Solve Coupled Curie -\nWeiss equation for FiM\nJFeCoGdD,θSH, αSolve LLB equation\nfit experimental MH, MI loop(c)(b)Atomistic LL equation \nwith spin torque for FiMFokker -Planck equation\nMean field model\nFinal form (Eq. 1)FIG. 1. (a) Schematic view of the device structure consist-\ning of a FiM layer deposited on top of a HM. The FiM in\nthis study is perpendicularly magnetized Gd X(FeCo) 1Xal-\nloy, where Gd and FeCo are antiferromagnetically coupled.\nProcedure of (b) equation derivation, (c) model validation\nand parameter determination. The derivation starts from the\nLandau-Lifshitz equation including the spin torque, followed\nby the corresponding Fokker-Planck equation to account for\nthe statistic behavior, and yields the \fnal form after using the\nmean \feld approximation. This model is validated by com-\nparing with the experimental MvsTtrajectory31,TMCvs\nGd concentration ( X), andM-HandM-I loops16.\nIII. DETERMINISTIC SWITCHING INDUCED\nBY THE SPIN-ORBIT TORQUE\nWe \frst study the SOT-induced deterministic switch-\ning in FiM using the LLB model. As shown in Fig. 2, the\nJCapplied along the xdirection generates spin torque\nacting on the FiM layer due to the spin-hall e\u000bect (SHE)\nor the inverse spin galvanic e\u000bect (ISGE)33{35. How-\never, the magnetization cannot be switched vertically\nsince the spin torque aligns the magnetization to yaxis.\nThis is similar to the perpendicular FM switched by\nin-plane current, where an external \feld along the cur-\nrent direction ( HX) is required to achieve determinis-\ntic switching36{39. The switching in FM system can be\nunderstood as follows: The switching direction is deter-\nmined by HXasL= \u0001m\u0002HX, and the spin torque,\n\u0001m=m\u0002(m\u0002HI), should be su\u000ecient to overcome the\nenergy barrier. Therefore, the switching direction will be\nreversed by reversing either HXor current direction37.\nRecently, by applying HX, the current-induced deter-\nministic switching in the FiM/HM bilayer has also been\ndemonstrated4,16. The measured M-Jloop clearly shows\nan opposite switching direction by reversing the current,\nwhereas the e\u000bect of HXhas not been investigated. In\nthis study, we show that the switching direction is also\nreversed under opposite HX[see Fig. 2], which can be\nexplained using the abovementioned two-torque analysis\ntogether with the exchange coupling between sublattices.\nAs shown in Fig. 2(a), the FiM at T= 300 K is FeCo\ndominant. The positive JCandHXswitch mFeCo from\ndown to up, and concurrently, the exchange interaction\nturns mGdfrom up to down. When the HXis reversed,\nmFeCo is switched from up to down [see Fig. 2(b)], re-\nsulting in an opposite M-Jtrajectory. To con\frm the\nunique role of HX, we have veri\fed that the equilibrium3\n-1 0 10.3\n0\n-0.3\nJC(1011A/m2)mZ_FeCo(a)\n-1 0 1\nJC(1011A/m2)(b)\nHXJC\nHXJC\nFIG. 2. Simulated SOT-induced switching in FiM\nGd21(FeCo) 79under (a) HX= 1 mT, (b) HX=\u00001 mT atT\n= 300 K. The HXonly breaks the symmetry for deterministic\nswitching. The reversal of switching direction under opposite\nHXis similarly explained using the theory in perpendicular\nFM.\nmagnetization is not altered when only HXis applied,\nand no switching event is observed when the current is\nswept with Hext= 0 or HY. Therefore, the SOT-induced\nswitching in FiM is determined by the dominant sublat-\ntice, followed by the reversal of the other sublattice via\nexchange interaction, and the HXonly breaks switching\nsymmetry. It is also worth noting that the maximum\nmZin Fig. 2 is around 0.3, which is an evidence of the\nT-induced magnetization-length reduction with mZ= 1\nde\fned atT= 0 K.\nAlthough SOT and HXhave similar e\u000bects in switch-\ning FiM and perpendicular FM, the time evolutions of\nmagnetization are very di\u000berent as shown in Fig. 3,\ni.e., the switching trajectory of FiM is precession free,\nwhereas it is precessional in FM. In the SOT-switched\nFM with initial mZ= 1, both anisotropy \feld and spin\ntorque align the magnetization to the + zdirection for\nmZ>0, resulting in a larger precession term compared\ntomZ<0, where the anisotropy \feld and spin torque\nare opposite. Consequently, more precession occurs when\nmZ>0 [see Fig. 3(b)]. Similarly, the precession-free tra-\njectory in FiM is attributed to the small precession term.\nAs illustrated using the 3D trajectories in Fig. 3(c), mGd\nandmFeCo are switched to opposite directions. Due\nto the strong exchange coupling, many studies assume\nthey are always collinear. However, as the time evolu-\ntion of each sublattice and their relative angle shown in\nFig. 4(a), a maximum deviation of 0.9 degree is observed\natt= 30 ns. This number is similar to a recent report\nfrom Mishra et al.4, where a cant of one degree is es-\ntimated from the strength of exchange \feld. Since the\nexchange coupling between sublattices is very strong ( >\n100 T40,41), even a very small cant deviates the behavior\nof FiM from FM, which might contribute to the di\u000berent\nmagnetization dynamics shown in Figs. 3(c) and 3(d).\nSimilar noncollinearity between sublattices is also pre-\ndicted in AFM42, with the deviation angle determined by\nthe strength of spin torque. To achieve a large-angle non-\ncollinearity in FiM, recent study shows that a magnetic\n\feld over 5 T is required43. By studying the \feld-induced\nswitching in FiM [see Fig. 4(b)], a similar trajectory is\nTime (ns)0 300\n-0.30.3(a)\nmZ_FeComX_FeComY_FeCom\nx\nyz(c)\nmGdmFeCoinitial\nfinal\n(b)\n(d)\nxyzTime (ns)0 10 20\nHXJCFiM\nHXJCFMFIG. 3. Time evolution of the SOT-induced switching (a) in\nFiM Gd 21(FeCo) 79and (b) in perpendicular FM at T= 300\nK. (c) and (d) are the 3D trajectories corresponding to (a)\nand (b) respectively. The dot lines in (c) are the projections\non thex-yplane. All simulations start from the equilibrium\nstate where mZ= 0.3. This reduced value re\rects the T-\ndependent magnetization, where mZ= 1 is de\fned at T= 0\nK.\n(b)\n0\n-0.40.4(a)\nTime (ns)5 25mZ_GdmZ_FeCo\n45179.9\n179.1\n25 45θ\nTime (ns)5\nm\nTime (ns)5 15 250\n-0.30.3\nHXJCFiM\nFIG. 4. (a) Time evolution of mZfor FeCo and Gd sublattices\natT= 300 K, with inset showing the magnetization angle\nbetween mGdandmFeCo. (b) Field-induced switching in\nGd21(FeCo) 79atT= 300 K, which has similar trajectories\nwith the current-induced switching.\nobserved compared to Fig. 3(a), indicating that a large\nspin-torque e\u000bective \feld would be required to get a large\nangle deviation. However, as discussed in the next sec-\ntion, large spin torque aligns the magnetization to the\nspin direction, hence no switching happens.\nIV. EFFECT OF TEMPERATURE AND\nMATERIAL COMPOSITION\nTandXare often tuned in experiments to control the\nproperties of FiM4,9,23,44. By measuring the M-Hloops\nas a function of T,TMCcan be identi\fed where the co-\nercive \feld ( HC) diverges. However, TMCmay not exist\nin another sample with a di\u000berent X16. In this study,\nthe LLB equation is used to investigate two samples, i.e.,4\nmagnetization magnitude vs Tmnet\nT (K)0 100 200 3000\n-0.10.10.2\nmGdmFeCo\nGd21(FeCo)79\nGd23(FeCo )77\nTime (ns)0 15 30\nmZ_FeCo\n0\n-0.80.80\n-0.80.8\nT = 130 K\nT = 70 K(b) (a)\n(c)\nFIG. 5. (a) E\u000bect of Ton the net magnetization of\nGd21(FeCo) 79(blue square) and Gd 23(FeCo) 77(red trian-\ngle) withTMC= 75 K below which Gd is dominant, where\nthe net magnetization is calculated using mnet= (1 \u0000\nX)mFeCo\u0016FeCo +XmGd\u0016Gdwith\u0016FeCo = 2:217\u0016Band\n\u0016Gd= 7:63\u0016B. Time evolution of mZ;FeCo under spin torque\nat (b)T= 130 K and (c) 70 K.\nGd21(FeCo) 79and Gd 23(FeCo) 77, and we show that the\nexistence of TMCis determined by the demagnetization\nspeed and the relative magnitude of mFeCo andmGd.\nAs reported in our recent study10, both mFeCo andmGd\ndecrease with Tand vanish at the same temperature lo-\ncated between TC;FeCo (1043 K) and TC;Gd (292 K). The\ncommon Curie temperature is induced by the strong ex-\nchange coupling which speeds up the demagnetization\nprocess in FeCo but slows down that in Gd. As shown\nin Fig. 5(a), the Gd 21(FeCo) 79shows FeCo dominant at\nall temperatures, whereas a transition from Gd to FeCo\ndominant is observed in the other sample. At low T,\nGd dominates due to the larger magnetic moment. As\nTincreases, mnetreduces and vanishes at TMC= 75\nK because of the faster demagnetization process in Gd.\nAboveTMC,mnetrises until a peak and then reduces to\nzero atTC. Furthermore, we \fnd that the magnetization\ndynamics near TMC[Fig. 5(c)] is similar to the one at\nhigherT[Fig. 5(b)], which can be understood by notic-\ning the gradual change in e\u000bective \felds such as HAand\nHI. It is only at TMCthat a sudden change occurs, and\nthe e\u000bective \felds diverge.\nAs shown in Fig. 6(a), the competition between mFeCo\nandmGdis also manifested in the T-dependent M-H\nloops23. In addition to the reversal of switching direction,\ntheHCreaches maximum at TMCto overcome the en-\nergy barrier ( E=\u0000M\u0001H). WhenTis further increased\n(i.e.,T > T MC), both mFeCo andmGdreduce, result-\ning in smaller exchange and anisotropy \felds [Eq. (3)]\nand hence a lower HC. Furthermore, we \fnd the TMC\nobtained from the M-Hloops is consistent with the equi-\nlibrium state calculation [Fig. 5(a)], which is another ev-\nidence that the LLB model captures FiM dynamics.\nFor practical reasons, Tis not preferred as the con-\ntrol parameter in device applications, whereas Xcan be\ntuned during the deposition process. The change of X\nshows similar results to that observed in the Tdepen-\ndence. AsXis increased, the FiM changes from FeCo to\nGd dominant, resulting in a reversal of both M-Hand\nM-Jloops4,16. Due to the vanishing mnetatXMC, the\nH (mT)-400 -200 200 400-0.60.6mZ_FeCo\n0-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6\n-0.60.610 K\n40 K\n60 K\n90 K\n120 K\n190 KGddominant\nFeCo dominant(a)\nJC(1011A/m2)-8 8-0.60.6mZ_FeCo\n0-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6X = 0.15\nX = 0.16\nX = 0.17\nX = 0.24\nX = 0.25\nX = 0.26FeCo dominant\nGddominant(b)Hcrit, Icritvs TFIG. 6. (a) M-Hloops at di\u000berent Tin Gd 23(FeCo) 77with\nHX= 2 mT. The blue dot line denotes the transition from\nGd to FeCo dominant. The switching-direction reversal and\nthe peak in HCobserved in experiments23are qualitatively\nreproduced. (b) M-JCloops at di\u000berent XwithHX= 2\nmT. Three dynamic regions are identi\fed, e.g., for X= 0.25,\nsuccessful switching happens for 5 :7\u00021011A/m2< J C<\n6:4\u00021011A/m2, oscillation region for 6 :4\u00021011A/m2<JC<\n8\u00021011A/m2, and distorted region for JC>8\u00021011A/m2\n(i.e.,maligns to the ydirection).\nspin torque diverges4,44. To show the capability of LLB\nmodel in capturing these e\u000bects, we have simulated the\nX-dependent current-induced switching at T= 300 K.\nAs shown in Fig. 6(b), the switching direction reverses\natXMC= 0.24 which separates FeCo and Gd dominant\nregions. In both regions, mnetis switched from down\nto up under positive current, indicating that the SOT-\ninduced switching is determined by mnet. This is di\u000ber-\nent with the anomalous Hall e\u000bect (AHE), where RAHE\nis determined by mFeCo. In contrast to the magnetic-\n\feld-induced switching in Fig. 6(a), no clear peak of crit-\nical switching current density ( JCrit) is observed, which\nis attributed to the increase of spin torque near XMC. In-\nterestingly, three dynamics regions are identi\fed in our\nsimulatedM-Jloops. According to the sub\fgure of X\n= 0.25 in Fig. 6(b), mFeCo is successfully switched from\nup to down for 5 :7\u00021011A/m2<JC<6:4\u00021011A/m2.\nWhen the current exceeds 8 \u00021011A/m2, the magneti-\nzation is aligned with \u001bdue to the dominance of spin\ntorque. The magnetization dynamics in these two regions\nare considered as typical behaviors which have also been\nobserved in FM systems45. For the JCin between, how-\never, an unexpected oscillation occurs. To understand\nthis, we have simulated the SOT-induced switching in\nFM using the LLB model, which is realized by setting X\n= 0, and no oscillation is observed. This indicates the im-\nportant role of exchange coupling, which competes with\nHIandHX, and the oscillation is obtained from the\nbalance of all these interactions, which is a unique prop-\nerty in the system of perpendicular FiM switched by an\nin-plane current.5\nV. CONCLUSION\nIn conclusion, we have developed a theoretical model\nbased on the modi\fed LLB equation to systematically\ndescribe the e\u000bect of temperature, material composition,\nand spin torque in FiM. The e\u000bect of spin torque on\nthe magnetization dynamics is studied in a FiM/HM bi-\nlayer, where the switching trajectory is found to be pre-\ncession free due to the small precession \feld. Similar to\nthe spin-torque switching in AFMs, the two sublattices\nare not always collinear, with a maximum angle deviation\nof 0.9 degree in our study. This cant between sublattices\ninduces an oscillation region between the magnetization\nswitching and the fully alignment with spin polarization,\nwhich is a unique behavior in the perpendicular FiM sys-\ntem. Our results of the spin-torque-induced magnetiza-\ntion dynamics can be helpful in understanding experi-\nmental results and in evaluating FiM-based devices.\nACKNOWLEDGMENTS\nThis work at the National University of Singapore\nwas supported by CRP award no. NRF-CRP12-2013-01,\nNUS FRC R263000B52112, and MOE-2017-T2-2-114.\nAPPENDIX A: NUMERICAL SIMULATION OF\nFIM USING THE LLG MODEL\nBefore we study the LLB model, the LLG equation is\nwidely used to qualitatively explain experimental \fnd-\nings in FiMs. As the FiM consists of antiferromagnetic\ncoupled TM and RE sublattices, it is straightforward to\napply one LLG equation to one sublattice19as\n_Ma=\u0000\raMa\u0002(Ha+hMb) +\u000baMa\u0002_ma\n\u0000\rMa\u0002(Ma\u0002Ha\nI);(10a)\n_Mb=\u0000\rbMb\u0002(Hb+hMa) +\u000bbMb\u0002_mb\n\u0000\rMb\u0002(Mb\u0002Hb\nI);(10b)\nwhere the three terms on the right hand side are preces-\nsion, Gilbert damping, and spin torque, respectively, and\nthese two equations are coupled through the exchange\nterms, i.e., hMbandhMa.\nDespite the clear physical picture behind the coupled\nequations, the analytical study of FiM using these equa-\ntions is complicated. Instead, another e\u000bective LLG\nmodel describing the net magnetization1,17,18is often\nused as\n_Meff=\u0000j\reffj(Me\u000b\u0002He\u000b) +\u000beff(Me\u000b\u0002_Me\u000b)=Meff\n\u0000j\reffj(Me\u000b\u0002(Me\u000b\u0002HI));\n(11)\n\reff(T) =MRE(T)\u0000MTM(T)\nMRE(T)=j\rREj\u0000MTM(T)=j\rTMj;(12)\u000beff(T) =\u0015RE=j\rREj2+\u0015TM=j\rTMj2\nMRE(T)=j\rREj\u0000MTM(T)=j\rTMj;(13)\nwhere\reffand\u000beffare the e\u000bective gyromagnetic ra-\ntio and damping constant, respectively. By assuming\na collinear TM and RE, we can demonstrate that the\ntwo LLG models are equivalent19. The e\u000bective LLG\nmodel is \frstly used to qualitatively explain the unex-\npected sign reversal of magnetoresistance (MR) in STT-\nswitched CoGd layer1, where the coexistence of magneti-\nzation and angular-momentum compensation is observed\nfor the \frst time. In addition, this model has been used\nin the ferromagnetic resonance (FMR) analysis18and\nin explaining the ultrafast spin dynamics in GdFeCo17.\nIn addition, the quantitative study of the spin-torque-\ninduced dynamics in FiM using the coupled LLG equa-\ntions has been studied without considering the di\u000berent g\nfactors or damping constant between sublattices4. In this\nstudy, we choose gFeCo = 2.2,gGd= 29,\u000bFeCo = 0.01,\n\u000bGd= 0.02, and perform numerical simulations of both\nmagnetic-\feld and spin-torque switching using the e\u000bec-\ntive LLG model, which is integrated via the fourth-order\nRunge-Kutta method46. The simulation results are then\ncompared with the experimental and the LLB results in\nthe aspect of critical switching current and switching tra-\njectory.\nLimited by the \fxed magnetization length in the LLG\nmodel, the simulations are performed at \fxed Twhich\nis less than TC;Gd due to the requirement of knowing\nbothMS;Gd andMS;FeCo in Eq. (11). Therefore, we \frst\nchooseT= 200 K and study the \feld-induced switch-\ning withtFiM = 2 nm, and FiM radius rFiM = 25 nm.\nTheMSof both sublattices are taken from their FM\ncounterparts at T= 200 K (MS;Gd = 2:02\u0002106A/m31\nandMS;FeCo = 1:73\u0002106A/m47), resulting in a FeCo-\ndominant sample. As the critical switching \feld is inde-\npendent on the damping constant, we are left with one\nfree parameter, i.e., the crystalline anisotropy \feld ( HK).\nThe magnitude of HKis determined by performing re-\nlaxation simulation, where the magnetization is required\nto return to the perpendicular direction, i.e., the magne-\ntization is \frstly tilted away from the zdirection (e.g.,\n70 degree tilt), and then, it is relaxed without any mag-\nnetic \feld or current. As a result, HKhas to be larger\nthan 0.9 T to maintain a perpendicular magnetization.\nNext, the e\u000bect of HKonHCis studied, where HCin-\ncreases linearly with HK. Consequently, the smallest HC\n= 200 mT is obtained using HK= 0.9 T, much larger\nthan the experimental measured HC= 50 mT23. Fur-\nthermore, when the above procedures are repeated at\notherT(0<T < 293K), we \fnd that the sample is al-\nways FeCo dominant, failing to explain the experimental\nobserved Gd-dominant region.\nHowever, the results from the LLG model can \ft the\nexperimental M-Hloop if the restrictions of taking MS\nfrom FM counterparts are removed [see Fig. 7(a)], which\ncan be justi\fed by the di\u000berence in exchange coupling6\nExp.Sim.\nH (mT)-15 0 15-1.501.5 RAHE(arb. unit)(a)\n-1.3 0 1.3-1.301.3 RAHE(arb. unit)\nJC(1011A/m2)(b)\n(d)\nxyz\n (c)\nxyz\nFIG. 7. Experimental RAHE (blue circle)16and simulated\nmZ;FeCo (red triangle) using the e\u000bective LLG model for (a)\nSOT-induced switching, and (b) magnetic-\feld switching with\nMS;Gd = 3:2\u0002105A/m andMS;FeCo = 1:73\u0002106A/m. (c)\nand (d) are the time evolutions of mFeCo correspond to (a)\nand (b), respectively.\nstrength and lattice occupations between FiM and FM.\nWe then simulate the current-induced switching using\nMSandHKobtained from the \ftted M-Hloop, and a\ngood \ftting shown in Fig. 7(b) is obtained with \u000bFeCo=\n0.01,\u000bGd= 0.02 and spin polarization P= 0.4.\nFigs. 7(c) and 7(d) show the trajectories of \feld and\ncurrent switching, respectively. Both of them are similar\nto that in FMs39, which is expected since the e\u000bective\nLLG model treats the FiM as a single magnet. How-\never, the current-induced magnetization dynamics pre-\ndicted by the LLG model [see Fig. 7(d)] is very di\u000berent\nto that in the LLB model as shown in Fig. 3(c), and more\nresults from time-resolved experiments48would be help-\nful to resolve this discrepancy. As a result, although the\nLLG model can reproduce the experimental M-Hand\nM-Jloops by \ftting MS, di\u000berent sets of MSwill be\ngenerated at di\u000berent T, and some of them might be un-\nrealistic. In addition, these MSare not correlated and\ncannot be explained using a uni\fed theory. In contrast,\nthe LLB model can reproduce the experimental results\nfor all temperatures by using one set of parameters10,\nwhich is attributed to its capability of capturing the T-\ndependent magnetization-length change. In this aspect,\nthe LLB model is more suitable in studying FiM proper-\nties.APPENDIX B: DERIVATION OF THE LLB\nEQUATION INCLUDING SPIN TORQUE\nThe derivation of Eq. (1) starts from the lattice-site\natomistic Landau-Lifshitz equation with an additional\nspin-torque term\n_ s=\r[s\u0002(H+\u0010)\u0000\u0015s\u0002(s\u0002H) + (s\u0002(s\u0002HI))];(14)\nH=Hext+ (2D=\u0016)sz\niez+X\nj2neigJijSij=\u0016; (15)\n<\u0010a(t)\u0010b(t0)>= 2\u0015T\u000eab\u000e(t\u0000t0)=(\r\u00160); (16)\nwhere sis the spin angular momentum, \u0010is the thermal\n\feld with the subscript representing di\u000berent Cartesian\ncomponents (i.e., x,y, andz), andtis the time. The\nthree terms on the right hand side of Eq. (14) represent\nprecession, damping, and spin-torque e\u000bect, respectively.\nThe exchange coupling in the last term of Eq. (15) only\nconsiders the in\ruence of nearest neighbors, and Eq. (16)\nindicates that the sublattice spin is uncorrelated with\nrespect to time and other Cartesian components. The\ndirect simulation using Eq. 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Degen4, Morgan Trassin1, Manfred Fiebig1 & Pietro Gambardella1,* 1 Department of Materials, ETH Zurich, 8093 Zurich, Switzerland 2 Departamento de Física de Materiales, Universidad Complutense de Madrid, 28040 Madrid, Spain 3 Alba Synchrotron Light Facility, 08290, Cerdanyola del Valles, Barcelona, Spain 4 Department of Physics, ETH Zurich, 8093 Zurich, Switzerland 5 Present address: Condensed Matter Physics Center (IFIMAC), Instituto Nicolás Cabrera, and Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain 6 Present address: Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany *e-mail: saul.velez@uam.es (S.V.); pietro.gambardella@mat.ethz.ch (P.G.)  Magnetic skyrmions are compact chiral spin textures that exhibit a rich variety of topological phenomena and hold potential for developing high-density memory devices and novel computing schemes driven by spin currents. Here, we demonstrate room temperature interfacial stabilization and current-driven control of skyrmion bubbles in the ferrimagnetic insulator Tm3Fe5O12 (TmIG) coupled to Pt. We track the current-induced motion of individual skyrmion bubbles. The ferrimagnetic order of the crystal together with the interplay of spin-orbit torques and pinning determine the skyrmion dynamics in TmIG and result in a strong skyrmion Hall effect characterized by a negative deflection angle and hopping motion. Further, we show that the velocity and depinning threshold of the skyrmion bubbles can be modified by exchange coupling TmIG to an in-plane magnetized Y3Fe5O12 layer, which distorts the spin texture of the skyrmions and leads to a directional-dependent rectification of their dynamics. This effect, which is equivalent to a magnetic ratchet, is exploited to control the skyrmion flow in a racetrack-like device.  2 Magnetic skyrmions are intensively investigated owing to their topological spin texture1,2,3,4,5 and sensitive response to electric currents in thin-film devices4,5,6,7. These properties, which can be tuned by interface engineering, open unprecedented prospects for the development of skyrmion racetrack memories4,8 and non-conventional logic devices9,10. Despite this surge of interest, skyrmion-based devices have so far only been realized in metallic systems. Controlling the formation and dynamics of skyrmions in magnetic insulators (MIs) is a requirement for enabling low-power spintronic and magnonic applications11,12,13. Among different classes of MIs, rare-earth garnets coupled to heavy metals have opened exciting possibilities for electrically driving and detecting magnon currents over large distances12,14,15 and for driving domain walls at high-speed16,17 due to their low damping and ferrimagnetic order. Current-driven domain wall motion in centrosymmetric MIs is possible thanks to the interfacial Dzyaloshinskii-Moriya interaction (DMI)16,17,18,19,20, which favours the formation of homochiral Néel walls17. These walls have the largest mobility in response to the spin-orbit torques (SOTs) generated by current flow in coupled heavy metal layers21,22,23,24. Recent measurements of the topological Hall effect in TmIG/Pt provided indirect evidence of the formation of skyrmion-like magnetic textures in thin-film MIs20,25,26,27. The possibility to vary the DMI, magnetic anisotropy, angular momentum, and damping of the rare-earth garnets by composition and strain engineering19,20,25,28 makes these systems extremely appealing for tuning the skyrmion properties29,30. Despite this progress, the direct imaging and current-induced manipulation of skyrmions in MIs remain to be demonstrated. We thus lack crucial information on the nucleation, current-driven dynamics, and stability of skyrmions in MIs and how they compare to metallic systems. In this work, we demonstrate interfacial-stabilization and current-driven motion of skyrmion bubbles in a ferrimagnetic garnet at room temperature. We provide first insights into the skyrmion nucleation process, current-induced dynamics in the presence of pinning and thermal diffusion, skyrmion Hall angle, and rectification effects. Our results demonstrate the potential of MIs for hosting skyrmions and tuning their motion, and highlight the limitations that need to be addressed to realize efficient skyrmion devices.   Sample design Our samples are Y3Fe5O12(YIG)/Tm3Fe5O12(TmIG)/Pt trilayers grown on (111)-oriented Gd3Sc2Ga3O12 (GSGG) with thickness of 10, 20, and 5 nm for YIG, TmIG, and Pt, respectively.  Current lines were patterned in the shape of Hall bars by optical lithography and etching of Pt (see Fig. 1a), leaving the YIG/TmIG layers unetched (Methods). Magnetic characterization confirms that the TmIG and YIG layers exhibit out-of-plane and in-plane magnetic anisotropy, respectively (Supplementary  3 Note 2), with TmIG having a smaller coercivity and remanent magnetization (Fig. 1b) compared to films directly grown on GSGG16,17,18,31. This indicates a reduction of the magnetic anisotropy of TmIG due to the exchange coupling with YIG, which favours the formation of skyrmions without compromising their stability with temperature26,29,30. Unless otherwise specified, the YIG film was demagnetized before starting the measurements.   Chirality and current-induced nucleation and motion of skyrmion bubbles Figure 1c shows magneto-optical Kerr effect (MOKE) microscopy images of the field-induced magnetization reversal of TmIG, which is characterized by the nucleation and expansion of labyrinthine stripe-like domains as commonly observed in garnets with out-of-plane magnetization28. By applying a current pulse, bubble domains nucleate out of a homogenous magnetic texture (Fig. 1d, upper image) as well as from the stripe domains (Fig. 1d, center and bottom images) in the region covered by the Pt current line. These bubble domains, which have an estimated radius 𝑅 of 0.5−1\tµm (Extended Data Fig. 1), are stable in time and are observed for magnetic fields |𝐻*|≲25 Oe, indicating that they are robust magnetic configurations. The best conditions to observe isolated bubbles correspond to a magnetic field |𝐻*|=20 Oe, for which the magnetization is close to saturation and domains have not yet formed (point “1” in Fig. 1b). Subsequent application of current pulses results in the motion of both bubbles and stripe domains in the direction of the current, accompanied by the elongation or contraction of stripe domains and the nucleation of bubbles from stripe domains (Fig. 1e,f). This directional motion is the result of the current-induced SOTs that are exerted on the walls that delimit the domains6,22,23.  According to the sign of the torques and spin Hall effect in TmIG/Pt16,17,31, we conclude that the domain walls of both bubbles and stripes have a right-handed Néel chirality. The closed geometry of the bubbles and the chirality of the domain walls point to a skyrmionic texture2,6, supporting previous reports of the topological Hall effect in TmIG/Pt20,25,26,27. Scanning nitrogen-vacancy (NV) magnetometry measurements of YIG/TmIG/Pt (Methods) confirm the skyrmionic nature of the bubbles. Figure 2 presents the stray field and reconstructed magnetization texture of a circular (Fig. 2a-c) and a deformed bubble due to pinning (Fig. 2d-f). The fits of the magnetic stray field reveal a domain wall width ∆/0~60 nm and confirm the right-handed Néel chirality of the walls. Analysis of the stray field of straight domains in YIG/TmIG further suggests that the right-handed Néel chirality of the walls is also favored in Pt-free regions (Supplementary Note 3), indicating that the YIG/TmIG interface has a positive DMI, unlike GSGG/TmIG17. As the top TmIG/Pt interface has a positive DMI17, we conclude that both YIG and Pt contribute to stabilize right-handed Néel walls and skyrmions in TmIG.  4 The threshold DMI strength required to stabilize chiral domain walls is given by16,17,21 𝐷45=2𝜇7𝑀9:𝑡ln2/𝜋:, where 𝑀9 and t are the saturation magnetization and thickness of the magnetic layer, respectively. For our TmIG film (Supplementary Note 2), we obtain 𝐷45~12 µJ m-2, a value that is compatible with the interfacial DMI found in TmIG-based heterostructures16,17,18,19,26 (𝐷45 further reduces if demagnetizing fields are taken into account32). Interestingly, the low 𝑀9 of TmIG lowers 𝐷45\tby two orders of magnitude compared to ferromagnets, evidencing the potential of ferrimagnetic garnets for stabilizing chiral structures even with weak DMI.  Skyrmion Hall effect Measurements of current-driven displacements of isolated skyrmion bubbles show that they exhibit a skyrmion Hall effect33,34 (Fig. 3), i.e., a transverse deflection relative to the current. The deflection direction depends on the magnetic polarity of the bubble’s core (Fig. 3a), confirming that the skyrmion bubbles carry a topological charge2. Interestingly, the sign of the deflection angle 𝜙9A is opposite to that encountered in metallic ferromagnets. We ascribe the sign of 𝜙9A to the net positive angular momentum of our TmIG films, 𝑠CD4=−EF𝜸>0, where 𝛾 is the effective gyromagnetic ratio (Methods and Supplementary Note 4). In metallic ferrimagnets, the reduced net angular momentum typically results in a small deflection angle35,36, but this is not the case of TmIG, for which |𝜙9A|~40° (Fig. 3b). In the absence of disorder, the deflection angle is given by35 (Supplementary Note 5)                     𝜙9A~tanNOP−QRSTQTUT:∆VWXYZ[,     (1) where 𝑠4\\4 is the total angular momentum, 𝛼~0.01 the magnetic damping parameter37, ∆/0~60\tnm and 𝑅~0.6 µm (Fig. 2 and Extended Data Fig. 1), and 𝑄=+1/−1 the topological charge for a skyrmion with core magnetization pointing up/down2. In our films we estimate QRSTQTUT~−0.06 (Methods), which leads to a large |𝜙9A|~50° due to the relatively low damping of TmIG. Another remarkable difference compared to metallic systems is that, in these, pinning strongly influences the deflection angle in the vicinity of the depinning threshold, resulting in a strong dependence of 𝜙9A on the current density33,36,38. This is not the case of TmIG as the average deflection angle is |𝜙`9A|~40° once the depinning threshold is reached (Fig. 3b). We emphasize that this result holds when averaging over several skyrmion trajectories. As discussed below, although disorder is lower than in polycrystalline metal films, the skyrmion motion in TmIG  is strongly affected by pinning and thermal diffusion.  Skyrmion trajectories and pinning effects  5 We now determine whether the skyrmion trajectories are deterministic or stochastic and investigate pinning effects. Figure 4a reports the trajectories of 35 different non-interacting bubbles following the injection of a series of current pulses of density 𝐽c~2×10OO A m-2 (Methods). Clearly, the displacements are neither linear in time nor in space. Further, histograms of the single-pulse displacements 𝛿𝑥 and 𝛿𝑦 along the 𝒙 and 𝒚 directions reveal a bimodal statistical distribution, which consists of a narrower peak centered at 𝛿𝑥,𝛿𝑦=0 and a broader one centered at 𝛿𝑥=𝑥7 and 𝛿𝑦=𝑦7 (Fig. 4b). The two modes of the distribution capture pulse events that did not lead to net bubble displacements (blue bars) and pulses that led to a net displacement (red bars), respectively. This behaviour indicates that the dynamics of the skyrmion bubbles driven by current pulses is strongly influenced by pinning at structural or magnetic defects. Because individual bubbles alternate between pinned and unpinned  states, and because the probability of depinning is 𝑃<1, the skyrmion bubbles move in the creep regime38,39.  Upon depinning, the bubbles preferably move towards the direction set by the driving force and the Magnus force (Fig. 3 and Supplementary Fig. 8). The analysis of their shape further reveals that the bubbles tend to deform in the direction of motion as well as perpendicular to it, indicating that both SOTs and the skyrmion Hall effect concur in the deformation process in the presence of pinning35,40,41,42 (Extended Data Fig. 2 and Supplementary Note 7). In most cases, however, the deformation is less  than 10% relative to the circular shape, and only in  less than 5% of the events clear deformations are observed (Fig. 2c,f and Extended Data Fig. 2a). The individual longitudinal and transversal displacements have standard deviations 𝜎co,𝜎po larger than the mean 𝑥7,𝑦7 values (Fig. 4b), indicating that the net skyrmion motion is accompanied by random hopping between pinning sites. We attribute this behaviour to the influence of disorder and current-induced thermal fluctuations on the displacements9. Measurements of the current threshold 𝐽c45 for bubble depinning show that 𝐽c45 decreases strongly upon increasing the pulse length 𝑡q, indicating that both SOTs and thermal effects concur in the depinning process (Extended Data Figs. 3 and 4). Applying stronger currents or longer pulses results in thermal motion dominating over directional motion and in the nucleation and annihilation of skyrmions9,43,44, which prevent us from driving the system into the flow regime as reported for metallic systems7,34,38,41,45. The analysis of the mean displacements shows that the bubbles move, on average, by an amount that increases with 𝑡q (Fig. 4c,d and Extended Data Fig. 5). However, the displacements tend to finite values 𝑥7≈100 nm and 𝑦7≈70 nm as 𝑡q drops below ~50 ns (Fig. 4c,d), which is unexpected. These values are independent of the amplitude and direction of the current as well as of the YIG magnetization (Extended Data Fig. 6). Thus, they likely reflect a characteristic length scale of TmIG, namely the average hopping distance between pinning sites. This idea is supported by the distribution of bubble deformations, which is consistent with an average distance between pinning centers on the  6 order of 100 nm (Fig. 2d-f and Supplementary Note 7). Because the bubble radius is larger than such a distance, the skyrmion dynamics is influenced by the pinning of the skyrmion wall to more than one pinning site. We also considered inertial and automotion effects46,47 to explain the finite bubble displacements observed as 𝑡q→0, but these effects appear unlikely in view of the properties of our system (Supplementary Note 8). The hopping motion has strong consequences on the mean skyrmion velocity 𝑣̅9A=w(∆𝑥````):+(∆𝑦````):/𝑡q calculated, as customary in skyrmion studies, using the mean bubble displacements ∆𝑥```` and  ∆𝑦```` averaged over all pulses, including those that did not lead to bubble motion. As shown in Fig. 4e, 𝑣̅9A increases from about 2\tm\tsNO at 𝑡q>50 ns to about 10\tm\tsNO at 𝑡q=10 ns. This increase is attributed to the reduction of the thermal load on TmIG at shorter pulses, which reduces the random hopping and leads to a more efficient directional flow of the skyrmions. In these conditions, 𝑣̅9A increases towards the flow regime limit, which we estimate as ~30 m\tsNO for 𝐽|=1×10OO\tA\tm-2 (Methods). This limit, however, is hard to reach given that 𝐽c45 increases with decreasing 𝑡q (Extended Data Fig. 3).  It is known from experimental33,34,35,38,40,43 and theoretical studies38,39,42 that disorder strongly impacts the current-driven motion of skyrmions. Our findings reveal that the behaviour of skyrmions in a MI exhibits remarkable differences compared to metallic heterostructures. First, the density of pinning sites in TmIG, estimated from (𝑥7𝑦7)NO for 𝑡q→0 is ≈10N\tnmN:, two orders of magnitude lower than in polycrystalline metal films43. Despite the lower disorder, the 𝐽c−𝑡q parameter space for skyrmion motion in TmIG is reduced to a narrow range due to current-induced heating dominating over the SOTs (Extended Data Figs. 3 and 4). We attribute this limitation to the exponential dependence of the skyrmion diffusion with temperature, which is expected to alter the skyrmion dynamics in materials with low damping and low disorder9, preventing the use of currents strong enough to reach the flow regime.  The thickness of our films also reduces the SOT driving force in comparison to ultrathin metallic systems38,41,43. Another difference with respect to metal films33,34,38,41,43 is that the average skyrmion Hall angle 𝜙`9A=tanNOPc````p````[ in TmIG is very large (Figs. 3 and 4f), even though the skyrmions move in the creep regime. Theoretical models of skyrmions interacting with random point defects39 or a granular magnetic anisotropy background38,42,43 predict a decrease of 𝜙`9A with disorder, leading to 𝜙`9A~0 in the creep regime. In TmIG, however, the density of defects is low compared to sputtered metal films, resulting in 𝜙`9A close to the flow limit given by Eq. 1, which only reduces from ~42° to ~34° upon decreasing 𝑡q from 100 to 10 ns (Fig. 4f). This reduction is another indication that the effects of pinning become more evident as 𝑡q is reduced. For long pulses, disorder has a small influence on the average  7 skyrmion deflection, as inferred from the large 𝜙`9A, but the thermal fluctuations are large, resulting in a reduction of 𝑣̅9A (Fig. 4e). Conversely, for short pulses the thermal fluctuations reduce, leading to a more efficient SOT motion but also to a stronger influence of pinning on the skyrmions’ trajectories, which results in the decrease of 𝜙`9A with decreasing 𝑡q (Fig. 4f). We also find that 𝜙`9A is nearly independent on 𝐽| (Fig. 3b), which we attribute to the competing action of SOTs and heating as 𝐽| increases above the depinning threshold. Future experimental and computational studies should aim at elucidating the influence of thermal diffusion9 on the current-induced dynamics of skyrmions in materials with low damping and low disorder.  Skyrmion ratchet effect We now investigate the influence of the in-plane magnetization of YIG (𝐌) on the skyrmion dynamics in TmIG. The skyrmion trajectories remain affected by pinning in a homogenous 𝐌, and the average deflection angle is about the same for different 𝐌 configurations (Figs. 4a,b and 5). However, the depinning probability and the mean bubble displacements depend strongly on the orientation of 𝐌 relative to 𝐉| (Fig. 5b,d). In particular, the bubble displacements with 𝐽c>0 are much larger for 𝐌 pointing along  −𝒚 (Fig. 5a,b) than for +𝒚 (Fig. 5c,d), with the demagnetized case lying in between the two (Fig. 4a,b). This asymmetry is only observed when 𝐌 is perpendicular to 𝐉|, with the motion of the bubbles being more (less) efficient when 𝐉|×𝐌~−𝒛(+𝒛) regardless of their topological charge (Supplementary Table 1). These observations suggest that 𝐌 modifies the escape probability of the bubbles from the pinning potential, whereas the density of pinning sites is not significantly influenced by 𝐌. We attribute this escape asymmetry to the distortion of the magnetic texture of the bubbles’ wall induced by the exchange coupling with YIG, which results in SOTs of different strength depending on the orientation of 𝐌 relative to 𝐉|. The mechanism that we propose can be explained as follows. In the absence of current, the magnetic moments in the wall of a bubble (𝐦/0) tilt towards 𝐌 (Fig. 6a,b). In the presence of current, 𝐦/0 acquires an additional tilt in the direction of the damping-like SOT (𝐓/)24,48, such that 𝑑𝐦/0/𝑑𝑡∝−𝐓/∝𝐽c𝒚 (Fig. 6c,d). Therefore, the distortion produced by 𝐉| opposes (favours) the one induced by 𝐌 when 𝐉|×𝐌 points towards −𝒛 (+𝒛), an asymmetry that is consistent with the experiments (Fig. 5) and holds for both signs of 𝑄 (Supplementary Note 10). Thus, given that 𝐓/ controls both the depinning and the displacement of the bubbles, and that 𝐓/∝𝐦/0×(𝐦/0×𝒚) is proportional to the 𝑥-component of 𝐦/0, the skyrmion bubbles move more (less) efficiently when the magnetic tilt towards 𝒚 is minimal (maximal). Importantly, no asymmetry in the skyrmion dynamics is observed when 𝐌 and 𝐉| are collinear (Extended Data Fig. 7), in agreement with our model. In addition, no  8 changes of 𝜙9A are expected due to a change in 𝐓/, which is also consistent with our observations (Figs. 4a and 5a,c).  We next exploit the asymmetry in the current-driven skyrmion depinning and displacements induced by 𝐌 to rectify the skyrmion motion. In the vicinity of the depinning threshold, the asymmetry of 𝐓/ induced by 𝐌 leads to the unidirectional motion of the skyrmions. Figures 6e,f and 6g,h show the trajectory of a few skyrmion bubbles for alternating sequences of positive and negative current pulses with\t𝐌 pointing along −𝒚 and +𝒚, respectively. Clearly, the bubbles move only for one polarity of the current, which depends on the orientation of 𝐌. This skyrmion ratchet effect is similar to a magnetic gate, which can be used to induce a net skyrmion displacement from random current excitations or for preventing the skyrmions to move along a particular direction when using an alternating current to generate SOTs. An analogous ratchet effect is observed for stripe domains due to the homochiral nature of the domain walls in TmIG (Supplementary Fig. 11). We remark that, whereas the ratchet effect requires the presence of pinning, the directional asymmetry of the skyrmion motion due to 𝐌 does not.  Conclusions We showed that skyrmion bubbles can be stabilized and driven by proximity charge currents in a centrosymmetric MI coupled to Pt. Despite the reduced density of defects in TmIG, we find that pinning and thermal skyrmion diffusion severely affect the motion of the skyrmion bubbles, which are constrained in the creep regime for currents below the emergence of thermal instabilities. The bubbles move by intermittent sequential jumps between nearby pinning sites, which result in a broad distribution of longitudinal and transverse displacements. Remarkably, the skyrmion Hall effect is large and opposite compared to ferromagnets, which we ascribe to the relatively low damping, low density of defects, and positive angular momentum of the TmIG film. In principle, the ferrimagnetic order of MIs allows for controlling the sign and amplitude of the net angular momentum, and therefore tune the skyrmion Hall effect35 and mobility49. Future realizations of skyrmions in MIs should aim at improving their thermal stability for a broader range of currents, possibly by a concomitant increase of the magnetic anisotropy and DMI in rare earth garnets19,20,25,26. Moreover, pinning effects should be minimized to achieve deterministic and efficient skyrmion motion. Finally, we demonstrated control over the skyrmion dynamics by exchange coupling TmIG to an in-plane magnetized YIG layer. As the skyrmion’s driving force depends on the polarity of the current relative to the magnetization of the in-plane layer, it is possible to rectify the skyrmion’s motion along a predefined direction. This new aspect of the skyrmion dynamics provides an additional tool for tailoring the mobility of skyrmions in spintronic devices.  9  Acknowledgements We acknowledge André Thiaville, Aleš Hrabec, and Christoforos Moutafis for useful discussions and Marvin Müller for technical assistance with the MOKE setup. This work was funded by the Swiss National Science Foundation (Grants No. 200020-200465 P.G., 200021-188414 M.T., 200021-178825 M.F., PZ00P2-179944 B.J.J., and 200020-175600 C.L.D.), by the European Research Council (Advanced Grant 694955-INSEETO M.F.), and by ETH Zürich (Career Seed Grant SEED-20 19-2 S.V.). S.R. acknowledges support from the Spanish Ministry of Economy and Competitiveness (FPI fellowship and Grant No. RTI2018-095303-B-C53). S.V. acknowledges financial support by the Ministry of Science, Innovation and Universities through the ‘Maria de Maeztu Program for Units of Excellence in R&D (Grant No. CEX2018-000805-M) and by the Comunidad de Madrid through the Atracción de Talento program (Grant No. 2020-T1/IND-20041).  Author contributions S.V. conceived the study and coordinated the experimental work. J.S., E.G., and M.T. grew and characterized the films. S.V. fabricated the devices. S.V. and S.R. performed the transport and MOKE experiments and analysed the data. B.J.J. assisted with time-resolved transport measurements. M.S.W. and P.W. performed the NV measurements with the help of S.V. S.V. and P.G. wrote the manuscript. P.G., M.T., C.L.D., and M.F. supervised the work. All authors contributed to the scientific discussion and manuscript revisions.  Competing interests  The authors declare that they have no competing interests.  10 FIGURES  \n Figure 1 | Current-driven nucleation and dynamics of bubble domains in TmIG. a, Schematic of the device structure with a superimposed wide-field differential MOKE image showing magnetic domains in TmIG. Bright (dark) contrast corresponds to regions with up (down) magnetization. The coordinate axis and the current line are indicated. b, Magnetic hysteresis loop of TmIG measured by MOKE microscopy in the region covered by the current line while sweeping the out-of-plane magnetic field 𝐻*. The films were first demagnetized by cycling the in-plane field in loops of alternating polarity and decreasing amplitude. c, Sequence of differential MOKE images taken at different magnetic fields (green dots in b) showing that the magnetization reversal proceeds by the formation of labyrinthine stripe-like domains. d, MOKE images taken after the application of a current pulse 𝐽c=6.2×10OO A m-2 with length 𝑡q=40 ns to the domain structures shown in c, revealing the nucleation of bubble domains (top image) as well as the nucleation and breaking of the stripe domains into bubble domains (center and bottom images). All magnetic configurations in c and d remained stable after one hour. e, f, From up to down, sequence of MOKE images showing snapshots of the current-driven dynamics for a sequence of positive and negative current pulses (see arrow direction), respectively. The domains move along the direction of the current and their dynamics is influenced by pinning, resulting in the deformation of bubbles into stripes or vice versa and the formation of new bubbles from stripes. Dashed lines of different colours evidence the position of selected domains before and after pulsing. We attribute domain motion outside the current line to repulsive dipolar interactions. The measurements are performed with 𝐻*=−10 Oe, |𝐽c|=3×10OO A m-2, and 𝑡q=40 ns. Scale bars in c, e are 15 and 5 µm, respectively. \n 11  \n Figure 2 | Nitrogen-vacancy magnetometry analysis of the skyrmion bubbles in TmIG. a, Normalized stray field map 𝐵(𝑋,𝑌) of a single skyrmion bubble in YIG/TmIG/Pt measured by scanning the NV tip over the 𝑋𝑌 plane. b, Best fit of the data in a, which corresponds to a circular bubble with diameter 2𝑅~950 nm and a right-handed Néel wall with ∆/0=60 nm. c, Reconstructed out of plane magnetic component 𝑀* corresponding to the data in b. d-f, Same as in a-c for a deformed skyrmion bubble due to wall pinning. Fit of the data in d assuming an ellipsoidal shape of the bubble with arbitrary orientation in the 𝑋𝑌 plane. The best fit is obtained for a right-handed Néel wall with ∆/0=60 nm and radial axes 940 and 830 nm. 𝐻*=−20 Oe in a and d. See Supplementary Note 3 for details regarding the fitting procedure.\n 12  \n Figure 3 | Skyrmion Hall effect in a ferrimagnetic insulator. a, Sequence of differential MOKE images showing the position of isolated skyrmion bubbles during a series of positive current pulses. The top and bottom images correspond to bubbles with 𝑄=+1\t and 𝑄=−1, respectively. The direction of 𝐉c, 𝐯9A and 𝜙9A are indicated in the rightmost image of each sequence. The images are selected from a sequence of current pulses with 𝐽c=2.0×10OO A m-2 and 𝑡q=20 ns. Bright (dark) contrast corresponds to regions with up (down) magnetization. In the bottom images, the yellow spots indicate a skyrmion bubble trapped at a defect site and the green ones a bubble driven into the imaging region during the pulse sequence. The scale bars are 3 µm. b, Average skyrmion deflection angle 𝜙`9A measured for different amplitude, length, and direction of the current pulses for skyrmion bubbles with 𝑄=+1\tand−1. Each data point is an average performed over ten independent skyrmion bubbles with error bars representing the standard deviation (Methods). Different symbols correspond to different pulse lengths. Data points with 𝜙9A=0 indicate no single skyrmion depinning events. The lowest |𝐽c| values shown for 𝑡q=20 and 10 ns indicate the current threshold for skyrmion depinning at these pulse lengths. The insets indicate the orientation of 𝐉c and\t𝐯9A, sign of 𝜙9A, and the magnetic configuration of the skyrmion bubbles. 𝐻*=−20\t(+20) Oe for 𝑄=+1\t(−1). \n 13  \n Figure 4 | Statistical analysis of the trajectories of the skyrmion bubbles and pinning effects. a, Representative data showing the trajectory of 35 different non-interacting skyrmion bubbles following the application of a sequence of current pulses with 𝐽c>0 for demagnetized YIG. The pulse length is 𝑡q=\t50 ns, the current density 𝐽c\t~\t2×10OO A m-2, 𝐻*=−20 Oe, and 𝑄=+1. The initial position of all the trajectories is set at (0,0); each data point represents the position of a bubble after a current pulse, labelled by a different symbol and colour. b, Histogram of individual bubble displacements 𝛿𝑥 and 𝛿𝑦 computed as the difference in position (𝑥,𝑦) between adjacent data points along the trajectories shown in a, representing bubble displacements per pulse. The grey bars represent the experimental data and the black lines the fits obtained using a double Gaussian distribution. The blue-shaded peak centered at 𝛿𝑥,𝛿𝑦=0 accounts for pulses that did not lead to bubble displacements. The red-shaded peak captures the distribution of pulses that led to bubble motion, centered at 𝛿𝑥=𝑥7 and 𝛿𝑦=𝑦7 with standard deviation 𝜎co and 𝜎po. 𝑃 indicates the relative weight of the area under the red curve relative to the total histogram area; ∆𝑥```` and ∆𝑦```` are the average bubble displacements per pulse along 𝒙 and 𝒚, respectively, including all pulse events. c-f, Analysis of the skyrmion trajectories as function of 𝑡q. c, d, Mean displacements 𝑥7 and 𝑦7 as a function of the pulse length; the error bars are the standard errors calculated from the variance of those parameters to the double Gaussian fit function. e, f, Pulse length dependence of the mean bubble velocity 𝑣̅9A and Hall angle 𝜙`9A calculated from ∆𝑥```` and ∆𝑦```` (Extended Data Fig. 5). Different symbols in c-f indicate the current density. \n 14  \n Figure 5 | Statistical analysis of the trajectories of the skyrmion bubbles for opposite orientations of 𝐌. a, Representative data showing the trajectory of 68 different bubbles following the application of a sequence of current pulses 𝐽c>0 for 𝐌∥−𝒚. b, Histogram of individual bubble displacements along the 𝒙 and 𝒚 directions extracted from the bubble trajectories shown in a. c, d, Same as a, b for 26 bubbles and 𝐌∥𝒚. The pulse length is 50 ns, the current density ~2×10OO A m-2, 𝐻*=−20 Oe, and 𝑄=+1. The orientation of 𝐌 is set by a constant in-plane magnetic field 𝐻p=10 Oe. The grey bars in b and d represent the experimental data and the black lines the fits obtained using a double Gaussian distribution, separately represented in blue and red colours (see caption of Fig. 4a,b). \n 15  \n Figure 6 | Skyrmion ratchet effect. a, Schematic of the domain wall magnetic moments 𝐦/0 in a right-handed 𝑄=+1 bubble. b, Distorted skyrmion bubble due to the exchange coupling with 𝐌=−𝑀𝒚. c, d, Additional distortion of the skyrmion bubble shown in b due to the dynamic action of the SOTs in TmIG/Pt. The dashed and solid arrows indicate 𝐦/0 before and during the pulse, respectively, with green and orange colours indicating the direction of 𝑑𝐦/0/𝑑𝑡 for opposite current directions. The reduced (enhanced) distortion of 𝐦/0 in c (d) results in a larger (smaller) current-induced driving force. e, Representative images showing the displacement of 𝑄=+1 skyrmion bubbles following the application of a sequence of current pulses of alternating polarity with 𝐌 pointing towards −𝒚. The bubbles are labelled by dashed coloured circles; dots indicate the initial positions of the corresponding bubbles (see image #I). f, Total displacement Δ𝑑=w∆𝑥:+∆𝑦: averaged for the trajectory of the three skyrmion bubbles shown in e (see image #VII). The sign of Δ𝑑 corresponds to the sign of ∆𝑥. The direction of 𝐉c and the corresponding image number are indicated in the pulse sequence. g, h Same as e, f but for 𝐌 pointing towards +𝒚. |𝐽||=1.8×10OO A m-2, 𝑡q=50 ns, and 𝐻*=−20 Oe in e, g. The orientation of 𝐌 is set by a constant in-plane magnetic field 𝐻p=10 Oe. Scale bars in e, g are 3 and 5 µm, respectively.  \n 16  References: 1. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009). 2. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013). 3. Tokura, Y. & Kanazawa, N. Magnetic Skyrmion Materials. Chem. Rev. 121, 2857–2897 (2021). 4. Fert, A., Reyren, N. & Cros, V. 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The YIG and TmIG films were epitaxially grown onto (111)-oriented Gd3Sc2Ga3O12 (GSGG) substrates (lattice constant a = 12.56 Å) by pulsed laser deposition to achieve high tensile strain (~2%) of the TmIG layer, which promotes TmIG to exhibit perpendicular magnetic anisotropy as earlier demonstrated17,50. The properties of the heterostructure were tuned by varying the thickness of the layers and the deposition conditions. For the sample investigated in this work, the YIG film was deposited at 720 °C with an oxygen background pressure of 0.15 mbar. The laser fluence was set to 0.9 J cm-2, and the repetition rate at 2 Hz. The TmIG film was consecutively grown at 650 °C with an oxygen pressure of 0.2 mbar, and the laser fluence and repetition rate were set to 1.35 J cm-2 and 8 Hz, respectively. After deposition, the sample was annealed at 750 °C under 120 mbar oxygen pressure for 30 min and cooled down to room temperature at a rate of -10 K/min in 200 mbar oxygen pressure. To ensure a high quality of the TmIG/Pt interface, the YIG/TmIG films grown were directly transferred to the sputter chamber without breaking vacuum, where the Pt layer was deposited at room temperature for 3 minutes at a power of 10 W in 0.05 mbar Ar plasma. The thickness of the layers was calibrated by X-ray reflectometry and determined to be 9.8, 20.2, and 5.1 nm for YIG, TmIG, and Pt, respectively. The uncertainty for the garnet layers is 2.0 nm. Atomic force microscopy measurements of the surface topography showed a root-mean-square roughness of about 0.18 nm over a 5 x 5 µm2 area. The films were magnetically characterized in a superconducting quantum interference vibration sample magnetometer (SQUID-VSM) system. We estimate the saturation magnetization of TmIG and YIG to be 60 and 175 kA m-1, respectively. See Supplementary Notes 1 and 2 for further details regarding the structural, topographic, and magnetic characterization of the films. Device fabrication. The Pt layer was patterned into Hall bars (consisting of three Hall crosses separated by 𝐿\t=50 µm and width 𝑊=10 µm) by photolithography and subsequent Argon plasma etching following the recipe of Ref. 17, leaving the YIG/TmIG films unetched. The required etching time for removing the Pt layer was calibrated in reference films. 48 devices were patterned in each sample. Measurements were performed in 6 different samples, from which 2 showed skyrmion bubble stabilization and motion driven by electric currents. All data presented in this work was acquired in the same sample. The current-induced skyrmion motion measurements were performed in the same device with the exception of the data presented in Extended Data Figs. 6 and 7 and Supplementary Fig. 9. The NV data was acquired in a third device. Additional measurements were performed in other  21 devices from the same sample, all showing same characteristics. The topography of the etched structures was characterized by atomic force microscopy, from which we determined the thickness of TmIG to be reduced by ~0.5 nm due to etching and the root-mean-square roughness in the etched region to be ~0.2 nm (Supplementary Fig. 2).  MOKE measurements. We used a home-built wide-field polar MOKE microscope with Koehler illumination to measure the out-of-plane component of TmIG17. As described in Ref. 17, the light source consists in a collimated light emitting diode from Prizmatix Ltd., model MIC-LED-455L, whose spectral emission is characterized by a maximum peak emission at 454 nm, centroid at 455 nm, and a full width at half maximum of 28 nm. The setup was equipped with two sets of orthogonal coils for the generation of out-of-plane and in-plane magnetic fields. For the skyrmion bubble generation and current-driven motion studies, current pulses were injected using an AGILENT 8114A (100V/2A) and a Kentech RTV40 sub-ns pulse generator. The impedance matching with the pulse generators was achieved by connecting a 50 W resistance in parallel to the Pt current line. Magnetic contrast was enhanced by taking differential MOKE images, i.e., each image was subtracted by a reference image captured in a fully magnetized state as employed in Ref. 17. The contrast is defined as the absolute value of the differential image, being black (bright) when the difference is zero (maximal). A change of the magnetic field applied reduces the contrast due to the Faraday effect. Accordingly, the reference image for a fully down (up) magnetized state was taken at 𝐻*=−20\t(+20) Oe. Consequently, optimized contrast is observed in the upper images of Fig. 1c,d and the images of Figs. 3 and 6. Conversely, contrast is reduced for the center (𝐻*=−10\tOe) and bottom (𝐻*=0\tOe) images of Fig. 1c,d as well as the images of Fig. 1e,f (𝐻*=−10\tOe). For the bottom images of Fig. 3a (𝐻*=+20\tOe;\t𝑄=−1), the contrast was inverted to preserve the colour-code definition black (bright) corresponding to down (up) magnetization. For the current-driven skyrmion dynamics experiments (Figs. 3-6), the skyrmion bubbles were prepared as follows. First, we demagnetized the films by cycling the in-plane field in loops of alternating polarity and decreasing amplitude (a similar result can be obtained by cycling 𝐇* instead). Second, for experiments with 𝐌 polarized (Figs. 5,6), we then applied a 𝐇p field. Third, we fully saturated the TmIG film up (down) by applying a magnetic field 𝐻*=+100 (−100) Oe, and then decreased the magnetic field to 𝐻*=+20 (−20) Oe. Fourth and last, we nucleated skyrmion bubbles by applying current pulses 𝐽c of amplitude and length between 5\tand\t8×10OO A m-2 and 20 to 40 ns, respectively, resulting in an apparent random nucleation of bubbles. The amplitude, length, polarity, and number of the pulses was adjusted to obtain a low-concentration of skyrmion bubbles (i.e., less than 1 bubble per 5×5 µm2 device area). Bubble nucleation, however, is favoured at defect sites40,44 as well as influenced by the current-induced Oersted field. Whereas the latter favours bubble  22 nucleation at the side where the out-of-plane component of the Oersted field opposes 𝐻*, the presence of defects can be inferred from the preferred nucleation of skyrmion bubbles at certain positions of the device. Moreover, we observe that the skyrmions located at a few of those spots remain trapped upon the application of current pulses. Therefore, we excluded those skyrmion bubbles for the analysis of the skyrmion dynamics.  For the acquisition of the skyrmion trajectories, differential MOKE images were taken at a rate of ~1 frame/pulse while the pulses were continuously applied at a repetition rate of 2 Hz. The trajectories of the bubbles were automatically determined by using a tracking software that binarizes the images and determines the position of the bubbles’ center during the image sequence. The standard deviation of the blue distributions in Figs. 4b and 5b,d, which is about 50 nm, indicates that the position of the bubbles’ center can be determined with sub-100 nm resolution. However, we remark that the internal structure and shape of the bubble’s walls cannot be resolved at this scale due to the optical resolution of the system, which we estimate to be ~300 nm from measurements performed in a reference sample. For the evaluation of the skyrmion trajectories (Figs. 3-5), only mobile bubbles separated by at least 5 µm from neighbour bubbles were considered; in other words, we exclude from the analysis those bubbles that are permanently pinned at defect sites or that might interact with each other, as well as bubbles close to the edge of the track. For the statistical analysis, different skyrmion bubbles from different repeats and moving across different areas of the device were considered. Each data point in Fig. 3b corresponds to the average of over 10 independent bubbles with the deflection angle computed from the initial and final positions of the bubbles measured after a sequence of current pulses. All data points presented in Fig. 4c-f are extracted from the trajectory of a minimum of 50 and up to 250 skyrmion bubbles. Data taken for both polarities of 𝐉c were considered for the analysis presented in Fig. 4c-f. For the experiments of Figs. 5 and 6e-h, in addition to 𝐻*, a permanent magnetic field 𝐻p=±10 Oe is applied to keep 𝐌 saturated along. For fields up to 𝐻p~25 Oe, condition at which the bubble domains transform into stripe domains, the impact of 𝐇p on the skyrmion dynamics is found to be negligible compared to 𝐌, indicating that the ratchet effect arise from the exchange coupling with YIG. See Supplementary Note 9 for more details. Scanning NV magnetometry. The stray field maps of domain walls and skyrmions in YIG/TmIG/Pt were acquired with a custom-built nanoscale scanning diamond magnetometer (NSDM) microscope17. This technique is based on a single NV defect located at the apex of a diamond tip. By scanning the tip over the 𝑋𝑌 surface of the sample, one can sense the magnetic stray field 𝐵(𝑋,𝑌) emanating from the surface with high magnetic sensitivity and nanometer spatial resolution, from which the spin texture of domain walls and skyrmions can be determined17,51,52,53. Experiments were carried out in ambient  23 environment by employing a commercial monolithic diamond probe tip from QZabre Ltd. (www.qzabre.com). The NV center spin resonance was monitored by optically detected magnetic resonance (ODMR) spectroscopy54 using a nearby microwave antenna (~2.9 GHz) for spin excitation and fluorescence microscopy (520 nm excitation, 630–900 nm detection) for spin state readout. The orientation of the NV center and its stand-off distance with respect to the sample surface were characterized beforehand using a reference sample, with the latter determined to be 114±17 nm. Due to the relatively large stray field of the film, measurements were conducted at an additional distance of 100 nm from the surface. Although this prevented us to resolve local features below 100 nm resolution, Δ/0 can be reliably fitted because the domain wall profile extends over a distance that is several times larger than the domain wall width (Supplementary Eq. 1). Measurements in Fig. 2 were performed on isolated bubble domains nucleated via the application of current pulses in the presence of a magnetic field 𝐻=−20 Oe. The straight domain in Supplementary Fig. 6 was nucleated by decreasing the field to 𝐻=−15 Oe from a fully down magnetized state. The field remained applied during the whole data acquisition. The details of the data analysis and determination of the magnetic texture of the domain walls and skyrmions are presented in Supplementary Note 3. Skyrmion Hall angle and velocity evaluation. For the estimates of 𝜙9A and 𝑣9A, we use the Wangsness relation55 and set the magnetic moment and the Landé g factors of the tetrahedral/octahedral Fe3+ and the dodecahedral Tm3+ sublattices of TmIG to be 𝑀9,O=175 kA m-1 and 𝑀9,:=115 kA m-1, and 𝑔O=2.0\tand 𝑔:=7/6, respectively (see Refs. 56,57 and Supplementary Note 4). Accordingly, in our TmIG film QRSTQTUT=¡EF,¢𝒈𝟏−EF,¥𝒈𝟐§/¡EF,¢𝒈𝟏+EF,¥𝒈𝟐§≈−0.06. Based on previous reports17, and considering that the magnetic anisotropy of our film is lower, we estimate the domain wall width of TmIG to be ∆/0\t~\t50\tnm. This value is consistent with the wall width inferred from NV magnetometry measurements, which is estimated to be about 60 nm (Fig. 2 and Supplementary Note 3). The skyrmion velocity in the flow regime is given by38,45 𝑣9A=−OwO¨©¥𝜉/𝐽c𝛾«Z, where 𝜂=−QTUTQRSTYZ:X∆VW~0.8, 𝜉/ is the effective field per unit current density associated to the damping-like SOT, and 𝛾=𝑔®ħ, with 𝜇° the Bohr magneton, ħ the reduced Planck constant, and 𝑔=EF¡±F,¢𝒈𝟏N±F,¥𝒈𝟐§≈−5.4 in our TmIG film. According to previous reports of SOT efficiency in TmIG/Pt heterostructures16,31,58, and considering the thickness of our films, we estimate 𝜉/~2×10NO² T A-1 m2 in our devices. Accordingly, we estimate 𝑣9A\t~\t35 m/s for |𝐽c|=1×10OO A m-2. See Supplementary Note 5 for more details.  50. Kubota, M. et al. Systematic control of stress-induced anisotropy in pseudomorphic iron  24 garnet thin films. J. Magn. Magn. Mater. 339, 63–70 (2013). 51. Tetienne, J. P.-P. et al. The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry. Nat. Commun. 6, 6733 (2015). 52. Dovzhenko, Y. et al. Magnetostatic twists in room-temperature skyrmions explored by nitrogen-vacancy center spin texture reconstruction. Nat. Commun. 9, 2712 (2018). 53. Gross, I. et al. Skyrmion morphology in ultrathin magnetic films. Phys. Rev. Mater. 2, 024406 (2018). 54. Gruber, A. et al. Scanning Confocal Optical Microscopy and Magnetic Resonance on Single Defect Centers. Science 276, 2012–2014 (1997). 55. Wangsness, R. K. Sublattice efects in magnetic resonance. Phys. Rev. 91, 1085–1091 (1953). 56. Collet, M. et al. Generation of coherent spin-wave modes in yttrium iron garnet microdiscs by spin-orbit torque. Nat. Commun. 7, 10377 (2016). 57. Paulevé, J. Ferromagnetic Resonance of Gadolinium Garnet at 9300 МC/S. C. R. Acad. Sci. 244, 1908 (1957). 58. Ding, S. et al. Identifying the origin of the nonmonotonic thickness dependence of spin-orbit torque and interfacial Dzyaloshinskii-Moriya interaction in a ferrimagnetic insulator heterostructure. Phys. Rev. B 102, 054425 (2020).  25 Additional Information Supplementary Information is available for this paper at https://doi.or/XXXX (to be inserted by the publisher).   26 Extended Data Figures  \n Extended Data Figure 1 | Skyrmion bubble radius. a, Differential MOKE image of a representative skyrmion bubble for 𝐻*=−20 Oe. The white (dark) contrast indicates regions with 𝐦 of TmIG pointing up (down). Scale bar, 1 µm. b, Line profile of the MOKE intensity taken along the red dashed line in a (red solid line) together with its fitting (light grey) assuming that the skyrmion is a square box function having an ellipsoidal shape convoluted by a Gaussian function with standard deviation ~300 nm, which represents the spatial resolution of the MOKE set up49. The domain wall width, which is Δ/0~60 nm (Fig. 2), is neglected in the fitting procedure. The main diameters 𝑎 and 𝑏 of the skyrmion bubble are extracted by fitting the two orthogonal axes of the ellipsoid. From the fit in b we estimate 𝑎~1.2 µm. c, Skyrmion bubble radius 𝑅 as a function of 𝐻*. The radius is estimated as 𝑅=(𝑎+𝑏)/4. Each data point corresponds to the average value obtained from fitting over 10 independent skyrmion bubbles with circular shape (𝑏/𝑎≳0.9). The error bars represent the standard deviation of the measurements. See Supplementary Note 6 for discussion on the field dependence of the skyrmion bubble radius and bubble stabilization with 𝐻*. \n 27  \n Extended Data Figure 2 | Skyrmion ellipticity and orientation. a, Statistical analysis of the skyrmion ellipticity 𝑏/𝑎 from fitting over 1000 skyrmion bubbles assuming an ellipsoidal shape (see schematic). The histograms are extracted from analyzing the MOKE images of the bubble trajectories presented in Fig. 4a. b, Analysis of the orientation of the skyrmion ellipsoids from the data in a. The angle 𝛽 defines the orientation of the longest axis of the ellipsoid with respect to the current (see schematic). \n 28  \n Extended Data Figure 3 | 𝑱𝒙−𝒕𝐩 threshold conditions for skyrmion depinning (blue circles), heat dominated motion (red triangles), and random nucleation and annihilation of skyrmion bubbles in TmIG (black squares). |𝐻*|=20 Oe and the YIG layer is demagnetized. No difference was observed between 𝑄=+1 and −1 skyrmion bubbles. Supplementary Note 9 presents the current threshold for skyrmion depinning in the presence of 𝐇p, i.e., with 𝐌 controlled with an in-plane magnetic field. All measurements of the skyrmion dynamics presented in this work were performed for 𝑡q,|𝐽c| conditions comprised between the curves defined by the blue circles and the red triangles. Above the threshold defined by the red triangles, the mean displacements ∆𝑥````,∆𝑦```` abruptly drop, indicating that the skyrmion dynamics are dominated by Joule heating induced random skyrmion motion rather than by SOTs. We attribute this behavior to the exponential increase of the skyrmion diffusivity with temperature, which is expected in materials with low disorder and low damping such as TmIG9. \n 29  \n Extended Data Figure 4 | Current-induced temperature increase. a, Schematic of the experimental setup. The voltage output of the pulse generator is applied through the device and both the voltage drop at the device and the pulse are monitored with an Oscilloscope with an internal impedance of 50 Ω. From these measurements, we can precisely determine the evolution of the sample resistance during the current pulse. b, Increase of the sample resistance during the application of a current pulse 𝑡q=100 ns (left axis). The pulse starts at 𝑡=0. The colour indicates different set currents computed from the base resistance of the device 𝑅7(295\tK)=1130 W. The right axis shows the increase of temperature calculated from calibration measurements. We estimate the threshold for heat dominated motion for temperature increases of about 20 K. \n 30  \n Extended Data Figure 5 | Mean ∆𝐱```` and ∆𝐲```` displacements for YIG demagnetized. a, b, Average bubble displacements per pulse ∆𝑥```` and ∆𝑦```` as a function of 𝑡q computed considering all pulse events, i.e., including those that did not lead to bubble displacements. ∆𝑥```` and ∆𝑦```` increase linearly with the pulse length, exhibiting a finite value as 𝑡q→0. The difference between ∆𝑥````,\t∆𝑦```` and 𝑥7,𝑦7 (Fig. 4c,d) arises from the decrease of the bubble depinning probability when decreasing 𝑡q. Different symbols indicate different current densities. 𝐻*=−20 Oe (𝑄=+1). YIG is demagnetized.    \n 31  \n Extended Data Figure 6 | Pulse length dependence of 𝒙𝟎,𝒚𝟎 with 𝐌𝐘𝐈𝐆 along −𝐲. a, b, Mean displacement values 𝑥7,𝑦7 extracted from the trajectory of several skyrmion bubbles with 𝐌 along −𝐲 (see Fig. 4a,b for details regarding the analysis). Data taken for 𝐻*=−20 Oe (𝑄=+1) and for both polarities of 𝐉c (indicated by an arrow). The sign of 𝑥7,𝑦7 corresponds to the sign of 𝐉c. Different colours indicate the current density. The error bars are the standard errors of 𝑥7, 𝑦7 calculated from the variance of these magnitudes to the double Gaussian distribution of 𝛿𝑥,𝛿𝑦.  As for the case of YIG demagnetized (Fig. 4c,d), 𝑥7 and 𝑦7 tend to finite values when 𝑡q→0. Remarkably, the 𝑥7,𝑦7(𝑡q→0) values are similar for both directions of 𝐉| and similar to the ones measured for YIG demagnetized. As 𝑡q increases, 𝑥7 and 𝑦7 start to increase from a pulse length threshold value that depends on the amplitude of the current. Larger (smaller) current densities are required for driving skyrmion bubbles with 𝐌 pointing to −𝒚 and 𝐽|<0 (𝐽|>0), which is in agreement with the ratchet effect reported in Figs. 5 and 6 and Supplementary Note 10. c, Velocity of the skyrmion bubbles computed as |𝑣7|=w𝑥7:+𝑦7:/𝑡q from the data shown in a and b. The sign of the velocity is defined by the sign of 𝑥7; the error bars are computed by error propagation. As observed for the mean bubble velocity 𝑣̅9A for YIG demagnetized (Fig. 4e), 𝑣7increase as 𝑡q reduces (note that 𝑣7(𝑡q) exhibits a stepper increase than 𝑣̅9A(𝑡q) when reducing 𝑡q). \n 32  Extended Data Figure 7 | Skyrmion trajectories with 𝐌𝐘𝐈𝐆 collinear to 𝐉𝒙. a, Skyrmion trajectories with 𝑄=−1 and YIG demagnetized (see inset’s schematic). Data taken in a different device from the same YIG/TmIG/Pt heterostructure. 𝐻*=+20 Oe, 𝐽c=3.5\t×\t10OO A m-2, and 𝑡q=20 ns. The measurement protocol is the same employed in Fig. 4a. b, c, Skyrmion trajectories for 𝐌 pointing to +𝒙 and −𝒙, respectively. |𝐻c|=10 Oe. In contrast to the difference observed between 𝐌 parallel to +𝒚 and −𝒚\t(Figs. 5a and 5b), the skyrmion dynamics for 𝐌 collinear with the current is, within the error, independent on the direction of 𝐌. No clear differences between 𝐌∥\t±𝒙 (b,c) and the demagnetized case (a) can be identified either.      \n 1 Supplementary Information for Current-driven dynamics and ratchet effect of skyrmion bubbles in a ferrimagnetic insulator Saül Vélez1,5,*, Sandra Ruiz-Gómez2,3,6, Jakob Schaab1, Elzbieta Gradauskaite1, Martin S. Wörnle4, Pol Welter4, Benjamin J. Jacot1, Christian L. Degen4, Morgan Trassin1, Manfred Fiebig1 & Pietro Gambardella1,* 1 Department of Materials, ETH Zurich, 8093 Zurich, Switzerland 2 Departamento de Física de Materiales, Universidad Complutense de Madrid, 28040 Madrid, Spain 3 Alba Synchrotron Light Facility, 08290, Cerdanyola del Valles, Barcelona, Spain 4 Department of Physics, ETH Zurich, 8093 Zurich, Switzerland 5 Present address: Condensed Matter Physics Center (IFIMAC), Instituto Nicolás Cabrera, and Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain 6 Present address: Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany *e-mail: saul.velez@uam.es (S.V.); pietro.gambardella@mat.ethz.ch (P.G.)  Table of contents: • Supplementary Note 1. Structural and topographic characterization of YIG/TmIG and YIG/TmIG/Pt • Supplementary Note 2. Magnetic characterization of YIG/TmIG • Supplementary Note 3. Chirality determination via nitrogen-vacancy magnetometry • Supplementary Note 4. Determination of the effective 𝑔 factor of TmIG • Supplementary Note 5. Thiele’s equation of a ferrimagnet and skyrmion velocity • Supplementary Note 6. Magnetic field dependence of the skyrmion radius • Supplementary Note 7. Skyrmion deformations due to pinning • Supplementary Note 8. Alternative explanations for the pulse length dependence: inertia and automation effects • Supplementary Note 9: Influence of the Oersted field and 𝐻# on the skyrmion dynamics • Supplementary Note 10. Skyrmion ratchet effect: supplementary data  2 Supplementary Note 1. Structural and topographic characterization of YIG/TmIG and YIG/TmIG/Pt \n Supplementary Figure 1 | Structural characterization of the YIG/TmIG films. a, Symmetric X-ray diffraction scan of the GSGG/YIG(10 nm)/TmIG(20 nm) sample investigated in this work. As the X-ray response of a 10-nm-thick YIG is relatively weak1, the signal in YIG/TmIG is dominated by the TmIG (444) diffraction peak and corresponding Laue oscillations. The peak shifts towards higher angles with respect to the bulk value2 (dashed grey line) because of a reduction of the out-of-plane lattice constant due to tensile strain, in agreement with previous reports of TmIG films grown on GSGG3,4. b, Reciprocal space maps of the same sample around the GSGG (486) substrate peak (𝑞% and 𝑞& are the in-plane and out-of-plane wavevectors along the crystal axes indicated). The in-plane lattice constants of the films and the substrate along the [1 -1 0] direction coincide, confirming full epitaxy. The colour code indicates the intensity of the diffraction peaks, with red (blue) corresponding to maximum (minimum) intensity. \n Supplementary Figure 2 | Topographic characterization of YIG/TmIG and YIG/TmIG/Pt. a, c, Atomic force microscopy (AFM) characterization of the surface topography of a bare YIG/TmIG and a Pt-capped YIG/TmIG heterostructure, respectively. The root mean square (RMS) roughness over a ~5×5 µm2 area is on the order of 0.182 and 0.184 nm, respectively. b, d, AFM line profiles along the blue lines indicated in a and c, respectively, confirming the low roughness of our YIG/TmIG and YIG/TmIG/Pt films. e, AFM scan of the sample studied in the main text in a region partially covered with Pt. f, Average topography scan of the Pt edge along the blue line indicated in e, showing that the step size is about 5.6 nm. Consequently, we estimate the etching of the TmIG thickness in ~0.5 nm. The RMS surface roughness is below 0.2 nm on both the etched and Pt covered regions.  \n 3 Supplementary Note 2. Magnetic characterization of YIG/TmIG The magnetic anisotropy and saturation magnetization of the YIG(10nm)/TmIG(20nm)/Pt sample investigated in this work have been characterized by a combination of superconducting quantum interference device (SQUID) magnetometry and spin Hall magnetoresistance (SMR) measurements3,5,6.  Magnetic anisotropy. Supplementary Figure 3 shows that the TmIG layer exhibits perpendicular magnetic anisotropy, with an effective anisotropy field 𝐻-\t~\t1.4 kOe, and that the YIG layer exhibits an easy-plane anisotropy with a part of the film rotating towards out-of-plane due to the exchange coupling with TmIG.  The smaller (larger) magnetic field required for saturating the magnetization in the out-of-plane (in-plane) configuration indicates that the dominant magnetic anisotropy of YIG/TmIG is out of plane (Supplementary Figs. 3a,3b). As SMR is only sensitive to the magnetic moments at the metal/insulator interface, the magnetic anisotropy of TmIG can be directly proven via transport measurements3,4. Supplementary Fig. 3c demonstrates that the magnetization of TmIG points out of plane at zero field, and gradually cants towards the plane as the in-plane field increases. From these measurements, we determine that the magnetic anisotropy of the bilayer is dominated by the perpendicular magnetic anisotropy of TmIG, which corresponds to an anisotropy field 𝐻-~1.4 kOe.  The small magnetic moment of the hysteresis loop for the in-plane measurement (~1.2\t×\t1023 A m2 at zero field, Supplementary Fig. 3b) relative to the saturation magnetic moment of the full heterostructure (~7.7\t×\t1023 A m2, Supplementary Fig. 3a) indicates that part of the YIG film magnetization lies in the plane of the film. This is not surprising, as YIG on GSGG is expected to exhibit in-plane anisotropy. From comparing the in-plane and out-of-plane data, we estimate that the magnetization of the first ~3 nm of YIG on GSGG lies in the plane of the film, while the rest gradually rotates towards out of the plane due to the exchange coupling with TmIG. The gradual increase of the magnetic moment with field above 𝐻~70 Oe, Supplementary Fig. 3b, is consistent with the in-plane magnetic field gradually canting the magnetic moments of both the exchange-coupled YIG and the TmIG layer towards the plane, eventually achieving full saturation at ~1.4 kOe (Supplementary Fig. 3c; the relatively large paramagnetic response of GSGG prevents us to extract the saturation field from SQUID measurements). Importantly, the magnetic jump observed around zero field for the in-plane configuration indicates that the bottom part of the YIG film can be oriented with relatively small in-plane fields (~5 Oe; see inset of Supplementary Fig. 3b). This allows for controlling the sign of the exchange field between YIG and TmIG, a result that is in agreement with the ratchet effect presented in Figs. 5 and 6 of the main text as well as Supplementary Notes 9 and 10. Saturation magnetization. From the measurements shown in Supplementary Fig. 3, we estimate that the saturation magnetic moment of YIG(10nm)/TmIG(20nm) is ~7.7\t×\t1023 A m2. Taking into account that the surface area of the films is ~26 mm2, and assuming that the saturation magnetization of the YIG film is 𝑀7(YIG)\t~\t175 kA m-1 (Ref. 7), we estimate the saturation magnetization of TmIG to be 𝑀7(TmIG)\t~\t60 kA m-1. Temperature dependence. The saturation magnetization reduces by about 18% when increasing the temperature from 300 to 350 K (Supplementary Fig. 4). We thus estimate that the TmIG magnetization  4 decreases by a maximum of 7% due to Joule heating in the current-induced skyrmion dynamics experiments (see Extended Data Fig. 4 for the analysis of the current-induced Joule heating).  \n Supplementary Figure 3 | Magnetic characterization of YIG(10nm)/TmIG(20nm). a, Magnetic moment of the heterostructure as a function of out-of-plane field 𝐻&. Both layers are fully saturated out of plane at 𝐻&\t~\t100 Oe. Note that the coercive field is smaller than the one shown in Fig. 1b of the manuscript. The larger coercive field in Fig. 1b is due to the pinning of domain walls at the device edges, resulting in a broadening of the hysteresis loop at the device area3. b, Magnetic moment as a function of in-plane field. Inset, magnification of the loop around zero field. The paramagnetic response of the GSGG substrate has been subtracted in a and b. c, Transverse SMR measurements as a function of in-plane field applied at an angle 𝛼 with respect to the current direction. From these measurements, we can extract the magnetic anisotropy of TmIG3,5,6. At 𝐻=0, 𝑅%#\t≈\t0, indicating that the local magnetic moments of TmIG point out of the plane and are mostly randomly oriented, which is consistent with the data in a and the bottom images of Fig. 1c,d of the manuscript. As 𝐻 increases, the magnetization of TmIG cants towards the plane, resulting in a change of the amplitude of 𝑅%#, which is maximum at 𝛼=45° (positive change) or 135° (negative change). The saturation of 𝑅%# above 𝐻-\t~\t1.4 kOe indicates that the magnetic moments of TmIG are saturated in-plane, thus identifying 𝐻- as the anisotropy field of TmIG. We remark that 𝐻- is about half the value measured for single-layer TmIG films of the same thickness on GSGG3,4, evidencing the role of the exchange coupling with YIG on the magnetic anisotropy of TmIG. Note that each measurement consists of superposed forward and backward field sweeps, indicating that the canting of the magnetic moments of TmIG does not exhibit hysteresis. A device-dependent constant offset is subtracted in c for convenience. \n Supplementary Figure 4. Temperature dependence of the out-of-plane magnetization. The data are normalized to the saturation magnetization 𝑀7 at 300 K. \n 5 Supplementary Note 3. Chirality determination via nitrogen-vacancy magnetometry We used nitrogen-vacancy (NV) scanning magnetometry to characterize isolated skyrmion bubbles. This technique measures the stray field 𝐵EF produced by the magnetic textures of the sample at the position of the NV center with high field and spatial resolution, from which the spin texture of domain walls and skyrmions can be determined3,8,9. Figures 2a,d of the manuscript and Supplementary Fig. 6a show representative stray field scanning maps 𝐵EF(𝑋,𝑌) of bubble domains located in YIG/TmIG/Pt and of a stripe domain intersecting a region partially covered by Pt. Skyrmion bubbles modelling. We model the skyrmion as a closed ring of ellipsoidal shape, arbitrary orientation 𝛽, diametral axes 𝑎 and\t𝑏, center 𝑋M,𝑌M, and wall width ∆OP. By adapting previous modelling of straight domain walls3 to the bubble case, the domain wall profile is described by 𝑀Q(𝑟)=𝑀Scos𝜓coshY𝑟−𝑟M∆OP[\t 𝑀\\(𝑟)=𝑀Ssin𝜓coshY𝑟−𝑟M∆OP[,                                                                    𝑀_(𝑟)=−𝑀StanhYb2bc∆de[,           (1) where 𝑟M indicate the position of the center of the wall for a given position along the wall ring of the bubble, 𝑟 the position in the direction perpendicular to the domain wall for the corresponding 𝑟M, and 𝜓 is the chiral angle that describes the magnetic texture of the wall. To fit the data, we normalized 𝐵EF(𝑋,𝑌) to the maximum value to remove the influence of YIG on the stray field of TmIG. The fitting procedure is done by first finding the best domain wall width ∆OP for a given domain wall type (Bloch, righ-handed Néel, and left-handed Néel), leaving 𝑎, 𝑏, and 𝛽 as free parameters. In a second step, ∆OP is fixed and 𝑎, 𝑏, and 𝛽 (if 𝑎≠𝑏) are fitted. As an example, Fig. 2b of the manuscript shows the best fit of the stray field data presented in Fig. 2a, which corresponds to a circular bubble with right-handed Néel chirality, ∆OP=60 nm, and 𝑎~𝑏=\t950 nm. The accuracy of the fitting is computed from the residual sum of squares (RSS) using following formula       ln\tℒj−ln\tℒk=−lk\tln\tmnnomnnp,          (2) where 1 and 2 indicate different sets of fit parameters with 2 being the one with smallest RSS, and 𝑛 the number of data points. Therefore, by using Supplementary Eq. (2) with different fit parameters, one can estimate the likelihood of a given type of domain wall. Supplementary Fig. 5 presents the likelihood of fitting the skyrmion data of Fig. 2a of the manuscript to different domain wall types characterized by 𝜓 and ∆OP. The likelihood plot clearly shows that the right-handed Néel chirality is the spin texture that best describes the skyrmions for wall widths in the range from  ∆OP\t~\t30 to ~\t100 nm, with best fit obtained with ∆OP= 60 nm. Taking into account the magnetic anisotropy of the film (Supplementary Note 2) and previous characterization of the domain wall width in TmIG3, we expect the domain wall width to be about 50 nm. We thus conclude that in YIG/TmIG/Pt the domain  6 wall texture is of right-handed Néel type. The same fitting procedure was followed for the deformed skyrmion presented in Fig. 2d-f of the manuscript.  \n Supplementary Figure 5 | Log-likelihood of the fits of the data of Fig. 2a computed using Supplementary Eq. (2). Different ∆OP values from 10 to 200 nm and different domain wall textures (Bloch, right-handed Néel, and left-handed Néel) were considered.  \n 7 Analysis of straight domain walls in YIG/TmIG/Pt and YIG/TmIG. To infer the contribution of the YIG/TmIG interface to the DMI, we performed measurements of the stray field of a narrow stripe domain running across a region partially covered by Pt (Supplementary Fig. 6a). Direct inspection reveals that the stray field in the Pt-covered region is stronger than in the Pt-free region (Supplementary Fig. 6b). However, as we demonstrated in an earlier work, Pt also contributes to the stray field due to the magnetization induced by proximity with TmIG3. Therefore, for comparing the data, we subtracted the stray field associated to the Pt polarization in the YIG/TmIG/Pt region. The normalized line scans are very similar (Supplementary Fig. 6c), indicating that the domain wall type in YIG/TmIG is also right-handed Néel (as determined to be for YIG/TmIG/Pt; Fig. 2 and Supplementary Fig. 5) or of intermediate right-handed Néel-Bloch favored by a positive DMI induced by the YIG interface. Note that changes in ∆OP between the Pt-capped and Pt-free regions are expected to be negligible10, and thus no significant influence of ∆OP on the stray field is expected. The relatively large uncertainty of the stray field data, however, does not allow us to conclude on the precise value of 𝜓. Nevertheless, the right-handed chirality in GSGG/YIG/TmIG is clearly different from that of TmIG directly grown on GSGG, which presents negative DMI and left-handed Néel domain walls3. As the TmIG/Pt interface has a positive DMI3, we conclude that both YIG and Pt interfaces contribute to stabilize right-handed Néel domain walls and skyrmions in TmIG with an overall DMI strength above 𝐷uv.  \n Supplementary Figure 6 | Comparison of the stray field of a stripe domain in a region partially covered by Pt. a, Stray field map 𝐵EF(𝑋𝑌) of a stripe domain running across a region partially covered with Pt. Note that the stray field arises from two parallel domain walls. |𝐻x|=15 Oe. b, Line scans of 𝐵EF along the dashed lines in a (open symbols; the colour code identifies the line scan). c, Same as in b with the stray field in the Pt region corrected by the contribution of the Pt polarization to the stray field3 (solid black symbols).\n 8 Supplementary Note 4. Determination of the effective 𝑔 factor of TmIG To estimate the effective g factor of TmIG we used the Wangsness relation11,12  yz{|z{−y}~|}~=y|,                (3) where 𝑀 and 𝑔 are the magnetic moment and g factor of the Fe3+ tetrahedral/octahedral sublattices,\t𝑀 and 𝑔 the magnetic moment and g factor of the Tm3+ dodecahedral sublattice, and 𝑀7=𝑀−𝑀 and 𝑔 the net magnetic moment and effective g factor of TmIG. The negative sign accounts for the antiferromagnetic coupling between the Fe3+ and Tm3+ sublattices with both 𝑀 and 𝑀 defined positive and 𝑀>𝑀. We take 𝑔=2 (Ref. 7) and estimate 𝑔\tfrom the expected 𝑔 factor of a free Tm+3 ion. At the lowest spin-orbit multiplet state, the total angular momentum of Tm3+ is 𝐽=6 with an orbital momentum 𝐿=1 and spin state 𝑆=5, resulting in 𝑔=7/6. The saturation magnetization of TmIG is estimated to be 𝑀7\t~\t60 kA m-1 (see Supplementary Note 2). In thin films, 𝑀 and 𝑀 may substantially deviate from the bulk values due to strain and finite size effects1,3,7,13, giving a wide range of possible 𝑀,𝑀 values for the solution of Supplementary Eq. (3). Supplementary Fig. 7 shows the value of 𝑔 computed by using Supplementary Eq. (3) and considering different combinations of 𝑀 and 𝑀 values. 𝑔 and 𝑔 are constrained to be 2 and 7/6, respectively. The dashed line indicates combinations with 𝑀−𝑀=\t60 kA m-1. The bluish-coloured area corresponds to solutions with negative\t𝑔 values, which is the case expected for our TmIG film according to the sign of the skyrmion Hall effect (Fig. 3 of the main text). By fixing 𝑀 to be 175 kA m-1 (Ref. 7), we estimate 𝑀~115 kA m-1 and 𝑔\t~−5.4 (solution indicated by a blue dot in Supplementary Fig. 7). \n Supplementary Figure 7 | Computed 𝒈 values of TmIG for different 𝑴𝐅𝐞 and 𝑴𝐓𝐦 combinations. Computed 𝑔 values by using Supplementary Eq. (3) and 𝑔=2 and 𝑔=7/6. The dashed line indicates 𝑀,𝑀 combinations with constant 𝑀7=60 kA m-1. The blueish region corresponds to solutions with 𝑔<0. The blue dot indicates the combination 𝑀=175 kA m-1 and 𝑀=115 kA m-1, which results in 𝑔=−5.4. See text for more details. \n 9 Supplementary Note 5. Thiele’s equation of a ferrimagnet and skyrmion velocity  Thiele’s equation. Under the approximation of point-like massless objects, the dynamics of skyrmions driven by current pulses is described by the modified Thiele’s equation14,15,16          𝐆×𝐯7-−𝛼𝓓𝐯7-+𝐅n=0,          (4) where 𝐆=𝐺𝒛 is the gyromagnetic vector, 𝐯7-=𝑣%𝒙+𝑣#𝒚 the skyrmion velocity, 𝛼 the damping parameter, 𝓓 the dissipative tensor, and 𝐅n=𝐹n𝒙 the SOT driving force generated by 𝐉%=𝐽%𝒙 (see Supplementary Fig. 8 for the schematics of the forces). 𝒙, 𝒚, and 𝒛 are unit vectors along the 𝑥, 𝑦, and 𝑧 directions, respectively. The gyromagnetic vector is given by 𝐺=−4𝜋y¢£\t𝑄,               (5) where 𝑀7, 𝑡, and 𝛾=𝑔§¨ħ are the saturation magnetization, thickness, and gyromagnetic factor of the magnetic layer, 𝑔 the Landé g factor, and 𝑄=jª«∬Y®𝐦®%×®𝐦®#[∙𝐦°𝑑𝑥𝑑𝑦 the topological charge with 𝐦(𝑥,𝑦) the magnetic moment at position (𝑥,𝑦). 𝜇³ and ħ are the Bohr magneton and the reduced Planck constant. Gyromagnetic vector in a ferrimagnet. In a ferrimagnet such as TmIG, the effective 𝛾 can be computed from the saturation magnetization 𝑀7,´ and the gyromagnetic factors 𝛾µ of the constituent sublattices (see Supplementary Eq. (3)), leading to11,17  𝛾=y¶·,o¸o2·,p¸p .            (6) In our TmIG films 𝛾<0 (see Supplementary Note 4), resulting in 𝐺>0 for 𝑄=+1 skyrmions. Note that Supplementary Eq. (5) can be rewritten as 𝐺=4𝜋𝑡𝑠ºu𝑄 with 𝑠ºu=−»y,o𝜸𝟏−y,p𝜸𝟐¿=−y𝜸. Skyrmion Hall angle. The skyrmion deflection angle 𝜙7- induced by the Magnus force 𝐆×𝐯7- is given by14,16,18 𝜙7-~tan2jYÁÂÃ𝒟[,              (7) where the components of the dissipative tensor 𝓓 are given by14,16 𝒟µÅ=−𝑠ºu𝑡∬®𝐦®µ∙®𝐦®Å°𝑑𝑥𝑑𝑦, with 𝒟′jk=𝒟′kj=0 and \t𝒟′jj=𝒟′kk≈−𝑠ºu𝑡k«Ç∆de under the approximation 𝑅≫∆OP, with 𝑅 and ∆OP being the radius of and domain wall with of the skyrmion bubble19,20. 𝛼É is the effective damping and is given by17 𝛼É=𝛼SÊËÊSÌ{Ê ,            (8) with 𝑠uÍu=−»y,o𝜸𝟏+y,p𝜸𝟐¿ the total angular momentum. By rewriting Supplementary Eq. (7) using the relations given above we obtain 𝜙7-~tan2jY−SÌ{ÊSÊËÊkÎ∆ÂÇ[ as given in Eq. (1) of the main text. Note that  10 because SÌ{ÊSÊËÊ is negative in TmIG, 𝜙7- is positive (negative) for 𝑄=+1(−1) skyrmion bubbles as opposed to skyrmions in ferromagnets (see Supplementary Fig. 8 and Fig. 3 of the main text). Skyrmion velocity. The skyrmion velocity 𝒗7-=𝑣%𝒙+𝑣#𝒚 in the flow regime is given by16,21  𝑣%=ÐjÑÐpÒÓÔ}Á, and 𝑣#=jjÑÐpÒÓÔ}Á,            (9) and hence, 𝑣7-=|𝒗7-|=jÕjÑÐpÒÓÔ}Á ,        (10) where  𝜂=\tÂÃ×Á=−SÊËÊSÌ{ÊÂÇk∆deÎ   and   ÒÓÔ}Á=−𝜉OÙ𝐽%𝛾«Çª,       (11) with 𝜉OÙ the effective field (per unit current density) associated to the damping-like SOT. Considering the values of 𝜉OÙ reported for TmIG/Pt (Refs. 4,22), we estimate a skyrmion mobility 𝜂=𝑣7-/𝐽% exceeding 3×1023 m3 A-1 s-1 in our TmIG devices, a value that is comparable to the mobility of ferrimagnetic domain walls near the angular momentum compensation3,17,22,23,24.  \n Supplementary Figure 8 | Schematic of the skyrmion Hall effect in TmIG. Schematics of 𝑄=−1 and 𝑄=+1 right-handed Néel skyrmion bubbles in TmIG and of the forces acting on the skyrmions due to the application of current pulses. The coordinate system and the direction of the current (𝐉%), the skyrmion velocity (𝐯7-), the sign of the deflection angle (𝜙7-), and the forces acting on the skyrmions (see Supplementary Eq. (4)) are indicated. The vectors indicate the direction of the magnetic moments. As the sign of 𝐆 is positive (negative) for 𝑄=+1\t(−1) skyrmion bubbles, the sign of 𝜙7- in TmIG is opposite to the one encountered in ferromagnetic materials. \n 11 Supplementary Note 6. Magnetic field dependence of the skyrmion radius.  The average skyrmion radius is analyzed from MOKE measurements taken on isolated bubbles and for different values of 𝐻& (Extended Data Fig. 1). We note that the average radius inferred from the MOKE analysis (Extended Data Fig. 1c) is consistent with the radius extracted from NV magnetometry measurements (Fig. 2a-c of the manuscript). Both the increase of the bubble radius (Extended Data Fig. 1c) and the destabilization of the bubble domains into stripe domains when decreasing |𝐻&| (Fig. 1d of the manuscript) are a consequence of the minimization of the magnetostatic energy of the bubble with field25. For fields below |𝐻&|≲10 Oe we cannot find only skyrmion bubbles by either current or field sweep protocols as the bubbles tend to expand and transform into stripe domains.  12 Supplementary Note 7. Skyrmion deformations due to pinning  Skyrmion ellipticity. The statistical analysis of the bubbles’ shape extracted from MOKE data show that most skyrmion bubbles present an ellipticity 𝑏/𝑎≳0.9 (Extended Data Figs. 1a and 2a), indicating that the bubbles tend to retain a circular shape after a current pulse. 𝑎 and 𝑏 define the larger and smaller axes of the ellipsoid. Such ellipsoidal deformations correspond to relative contractions/elongations of the skyrmion diameter of about 50±50 nm between each other. These results are consistent with an average distance between pinning centers of about 50 to 100 nm if the deformations are assumed to arise from the hopping of the skyrmion wall between two adjacent defects. Current-induced bubble deformations. We found a correlation between the most preferred direction of bubble deformation relative to the direction of the current pulses. Concretely, we found that the bubbles exhibit a larger probability to exhibit deformation in the direction of motion as well as perpendicular to it (Extended Data Fig. 2b). Whereas the former indicates that the deformations correlate with the direction of skyrmion motion set by the skyrmion Hall effect and pinning20,26,27, the latter is consistent with bubble distortions induced by SOTs28.        13 Supplementary Note 8. Alternative explanations for the pulse length dependence: inertial and automotion effects   Inertial effects. A possible explanation for the finite bubble displacements observed as 𝑡Ý→0 (Fig. 4c,d of the main text as well as Extended Data Figs. 5 and 6) is that the skyrmion bubbles behave as objects with a finite mass and thus keep moving after the end of a current pulse. This effect would be detected in MOKE experiments if there is an asymmetry in the acceleration and deceleration times of the skyrmion bubbles, as reported for Néel domain walls in low-damping media and moderate DMI29. However, such inertia effects are expected to emerge for domain walls and skyrmions driven in the flow regime. In the presence of random hopping produced by disorder and thermal fluctuations, as observed in our experiments for skyrmion bubbles driven in the creep regime (Figs. 4 and 5 of the main text), we expect that the inertia effects would be negligible.   Automotion. Another explanation for the pulse length dependence of ∆𝑥ßßßß,∆𝑦ßßßß and 𝑣̅7- (Fig. 4c-e of the main text) is that the skyrmion bubbles exhibit automation effects as the result of the displacement of vertical Bloch lines around the bubble boundary, in analogy with the behaviour of magnetic bubbles in a field gradient observed in thick garnet layers16,30,31. The displacement of vertical Bloch lines would be driven by the reversal of 𝐌âãä induced by the in-plane Oersted fields generated by the pulses (𝐇,æ∝+𝐽è𝐲 at the YIG plane), and therefore inertia effects may only emerge for one polarity of 𝐉è for a given 𝐌âãä||𝒚 configuration. This scenario, however, is ruled out because a similar behaviour is observed for all orientations of 𝐌âãä and 𝐉è. Extended Data Figure 6 shows representative data taken for 𝐌âãä=−𝑀âãä𝐲, revealing finite displacements for both polarities of 𝐉è as 𝑡Ý→0. In addition, analysis of the current threshold for bubble depinning as function of 𝐉è amplitude, in-plane field 𝐻æ, and pulse length 𝑡Ý suggests that the in-plane Oersted fields are not capable to produce significant changes to 𝐌âãä (Supplementary Note 9), which we ascribe to the relatively large thickness of TmIG and the moderate current densities employed in the experiments.   14 Supplementary Note 9: Influence of the Oersted field and 𝑯𝒚 on the skyrmion dynamics As demonstrated in Figs. 5 and 6 of the manuscript, both the skyrmion depinning probability and the velocity of the skyrmion bubbles strongly depend on the orientation of 𝐌âãä relative to 𝐉%. To control the orientation of 𝐌âãä, a small in-plane magnetic field 𝐇#=𝐻#𝒚 is applied (see Supplementary Note 2 for more details regarding the magnetic properties of the films; in particular, Supplementary Fig. 3b shows that an in-plane field as small as 3 Oe can significantly modify 𝐌âãä). It is therefore crucial to determine whether the asymmetry in the dynamics of the skyrmion bubbles with 𝐉è may arise from the influence of the in-plane Oersted field 𝐇,#=𝐻,#𝒚∝+𝐽è𝐲 generated by the pulses on 𝐌âãä, as well as to determine whether 𝐇# itself influence the dynamics. To investigate these questions, we determined the current threshold 𝐽%uv for skyrmion depinning as function of 𝐉è orientation, pulse length 𝑡Ý, and 𝐻# strength. Supplementary Figure 9 shows representative data taken for 𝐻#<0 and 𝑄=+1 (𝐻&=−20 Oe). At magnetic fields below ë𝐻#ë~4 Oe, the current threshold is rather independent on the magnetic field and the polarity of the current, but above that field value, a strong asymmetry with 𝐉% emerges, with the current threshold becoming larger for 𝐽% < 0, an asymmetry that is in agreement with the results presented in Fig. 6e,f of the main text. Moreover, when reversing the direction of 𝐻# (not shown here), we observe that the current threshold becomes larger for 𝐽% > 0, while it stays rather constant for 𝐽% < 0, also in agreement with the asymmetry with 𝐉è for 𝐌âãä=+𝑀âãä𝒚 presented in Fig. 6g,h of the main text (see Supplementary Note 10 for more details regarding the asymmetries of the ratchet effect). We note that the results shown in Supplementary Fig. 9 are not dependent on the particular sequence followed with the magnetic field 𝐇# before starting the measurements, suggesting that the application of current pulses randomize the domains towards the equilibrium configuration set by the external field for values in the range ë𝐻#ë<−10 Oe. The rather constant 𝐽%uv with 𝐻#<0 observed for 𝐽%>0 (Supplementary Fig. 9b) is attributed to the fact that no significant difference should be observed in 𝐽%uv between a demagnetized case (𝐻#=0 Oe) and a saturated one along the favoured 𝐌âãä direction. That is because a demagnetized case presents domains with both favoured and unfavoured 𝐌âãä\torientations as well as intermediate ones aligned with the current. The average skyrmion velocity for 𝐽%>0, however, increases when 𝐌âãä saturates along −𝒚, in agreement with the data shown in Figs. 4a,b and 5a,b of the main text. Further, the velocity is found to be weakly dependent on the external field for fields from ë𝐻#ë\t~−10 Oe to ~−25 Oe, the later defining the threshold for the destabilization of the bubble domains into stripe domains.  We now focus our attention on the field dependence of the current threshold for 𝐽%<0 and\t𝐻#<0 (Supplementary Fig. 9a). Remarkably, while the depinning current density increases by a factor ~\t4 when decreasing the pulse length from 100 to 15 ns (blue up triangles and green down triangles, respectively; note that the 𝑡Ý-dependence of 𝐽%uv is in agreement with the data presented in Extended Data Fig. 3), the field-dependence remains qualitatively the same, with a field-independent regime  15 observed from ë𝐻#ë\t~ 10 to 20 Oe. The fact that the field independent regime is reached at the same value for all current conditions (despite the associated 𝐻,# field differs by a factor ~4 between the extreme cases: 𝐽%\t~\t8.5\t×\t10jj A m-2 for 𝑡Ý=15 ns, while 𝐽%\t~\t2.2\t×\t10jj A m-2 for 𝑡Ý=100 ns), indicates that the influence of 𝐇,# on 𝐌âãä is negligible and that the magnetization of the YIG film is saturated above ë𝐻#ë=𝐻S=10 Oe (indicated by a vertical line), further indicating that 𝐻# has a negligible effect on the skyrmion dynamics compared to 𝐌âãä. We thus conclude that the ratchet effect arises from the exchange coupling of 𝐌âãä with the skyrmions in TmIG.  \n Supplementary Figure 9 | Current threshold for the depinning of skyrmion bubbles as function of 𝐉𝐱, 𝒕𝐩, and 𝑯𝐲. a, Current threshold as function of 𝐻# for different pulse lengths. The polarity of the magnetic field and the current applied are 𝐻#<0 and 𝐽%<0, which corresponds to the unfavoured configuration for bubble motion. Same field dependence is observed for 𝐻#>0 and 𝐽%>0. The dashed lines are guides to the eye. A field-independent regime is reached at 𝐻7\t~\t10 Oe for all 𝑡Ý,|𝐽%| conditions (indicated with a vertical solid line). b, Same as a, but for 𝐽%>0 and 𝐻#<0, i.e., a favoured configuration for skyrmion motion. Same data is obtained for 𝐽%<0 and 𝐻#>0. The solid lines are guides to the eye. Data in a and b correspond to 𝑄=+1 skyrmion bubbles with 𝐻&=−20 Oe. Same behaviour for the current depinning threshold is observed for 𝑄=−1 skyrmion bubbles and 𝐻&=+20 Oe. The inset indicates the orientation of 𝐉% and 𝐇# relative to the current line. See Supplementary Note 10 and Figs. 5 and 6 of the main text for more details regarding the symmetries of the skyrmion dynamics with 𝐌âãä, 𝐉%, and 𝑄.  \n 16 Supplementary Note 10. Skyrmion ratchet effect: supplementary data  \n Supplementary Table 1 | Skyrmion dynamics with 𝐌𝐘𝐈𝐆, 𝐉𝒙, and 𝑸: symmetry of the ratchet effect. Comparison of the skyrmion dynamics for different 𝐌âãä, 𝐉%, and 𝑄 configurations relative to the YIG demagnetized case. The vectors 𝒙 and 𝒚 indicate the orientation of the vectors 𝐌âãä, 𝐉%, 𝐓OÙ, and the ratchet effect, and the sign + or – their polarity (see Fig. 1a of the main text for the definition of the sample coordinates). Note that the polarity of 𝐓OÙ is given by 𝐉% and that here we only consider the sign of the 𝒚 component at the bubbles wall as is the relevant one for the ratchet effect. Further, we only consider configuration with 𝐌âãä orthogonal to 𝐉% as no asymmetry in the dynamics is observed when 𝐌âãä and 𝐉% are collinear (Extended Data Fig. 7). When the polarity of 𝐉𝒙(𝒙) and 𝐌âãä(𝒚) are the same, the dynamics of the skyrmions are slow or pinned (combinations indicated by orange colour), but when they are opposite, the skyrmion motion is efficient and faster (indicated by green colour) relative to the demagnetized case. See also Figs. 5 and 6 of the main text, which present representative data of the dynamics of the skyrmion bubbles with 𝐉𝒙(𝒙) and 𝐌âãä(𝒚). For a given 𝐌âãä(𝒚) orientation, the asymmetry in the skyrmion dynamics with 𝐉𝒙(𝒙) leads to the ratchet effect indicated in the last column, with the bubble motion being preferred towards +𝒙 for 𝐌âãä aligned to −𝒚, while motion is preferred towards −𝒙 for 𝐌âãä saturated along +𝒚 (note that here the deflection of the skyrmion bubbles towards ±𝒚 due to the topological Hall effect is not considered for simplicity). The same asymmetric motion is observed for both\t𝑄=+1 and −1 skyrmion bubbles. As schematized in Fig. 6a-d of the main text, the asymmetric dynamics with 𝐉% originates from the distortion of the magnetic configuration of the skyrmion bubbles produced by 𝐓OÙ\t(which\tdepends\ton\t𝐉%) relative to the one induced by the exchange coupling with 𝐌âãä. When the distortions oppose (favour) each other, the net distortion towards 𝒚 becomes smaller (larger), resulting in stronger (weaker) 𝐅n driving forces. See also Supplementary Fig. 10, which provide additional sketches of the expected distortion of the skyrmion bubbles for other 𝐌âãä, 𝐉è, and 𝑄 configurations than the ones presented in Fig. 6a-d of the main text.  \n 17  \n Supplementary Figure 10 | Schematics of the magnetic distortion of the skyrmion bubbles with 𝐌âãä and 𝐉%: additional 𝐌âãä and 𝑸 configurations. a-d, Same as Fig. 6a-d of the main text, but for 𝐌âãä=+𝑀âãä𝒚. As opposed to the case of Fig. 6a-d, the distortion of the skyrmion bubble is enhanced (reduced) for 𝐽%>0 (𝐽%<0). Consequently, for 𝐌âãä=+𝑀âãä𝒚 the skyrmion motion is more efficient for 𝐽%<0 than for 𝐽%>0, in agreement with the results presented in Fig. 6g,h of the main text. e-h, Same as a-d, but for a 𝑄=−1 skyrmion bubble. The symmetry of the resulting torques with 𝐉% is the same as for 𝑄=+1 skyrmion bubbles, i.e., larger (smaller) distortions are observed for 𝐽%>0 (𝐽%<0). See c,d and g,h. In the later, only the direction of the 𝒚 component of the 𝑑𝐦OP/𝑑𝑡 induced by the torques is depicted for simplicity. Fast (slow) motion is expected when 𝑑𝐦OP/𝑑𝑡(𝒚) opposes 𝐌âãä.    \n 18  \n Supplementary Figure 11 | Ratchet effect for stripe domains. a, Top, Differential MOKE image showing stripe domains in TmIG. The white (dark) contrast indicates domains with the magnetization pointing up (down). An in-plane magnetic field 𝐻#=+30 Oe is applied along +𝒚 to stabilize stripe domains oriented perpendicular to the current line. 𝐻&=−10 Oe. Bottom, Schematics indicating the position of the current line in the MOKE image. When applying current pulses 𝐉%, as for the case of skyrmion bubbles with 𝐌âãä=+𝑀âãä𝒚, we observe that the dynamics of the stripe domains is more efficient for 𝐽%<0 than for 𝐽%>0 (see Fig. 6g,h of the main text). When the orientation of 𝐇# is reversed, the asymmetry in the dynamics of the stripe domains with 𝐉% is also reversed, agreeing with the asymmetry with 𝐌âãä (which is set by 𝐇#) and 𝐉% as summarized in Supplementary Table 1. The explanation of the effect is the same as for the case of skyrmion bubbles. The motion of the stripe domains is more (less) efficient when the distortion of the magnetic moments at the stripe’s walls by 𝐌âãä and 𝐉% is minimal (maximal). See sketches from b to e. The bubble domains surrounding the current line are induced by the out of plane component of the Oersted field for currents 𝐉%≳8×10jj A m-2 at this field values. b, Schematics of the expected orientation of the magnetic moments at the walls of the stripe domains in TmIG (as discussed in the main text, the domain walls should exhibit right-handed Néel chirality). c, Magnetic distortion of the walls due to 𝐌âãä=+𝑀âãä𝒚. d, e, Additional magnetic distortion due to 𝐉%, showing that a more efficient wall motion is expected for 𝐽%>0 (d) than for 𝐽%<0 (e).  \n 19 References 1. Mendil, J. et al. Magnetic properties and domain structure of ultrathin yttrium iron garnet/Pt bilayers. Phys. Rev. Mater. 3, 034403 (2019). 2. Espinosa, G. P. Crystal chemical study of the rare-earth iron garnets. J. Chem. Phys. 37, 2344–2347 (1962). 3. Vélez, S. et al. High-speed domain wall racetracks in a magnetic insulator. Nat. Commun. 10, 4750 (2019). 4. Avci, C. O. et al. 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Ferrimagnetic Skyrmions in Topological Insulator/Ferrimagnet Heterostructures. Adv. Mater. 32, 2003380 (2020). 26. Zeissler, K. et al. Diameter-independent skyrmion Hall angle observed in chiral magnetic multilayers. Nat. Commun. 11, 428 (2020). 27. Kim, J.-V. & Yoo, M.-W. Current-driven skyrmion dynamics in disordered films. Appl. Phys. Lett. 110, 132404 (2017). 28. Litzius, K. et al. The role of temperature and drive current in skyrmion dynamics. Nat. Electron. 3, 30–36 (2020). 29. Torrejon, J., Martinez, E. & Hayashi, M. Tunable inertia of chiral magnetic domain walls. Nat. Commun. 7, 13533 (2016). 30. Slonczewski, J. C., Malozemoff, A. P. & Voegeli, O. Statics and Dynamics of Bubbles Containing Bloch Lines. AIP Conf. Proc. 10, 458 (2008). 31. Tabor, W. J., Bobeck, A. H., Vella-Coleiro, G. P. & Rosencwaig, A. A New Type of Cylindrical Magnetic Domain ( Hard Bubble ). AIP Conf. Proc. 10, 442–457 (1973).  "    },    {        "title": "1806.06334v1.Skyrmion_Formation_Induced_by_Antiferromagnetic_enhanced_Interfacial_Dzyaloshinskii_Moriya_Interaction.pdf",        "content": "Skyrmion Formation Induced by Antiferromagnetic -enhanced Interfacial Dzyaloshinskii Moriya \nInteraction  \nAuthor : Marco Chung Ting Ma1, Yunkun Xie2, Howard Sheng3, S. Joseph Poon1, and Avik Ghosh2  \n1Department of Physics , University of Virginia, Charlottesville, Virginia 22904 USA  \n2Department of Electrical and Computer Engineering , University of Virginia, Charlottesville, \nVirginia 22904 USA  \n3Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22 030 USA  \nAbstract  \nNeél skyrmions originate  from interfacial Dzyaloshinskii Moriya interaction  (DMI) . Recent \nstudies have explored using ferromagnet to host Neél skyrmions for device applications.  \nHowever, challenges remain  to reduce the size of skyrmion to near 10 nm. Amorphous rare -\nearth -transitional -metal ferrimagnet s are  attractive alternative  material s to obtain  ultrasmall  \nskyrmion s at room temperature . Their  intrinsic  perpendicular magnetic  anisotropy and tunable \nmagne tization provides a favorable environment  for skyrmion stability. In this work , we employ  \natomistic stochastic  Landau -Liftshitz -Gilbert (LLG) algorithm to investigate  skyrmion s in GdFe  \nwithin the interfacial DMI model . Despite the rapid decay of DMI away from the interface, small \nskyrmions of near 10 nm are found in thick ~ 5 nm amorphous GdFe film at 300K . We have also \nconsidered three scenarios  for the sign of DMI between Gd -Fe pa ir. It is revealed that \nantiferromagnetic coupling in the ferrimagnet plays an important role in enhanc ing the effect of \ninterfacial DMI  and to stabilize  skyrmio n. These results show  that ferrimagnets and \nantiferromagnets with intrinsic  antiferromagnetic couplings are appealing materials to host \nsmall skyrmions  at room temperature , which is crucial to  improve density and energy efficiency \nin skyrmion based devices.  \nIntroduction  \nMagnetic skyrmions are t opologically protected spin textures. Their potentials in advancing  \nmemory density and efficiency  have drawn extensive  investigation s in recent years1-24. In \nmagnetic materials, skyrmions are stabilized th rough Dzyaloshi nskii Moriya interaction (DMI)25-26. \nDMI is generated by  either  intrinsic or interfacial effect. Intrinsic  DMI arises in non-\ncentrosymmetric crystal , such as B20 alloys , where  Bloch skyrmions are found to exist in MnSi \nand FeGe  at low temperature13-14. Interfacial DMI originates  from interf acial layer with strong \nspin-orbit coupling . Multilayer stacks, such as Ir/Fe/Co/Pt  and Pt/Co/Ta , are found to host  ~ 50 \nnm Neél skyrmions at room temperature15-16. Several challenges remain in developing skyrmion \nbased memory  and logic  devices. For example, further reduction in skyrmion sizes is needed to \noptimize skyrmion based devices . However, the stability  of small skyrmion at room temperature  \nbecomes a problem . Thicker magnetic layers are required  to increase stability17-18. For \nferromagnet /heavy metal multilayer stacks, increase in thickness of magnetic layer can lead to \nthe loss of interfacial anisotropy  and the reduction of the strength of DMI52-55. Both are critical \nfor skyrmion formations. Moreover, skyrmions Hall effect can provide great challenges on \nmoving skyrmions in electronics  devices19-22. To overcome these challenges, one need s to \nexplore more  materials.  \nAmorphous rare -earth-transitional -metal ( RE-TM) ferrimagnet is one of the  pote ntial material s \nto overcome  these challenges.  Several properties of RE -TM alloys provide favorable environment  to host small skyrmion s at room temperature. Their Intrinsic  perpendicular \nmagnetic anisotropy (PMA) 27-30 gives a crucial advantage in stabilizing  small skyrmion by \nallowing the use of thicker films (~  5 nm) . However, the effectiveness of interfacial DMI  \ndecreases significantly away from the interface52-55. Besides PMA, the magnetization of RE -TM \nalloys vanishes at the compensation temperature31. With near zero magnetization, the skyrmion \nHall effect is vastly  reduced . Another advantage of RE -TM alloys is the access to ultrafast \nswitching32-39. Recently,  all-optical switching  helicity -dependent  has been demonstrated in RE-\nTM alloys  using a circularly polarized  laser32-35. This gives an additional tool to control spins in \nfuture devices . RE-TM alloys have begun to draw  interest in the field of skyrmions  research . \nLarge skyrmions of ~  150 nm have  been observed in Pt/GdFeCo/MgO23, and skyrmion bound \npairs are found in Gd/Fe multilayers24. Further tuning is needed to  reduce the size of skyrmion in \nRE-TM alloys . To guide experiments , numerical model has served  as a n important tool , \nespecially for complex systems such as RE-TM alloys34,40 -44. Several  methods, such as atomistic \nLandau -Liftshitz -Gilbert (LLG) algorithm34,40 -43 and micromagnetic Landau -Lifshitz -Bloch (LLB) \nalgorithm44, has been employed  to provide deeper understanding of magnetic properties in RE-\nTM alloys .  \nIn this study , atomistic LLG algorithm34,40 -43 is employed to study properties  of skyrmions  in GdFe  \nwith interfacial DMI . Although the sign of DM I at ferromagnet s/heavy metal interface  is well \nstudied44-51, the sign of DMI involved  ferrimagnet remains complex . Here,  we consider three \nscenario s for the DMI between Gd and Fe (D Gd-Fe). First, the influence  of DMI between \nantiferromagnetic pair is excluded by setting it to zero (D Gd-Fe = 0) . Second, DMI  between \nantiferromagnetic  pair is set to the same sign as DMI  between ferromagnetic  pair, where  DGd-Fe > \n0. Finally, the case of D Gd-Fe < 0 is considered. Furthermore , to incorporate  DMI being an \ninterfacial effect, an exponential decay DMI is utilized. Simulation results find that near 10 nm  \nskyrmions remain robust in ~ 5 nm GdFe at room temperature . This demonstrates  that \ninterfacial  DMI remain s prominent  in thicker ferrimagnet samples, which is critical in stabilizing  \nsmall skyrmions at room temperat ure.  \nSimulation  Model  \nThe classical atomistic Hamiltonian H  in Eq. (1) is employed to investigate magnetic textures  in \namorphous ferrimagnets .  \n    \n ∑        \n       \n ∑    (     )\n        (     ̂)  \n                                \nWhere       are the normalized spin at site i, j respectively,       are the atomic moment at \nsite i, j respectively . Atomic  moment is absorbed  into the following constant,              is the \nexchange interaction,              is the DMI interaction and          is the anisotropy.      \nand        is the external field and demagnetization field respectively.  \nOnly nearest neighbor interactions are considered in exchange and DMI interactions. Periodic \nboundary condition  is enforced in x and y direction. To find the ground state, spins are evolved \nunder  the stochastic Landau -Lifshitz -Gilbert (LLG) Equation as sho wn in Eq. ( 2), and the constant \nparameters used in the simulation are listed in Table 1 .   \n     \n      (      )   \n          [  (      )]     \nWhere   is the gyromagnetic ratio ,   is the Gilbert damping constant,       is the effective \nfield,   is the Gaussian white noise term for thermal fluctuation and    is the saturation \nmagnetization.  \nTo incorporate the amorphous short range order, a n amorphous structure of a 1. 6 nm x 1. 6 nm \nx 1.6 nm box containing  250 atoms is generated from ab initio molecular dynamic s calculations \nby Sheng et al.56. Fig. 1 shows a plot of RE and TM atoms in the amorphous structure. Replicas of \nthis box (32 x 32  x 1) are place d next to each other  to expand the simulated sample  to 50.7 nm x \n50.7 nm x 1. 6 nm and 256000 atoms.  For a 4.8 nm thick sample, replicas of the box are also \nplaced in z -direction, and the total number of atoms is 768000.  \nResults and Discussion  \nWith  ferromagnetic DMI (D Gd-Gd and D Fe-Fe) remains positive, th ree scenarios  of antiferromagnetic \nDMI (D Gd-Fe) are considered.  Samples of 50.7 nm x 50.7 nm x 1.6 nm are simulated using \natomistic LLG equation from Eq. (2) at 0 K.  Fig. 2 shows the equilibrium spin configurations at 0 K. \nFor DGd-Fe = 0 and DGd-Fe < 0, skyrmion ’s radius  increase s as DMI increases and be come s stripe at \nlarge DMI value, which behaves  similar to a ferromagnet17-18. On the other hand, with DGd-Fe > 0, \nsame sign as DGd-Gd and D Fe-Fe, the trend of skyrmion sizes is somewhat  different. At s mall DMI \nvalue, skyrmion’s radius increases  as DMI increases. However, at large DMI value, skyrmion’s \nradius decreases as DMI increases, which is different from what observed in DGd-Fe = 0 and DGd-Fe \n< 0, and in a ferromagnet. For a given DMI value, the ra dius of skyrmion is also different for the \nthree scenarios , where smallest skyrmions are found with DGd-Fe > 0, and the largest skyrmions \nare found with DGd-Fe < 0.   \nTo understand the intriguing behavior of skyrmion’ s size in ferri magnet, in -plane spin \nconfigurations and the chirality of skyrmion ’s wall are investigated. Fig. 3  summarizes the \nchirality of skyrmion wall at 0 K.  With DGd-Gd, DFe-Fe > 0 and DGd-Fe = 0, for Fe sublattice, the spins in \nthe skyrmion’s wall are turning cou nter -clockwise. For Gd sublattice, the spins in the skyrmion’s \nwall are also turning counter -clockwise. This can be explained by the dominance of exchange \ninteraction in the system . Antiferromagnetic  coupling s between Gd and Fe align the spins of Gd \nand Fe  in nearly antiparallel direction, with small canting due to presence of DMI. Identical \nbehavior is observed with DGd-Gd, DFe-Fe > 0 and  DGd-Fe < 0, where spins in both Gd and Fe \nsublattice are turning counter -clockwise across the skyrmion’s wall. With DGd-Gd, DFe-Fe > 0 and \nDGd-Fe > 0, the chirality of skyrmion ’s wall is opposite to what observed in DGd-Fe = 0 and DGd-Fe < 0. \nThe spins in both Gd and Fe  sublattice  are turning clockwise across the skyrmion’s wall.  \nIn order t o determine the reason behind the change in chirality, the total DMI energies of each \nnearest neighbor  pair are computed using equilibrium configurations at 0 K. Table 2 summarizes \nthe sign of total DMI energies of different nearest neighbor pair. With DGd-Gd, DFe-Fe > 0 and DGd-Fe \n= 0, the total DMI energy between Gd and Gd pair E DMI(Gd-Gd) and Fe and Fe pair E DMI(Fe-Fe) is \nnegative, and the total DMI energy between Gd and Fe pair E DMI(Gd-Fe) is zero. This means that \nwith DGd-Gd, DFe-Fe > 0, it is energetically favorable for spins to turn counterclockwise across \nskyrmion’s wall.  EDMI(Gd-Fe) is zero because DGd-Fe is set to zero. With DGd-Gd, DFe-Fe > 0 and DGd-Fe > \n0, E DMI(Gd-Gd) and E DMI(Fe-Fe) is positive, while E DMI(Gd-Fe) is negative.  This implies  that it is \nenergetically favorable for Gd -Fe pair to turn clockwise across skyrmion’ s wall, but  it is \nenergetically unfavorable for Gd -Gd and Fe -Fe pair  to do so . This means that in a ferrimagnet, if the DMI of ferromagnetic pair and antiferr omagnetic pair has the same sign, cancellation of DMI \noccurs because it is preferable  for ferromagnetic pair to turn in opposite direction of \nantiferromagnetic pair. With DGd-Gd, DFe-Fe > 0 and DGd-Fe < 0, all three terms E DMI(Gd-Gd), E DMI(Fe-\nFe) and E DMI(Gd-Fe) is negative, so turning counterclockwise is energy favorable for both \nferromagnetic pair and antiferromagnetic pair in a ferrimagnet . These differences in sign of total \nDMI energy also explain the size of skyrmion in all three scenarios . For a give n DMI, skyrmions \nare smallest for DGd-Fe > 0 because cancellation in DMI leads to reduction in DMI effectiveness in \nthe sample . DGd-Fe < 0 scenario has the largest skyrmions because both ferromagnetic and \nantiferromagnetic are contributing to formation of a skyrmion, which means DMI is stronger  \noverall. The trend of skyrmion’s radius in  DGd-Fe > 0 scenario can also be explained by \ncancellation of DMI between ferromagnetic and antiferromagnetic pair. As DMI becomes larger, \nmore cancellation in DMI leads to s maller skyrmion. Thus, with large DMI, skyrmion decreases \nas DMI increases  in the case of DGd-Fe > 0.  \nTo determine the viability  of using RE -TM alloys for skyrmion devices, simulations are also \ncarried out at 300 K.  Samples of 50.7 nm x 50.7 nm x 4.8 nm are simulated using atomistic \nstochastic LLG equation  in Eq. (2) . Since DMI is known decay away from the interface52-55, an \nexponential decay DMI is employed in the simulation. Fig. 4 shows the functional form of \nexpo nential decay DMI used in the simulation. In this model, DMI remains constant within 5 Å of \nthe top and bottom interface , and start to decay exponential at 5 Å away from the interface.  \nFig. 5 summarizes the results of equilibrium spin configuration at 300  K. only ferrimagnetic \nstates are observed with DGd-Fe > 0. As discussed earlier, with DGd-Fe > 0, cancellation of DMI \nbetween ferromagnetic and antiferromagnetic pair leads to unfavorable conditions for skyrmion \nformation. For the case of DGd-Fe = 0 and DGd-Fe < 0, s mall skyrmions of near 10  nm are found in \nthe case of DGd-Fe = 0 and DGd-Fe < 0.  Skyrmions this small are very promising for improving \ndensity and efficiency in skyrmion based devices. Fig. 6 shows a comparison between atomistic \nsimulation of GdFe and micromagnetic simulation  of an equivalent ferromagnet. Using the same \nexponential decay DMI, m uch larger interfacial DMI is required to obtain skyrmion in the \nmicromagnetic simulation of an equivalent ferromagnet. This demon strates that internal spin \nstructure in a ferrimagnet is essential to prolong the effect of DMI away from the interface. This \nDMI robustness in a ferrimagnet can be explained by the antiferromagnetic coupling between \nthe two sublattices. Even without  the p resence of DMI, the spins in the Gd sublattice are known \nto be canted at room temperature31. With the presence of DMI, the spins in the Gd sublattice \nare easily guided by DMI and leads to formation of skyrmions. Thus, antiferromagnetic couplings  \ncan help to extend the influence of DMI, and increase stability of small skyrmions at room \ntemperature.  \nConclusions  \nEffect of interfacial DMI is investigated in amorphous ferrimagnetic GdFe using atomistic  \nstochastic  LLG algorithm. Three scenarios for the sign of DMI between Gd and Fe are considered. \nIt is revealed  that for a ferrimagnet, if  the DMI between ferromagnetic pair and \nantiferromagnetic pair has the same sign, it lead s to cancellation in  DMI , and it is unfavorable \nfor skyrmion formations. If the DMI between ferromagnetic pair and antiferromagnetic pair has \nopposite sign, it is advantageous  for skyrmion formation, and small skyrmions of near ~10 nm \nare found to be stable at room temp erature with exponential decay DMI. The antiferromagnetic \ncouplings  in ferrimagnet are uncovered to help extend the influence of DMI in thicker sample s \nof ~ 5 nm . 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B 97, 134405 \n(2018).  \n52. Belmeguenai, M. et al.  A. Interfacial Dzyaloshinskii –Moriya interaction in  \nperpendicularly magnetized Pt/Co/AlOx ultrathin films measured by Brillouin  light  \nspectroscopy. Phys. Rev. B  91, 180405(R) (2015).  \n53. Nembach, H.T., Shaw, J.M, Weiler, M., Jué E. & Silva, T.J. Linear relation between \nHeisenberg exchange and  interfacial Dzyaloshinskii –Moriya interaction in  metal films . \nNature Phys. 11, 825 -829 (2015).  \n54. Yang, H., Thiaville, A., Rohart, S., Fert, A. & Chshiev, M. Anatomy of Dzyaloshinskii -\nMoriya Interaction at Co/Pt Inter faces . Phys. Rev. Lett.  118, 219901 (2017)  \n55. Belmeguenai, M. et al.  Thickness Dependence of the Dzyaloshinskii -Moriya Interaction \nin Co2FeAl  Ultrathin Films: Effects of Annealing Temperature and Heavy -Metal Material . \nPhys. Rev. Appl. 9, 044044 (2018).  \n56. Sheng , H.W., Luo, W.K., Alamgir, F.M., Bai, J.M., & Ma, E. Atomic packing and short -to-\nmedium -range order in metallic glasses . Nature 439, 419 -425 (2006).  \n \nParameter  Value  \nGyromagnetic ratio (ϒ) 2.0023193  Gilbert Damping (α) 0.05 \nGd moment ( μGd) 7.63 μB \nFe moment ( μFe) 2.217 μB \nGd-Gd exchange constant (J Gd-Gd)  1.26 x 10-21 J \nFe-Fe exchange constant (J Fe-Fe) 3.82 x 10-21 J \nGd-Fe exchange constant (J Gd-Fe) -1.09 x 10-21 J \nTable. 1 Values of parameters used in the simulation.  \n  \nFigure 1 Amorphous structure of RE 25TM 75 from ab initio molecular dynamics calculations . Red \natoms are rare -earth, and blue atoms are transitional -metal.  \nResults and Discussion  \n \nFigure 2 Equilibrium spin configurations  for various DMI (uniform DMI) at 0K for three \nscenarios of D Gd-Fe. Parameters used here are listed in Table 1 with Magnetic field     is 0.01 T, \nanisotropy energy K is 0.3 x 105 J/m3(distributed within a 45 degree cone), and simulation space \nis 50.7 nm x 50.7 nm x 1.6 nm. For DFe-Fe = 0.25 x 10-22 J, DGd-Gd = 2.89  x 10-22 J, and | DGd-Fe| = 0 or \n0.85  x 10-22 J. For DGd-Fe = 0 and DGd-Fe < 0, the skyrmions size increases with DMI. On the other \nhand, for DGd-Fe > 0, the skyrmion size first increases with DMI at small DMI, then decreases at \nlarger DMI value.  \n \nFigure 3 Skyrmions wall chirality for three scenarios of D Gd-Fe. For DGd-Fe = 0 and DGd-Fe < 0, the \nskyrmions wall is rotating counter -clockwise. On the oth er hand, for DGd-Fe > 0, the skyrmion wall \nis rotating clockwise.  \n \nScenario  EDMI(Gd-Gd)   EDMI(Fe-Fe)   EDMI(Gd-Fe)   \nDGd-Gd, DFe-Fe > 0, DGd-Fe = 0 - - 0 \nDGd-Gd, DFe-Fe > 0, DGd-Fe > 0 + + - \nDGd-Gd, DFe-Fe > 0, DGd-Fe < 0 - - - \nTable.  2 Sign of total DMI energy E DMI computed from equilibrium spin configurations at 0 K.   \n \nFigure 4  Plot of exponential decay DMI as function of distance from bottom interface (z) . In \nthis model, DMI remains constant within 5 Å of the top and bottom interface, as indicated  by \nthe red line. At the center of the 4.8 nm sample, the strength of DMI decays exponentially as \nshown.  \n \nFigure 5  Equilibrium spin configurations for various DMI (exponential decay DMI) at 300K for \nthree scenarios of D Gd-Fe. Parameters used here are listed in Table 1 with Magnetic field     is \n0.01 T, anisotropy energy K is 0.3 x 105 J/m3(distributed within a 45 degree cone), and simulation \nspace is 50.7 nm x 50.7 nm x 4.8 nm. For DFe-Fe = 0.25 x 10-22 J, DGd-Gd = 2.89 x 10-22 J, and | DGd-Fe| = \n0 or 0.85 x 10-22 J. At 300K with exponential decay DMI, skyrmions are only found to exist with \nDGd-Fe = 0 and DGd-Fe < 0.  \n \nFigure 6 Equilibrium spin configurations from atomistic simulation of GdFe at 300 K  (left)  and \nmicromagnetic simulation of an equivalent ferromagnet  at 0 K  (right) .  \n \n"    },    {        "title": "1301.2541v1.Spin_filtering_efficiency_of_ferrimagnetic_spinels_CoFe2O4_and_NiFe2O4.pdf",        "content": "arXiv:1301.2541v1  [cond-mat.mtrl-sci]  11 Jan 2013Spin-filtering efficiency of ferrimagnetic spinels CoFe 2O4and NiFe 2O4\nNuala M. Caffrey,1,∗Daniel Fritsch,2Thomas Archer,1Stefano Sanvito,1and Claude Ederer3,†\n1School of Physics and CRANN, Trinity College, Dublin 2, Irel and\n2H. H. Wills Physics Laboratory, University of Bristol,\nTyndall Avenue, Bristol BS8 1TL, United Kingdom\n3Materials Theory, ETH Z¨ urich, Wolfgang-Pauli-Strasse 27 , 8093 Z¨ urich, Switzerland\n(Dated: January 14, 2013)\nWe assess the potential of the ferrimagnetic spinel ferrite s CoFe 2O4and NiFe 2O4to act as spin\nfiltering barriers in magnetic tunnel junctions. Our study i s based on the electronic structure\ncalculated by means of first-principles density functional theory within different approximations\nfor the exchange correlation energy. We show that, in agreem ent with previous calculations, the\ndensities of states suggest a lower tunneling barrier for mi nority spin electrons, and thus a negative\nspin-filtereffect. However, amoredetailed analysis basedo nthecomplexband-structurereveals that\nboth signs for the spin-filtering efficiency are possible, dep ending on the band alignment between\nthe electrode and the barrier materials and depending on the specific wave-function symmetry of\nthe relevant bands within the electrode.\nI. INTRODUCTION\nThe ability to generate and detect spin-polarized cur-\nrents is a central requirement for any practical spintron-\nics device. A promising approach to achieve this goal\nis to use tunnel junctions containing ferro- or ferrimag-\nnetic barrier materials, thus presenting different tunnel-\ning probabilities for majority (spin-up, ↑) and minority\n(spin-down, ↓) electrons. Efficient spin-filtering has been\ndemonstrated for ferromagnetic insulators such as EuS,1\nEuO,2and BiMnO 3.3However the magnetic ordering\ntemperatures of these magnets are rather low. There-\nfore the identification of suitable barrier materials that\noperateat roomtemperatureoraboveisofgreatinterest.\nSpinel ferrites are insulating ferrimagnets with high\nCurie temperatures ( TC=790 K for CoFe 2O4and 865 K\nforNiFe 2O4),4andthereforearepromisingcandidatesfor\nefficient room temperature spin-filtering. A measure of\nthe ability of a material or a device to select a particular\nspin direction is the spin-filtering efficiency, Psf, which is\ndefined as\nPsf=I↑−I↓\nI↑+I↓,\nwhereIσis the spin- σcomponent of the current, which\nis assumed to be carried by the two spin species in paral-\nlel. Recent experiments on ferrimagnetic spinels appear\npromising, as a spin-filtering efficiency of +22% has been\nmeasured for NiFe 2O4at low temperatures.5The mea-\nsured positive sign of Psfis in apparent contradiction\nwith results of band-structure calculations, demonstrat-\ning that the bottom of the conduction band is lower for\nspin-down electrons than for spin-up,6which would lead\nto a lower tunneling barrier for minority spin electrons.\nIt was suggested that this apparent discrepancy could\nbe due to effects related to the wave-function symmetry\nof the tunneling states.5Furthermore, for CoFe 2O4both\npositive and negative Psfhave been reported in junctions\nmade of different electrode materials and where Psfwas\nmeasuredwithdifferentexperimentaltechniques. There-ported values of Psfrange from −44% to +26%.7–12Due\nto these large variations in experimental results (with\nboth signs occurring for the spin-filtering efficiency) a\nconclusive picture of spin-filtering in spinel ferrites has\nnot emerged, yet. As such, a first-principles investigation\nof the spin-filtering efficiency in these materials is highly\ndesirable, in order to provide a reference for future ex-\nperimental studies and to allow further optimization of\nthe corresponding devices.\nSo far, theoretical predictions for the spin-filter effect\nin CoFe 2O4and NiFe 2O4are almost exclusively based\non density of states (DOS) calculations within a self-\ninteraction corrected (SIC) local spin-density approxi-\nmation (LSDA).6The spin-splitting of the conduction\nband minimum (CBM) in these calculations suggests a\nlower tunnel barrier for minority spin electrons and thus\na negative sign for the spin-filtering efficiency. However,\nit is well known that in many cases this simple density\nof states argument can be misleading, and the tunnel\nprobability can be strongly dependent on the specific\nwave-function symmetry.13The implications of this were\nfirst noticed in a Fe/MgO/Fe heterostructure,14,15where\nsymmetry-dependent tunneling results in half-metallic\nbehaviour of the Fe/MgO(001) stack. Since then, the\nso-called complex band-structure , which determines the\ndecay length of Bloch states with different wave-function\nsymmetries inside an insulating barrier, has been used\nto account for many, otherwise unexplained, experimen-\ntal results in spin-dependent tunnel junctions. Further-\nmore, it is of interest to compare the SIC-LSDA result of\nRef. 6 to the electronic structure obtained by using alter-\nnative approaches such as LSDA+ U, hybrid functionals,\nor other SIC approaches.\nHerewe presenta detailed comparisonofthe electronic\nstructure of CoFe 2O4and NiFe 2O4calculated within dif-\nferent approximations for the exchange-correlation po-\ntential. This allows us to identify features of the DOS\nthatarefairlyrobustwith respecttothe specific choiceof\nexchange-correlationpotential and features that are very\nsensitivetothischoice. Inaddition,wecalculatethecom-2\nplex band-structure for both materials within the atomic\nSIC method (ASIC)16,17, which facilitates the identifica-\ntion of suitable electrode materials that can lead to high\nspin-filtering efficiency. We show that, for both CoFe 2O4\nand NiFe 2O4and the two transport directions [001] and\n[111], electrons tunnel with the highest probability at the\ncenter of the two-dimensional Brillouin zone in the plane\northogonal to the transport direction. Furthermore, de-\npending on the exact alignment of the electrode Fermi\nlevel relative to the CBM of the barrier, the tunneling\ncurrent may present either a predominant majority or a\npredominant minority contribution, i.e. Psfmay change\nsign depending on the level alignment.\nThe paper is organized as follows. After having briefly\npresented the computational method and the details of\nthe crystallographic unit cell used for this study, we pro-\nceed to describe the electronic structure of CoFe 2O4and\nNiFe2O4. In particular, we first discuss the DOS and\nreal band-structures, and then move on to present the\ncomplex ones. The final section summarizes our main\nconclusions.\nII. METHODS\nWe employ the vasp18andsiesta19density functional\ntheory (DFT) code packages for the calculation of DOS\nand real band-structures and the smeagol code20,21to\ncalculate the complex band-structure. The vaspcal-\nculations have been performed by using the projector-\naugmented wave (PAW) method22with standard PAW\npotentials supplied with the vaspdistribution, a 500 eV\nplane wave energy cutoff, and a Γ centered 6 ×6×6k-\npoint mesh for the Brillouin zone sampling. We employ\nthegeneralizedgradientapproximation(GGA) according\nto the Perdew-Burke-Ernzerhof formulation23together\nwith the Hubbard “+ U” correction,24whereU= 3 eV\nandJ= 0 eV is applied to the dstates of all transition\nmetal cations, as well as the hybrid functional approach\naccording to Heyd, Scuseria and Ernzerhof (HSE),25us-\ning the standard choice for the fraction of Hartree-Fock\nexchange ( α= 0.25) and a reduced plane wave energy\ncutoff of 400eV. When using the localised basis set code\nsiesta, structural relaxations were performed using the\nGGA while the atomic self-interaction correction (ASIC)\nscheme was used to determine the electronic structure,\nincluding the complex band-structure. A 6 ×6×6k-point\nMonkhorst-Pack mesh was used to converge the density\nmatrix to a tolerance of 10−5and a grid spacing equiva-\nlent to a plane-wave cutoff of 800eV was used.\nFor most of our calculations we use the smallest possi-\nble unit cell (containing 2 formula units) to describe the\ninverse spinel structure. The corresponding distribution\nofcationsonthespinel Bsitelowersthespacegroupsym-\nmetry from Fd¯3mtoImma.26We also present some re-\nsults obtained for a cation distribution with P4122 sym-\nmetry, which requires a doubling of the unit cell to 4 for-\nmulaunits (the k-pointsamplingisthen adjustedaccord-TABLE I. Band gap (E g) and spin-splitting of the CBM\n(∆CBM) for CoFe 2O4and NiFe 2O4calculated with different\nexchange-correlation functionals. All values are in eV.\nCoFe2O4 NiFe2O4\nEg ∆CBM Eg ∆CBM\nGGA+U 0.52 0.92 0.83 0.86\nHSE 1.60 1.09 2.32 1.00\nASIC 1.08 1.00 2.07 0.46\ningly). We have previously shown that both Immaand\nP4122arelowenergyconfigurationsfor the inversespinel\nstructure in CoFe 2O4and NiFe 2O4, and that the specific\ncation arrangement has only a minor influence on the\nglobal electronic structure of these systems.27We note\nthatexperimentallyadisordereddistributionofFe3+and\nCo2+/Ni2+cations over the spinel Bsite with effective\ncubicFd¯3msymmetry, i.e. with no long-range cation\norder, is generally observed, even though recently indica-\ntions for short range cation order in both NiFe 2O4bulk\nandthinfilmsampleshavebeen reported.28,29Foramore\ndetailed comparison between the different cation config-\nurations see Refs. 27 and 30.\nStructural relaxations have been performed at the\nGGA level, with all cations being fixed to their ideal\ncubic positions.26The relaxed bulk lattice constants a0\nobtained by using vasp(siesta) are 8.366 ˚A (8.360 ˚A)\nand 8.346 ˚A (8.356 ˚A) for CoFe 2O4and NiFe 2O4, respec-\ntively, and are in very good agreement with experimental\ndata (see Ref. 27 and references therein).\nIII. RESULTS AND DISCUSSION\nA. Electronic structure\nIt has been previously shown that GGA leads to a\nhalf-metallic solution for CoFe 2O4and results in only\na very small insulating gap in the case of NiFe 2O4(see\ne.g. Refs. 26 and 31 and references therein). The DOS of\nCoFe2O4and NiFe 2O4calculated by using a selection of\nbeyond-GGAfunctionals aredepicted in Fig.1. It can be\nseen that all the studied exchange-correlation potentials\nlead to an insulating state for CoFe 2O4and an enhanced\nband gap for NiFe 2O4. When compared to the GGA+ U\nband gaps, both the inclusion of Hartree-Fock exchange\nwithin the HSE calculation as well as the ASIC treat-\nment leads to a large increase in the band gap values for\nboth the Co and Ni based ferrite, with the largest band\ngaps obtained for HSE (see Table I). We also note that\nour results are consistent with recent HSE and LSDA+ U\ncalculations for NiFe 2O4.32\nGoing into more details we notice that, while the oc-\ncupied DOS are very similar for GGA+ Uand HSE, the\nASIC methodplacesthe localFespin-majoritystatessig-\nnificantly lower in energy. This results in a gap between\ntheseFe statesandthe higher-lyingCo(Ni) dandoxygen3\n-5.00.05.0DOS [eV-1]Co (Oh)t2g\negGGA+UCoFe2O4\nHSE ASIC\nNi (Oh)t2g\negGGA+UNiFe2O4\nHSE\n-5.00.05.0\nDOS [eV-1]ASIC\n-5.00.05.0DOS [eV-1]Fe (Oh)t2g\negFe (Oh)t2g\neg\n-5.00.05.0\nDOS [eV-1]\n-10-505\nE [eV]-5.00.05.0DOS [eV-1]Fe (Td)e\nt2\n-10-505\nE [eV]-10-505\nE [eV]-10-505\nE [eV]Fe (Td)e\nt2\n-10-505\nE [eV]-10-505\nE [eV]-5.00.05.0\nDOS [eV-1]\nFIG. 1. (Color online) Total and projected DOS per formula un it for CoFe 2O4(left panels) and NiFe 2O4(right panels)\ncalculated with different exchange-correlation potential s (from left to right: GGA+ U, HSE and ASIC). The t2gandegstates of\nFe, Co, and Ni on the Ohsites and the eandt2states of Fe on the Tdsites are shown as black (blue) and dark grey (red) lines,\nrespectively. The shaded grey area in all panels depicts the total DOS. Minority spin projections are shown using negati ve\nvalues. The zero energy is set to the middle of the band gap.\npvalence bands. Interestingly, for CoFe 2O4the valence\nband maximum in ASIC is made up of the majority spin\nCoegstates, whereasforboth GGA+ UandHSEthe cor-\nresponding minority spin t2gstates are slightly higher in\nenergy. We also note that the difference in the calculated\nGGA+Uband gap of CoFe 2O4(NiFe2O4) compared to\nthe previously obtained values of 0.9 eV (0.97 eV) for the\nImmastructure,26and 1.24 eV (1.26 eV) for the P4122\nstructure27, is due to the fact that in the present work all\ncalculations are performed at the GGA volume, whereas\nthe calculations in Refs. 26 and 27 have been performed\nat the larger GGA+ Uoptimized volume. In addition\nto the expected dependence on the exchange correlation\npotential, ourresults thus indicate a strongvolume sensi-\ntivity in particular of the calculated CoFe 2O4band gap.\nExperimental estimates for the band gaps of spinel fer-\nrites are sparse and vary over a broad range comprised\nbetween 0.11eV and 1.5eV for CoFe 2O4and between\n0.3eV and 3.7eV for NiFe 2O4.33,34A recent optical ab-\nsorption study of NiFe 2O4suggests an indirect gap of\n1.6eV in the minority spin channel,32which thus repre-\nsents an upper bound for the corresponding fundamental\nband gap.\nIn all cases, and for both CoFe 2O4and NiFe 2O4, the\nCBM is lower in energy for the spin-down states than for\nspin-up ones, in agreement with the SIC-LSDA calcula-tions of Ref. 6. In the case of CoFe 2O4, all the three ap-\nproaches used in our work predict a spin-splitting of the\nCBM (∆CBM in Table I) of around 1 eV. For NiFe 2O4\nhowever,GGA+ UandHSEyielda∆CBMofaround0.9-\n1.0 eV, while ASIC gives a somewhat smaller splitting of\nonly 0.46eV. In all the cases, the obtained spin-splittings\nof the CBM are smaller than those reported in Ref. 6,\n1.28 eV (1.21 eV) for CoFe 2O4(NiFe2O4). We note,\nhowever, that even smaller values, namely of 0.47 eV\nfor both CoFe 2O4and NiFe 2O4, have been obtained in\nprevious GGA+ Ucalculations at the relaxed GGA+ U\nvolume.27Recent experiments estimate the spin-splitting\nof the CBM in the tens of meV range for CoFe 2O4-\ncontaining junctions.9\nInordertoshedfurtherlightonthenatureofthebands\naround the gap, the calculated GGA+ Uand ASIC band-\nstructures for both CoFe 2O4and NiFe 2O4are shown in\nFig. 2. Apart from the larger band gaps obtained by the\nASIC approach, it can be seen that the relative energies\nof the minority and majority spin bands in the upper va-\nlence band region for CoFe 2O4differ between GGA+ U\nand ASIC. This is consistent with our previous discus-\nsion of the DOS. For the calculation that is performed\nwith GGA+ U, the top of the valence band is formed\nby a minority spin band with maximum at the X point,\ni.e. the minority spin band gap is indirect. In contrast4\n-2-1012E [eV]\nL Γ X-2-1012E [eV]-2-1012\nL Γ X-2-1012(a)  CoFe2O4 - GGA+U (b)  CoFe2O4 - ASIC\n(d) NiFe2O4 - ASIC (c)  NiFe2O4 - GGA+U\nFIG.2. (Coloronline)Bandstructuresforenergies aroundt he\nband gap of CoFe 2O4[upper panels (a) and (b)] and NiFe 2O4\n[lower panels (c) and (d)] calculated by using the GGA+ U\nexchange-correlation functional [left panels (a) and (c)] and\nthe ASIC scheme [right panels (b) and (d)]. Majority and\nminorityspinbandsare shownas full (black)anddashed(red )\nlines.\na direct gap with mixed spin character at Γ is obtained\nby ASIC. Since, as we will show in the following, the\ntunneling probabilities are dominated by states around\nthe Γ point, we do not expect that this qualitative dif-\nference between the two exchange-correlationfunctionals\nwill critically affect the transport properties.\nBased on our analysis of the DOS and the band-\nstructure in the vicinity the gap, we can conclude that\ndespite some differences, all computational methods con-\nsistently predict a lower tunnel barrier for the minority\nspin electrons and therefore a negative spin-filtering ef-\nficiency for both CoFe 2O4and NiFe 2O4. However, as\nshown in Fe/MgO/Fetunnel junctions,14,15in the case of\nhigh quality epitaxial interfaces between the electrodes\nand the barrier material such DOS considerations are\nonly of limited value for the description of actual trans-\nport properties. Instead, the specific symmetry of the\ndecaying wave-functions inside the barrier has to be con-\nsidered. This can be achieved through calculation of the\ncomplex band-structure.13\nB. Complex band structure\nThe complex band-structure along a particular crys-\ntalline direction is calculated with the DFT non-\nequilibrium Green’s function code smeagol .20,21Thecomplex band-structure is nothing but the solution of\nthe secular band equation extended to imaginary wave-\nvectors. Let us assume that the transport direction of a\ngiven tunnel junction is along the zdirection and that\nthe material composing the barrier has a particular crys-\ntalline axis aligned along that direction. For any given\nk-vector in the transverse x-yplane,k/bardbl= (kx,ky), and\nfor any energy, E, the band equation E=E(kx,ky,kz)\ncan be solved for kzif one admits imaginary solutions\nkz=q+iκ. This means that the wave-function of an\nelectron approaching the tunneling barrier with trans-\nverse wave-vector k/bardblexponentially decays into the bar-\nrier along the zdirection over a length-scale given by\n1/κ. Clearly such decay rate depends on the transverse\nk-vector and the energy, i.e. κ=κ(kx,ky;E). Here we\nconsider the situation of electron transport along both\nthe [001] and [111] directions of the cubic spinel struc-\nture.\nIn Fig. 3 we plot the minimal value of κas a function\nofkxandky(calculated on a 100 ×100 grid) at differ-\nent energies within the gap. We include data for both\nCoFe2O4and NiFe 2O4considering both transport direc-\ntions for the Immaconfiguration, and we also present\ndata for the P4122 configuration and transport along the\n[001] direction. The crucial result emerging from Fig. 3\nis that in all cases κis smallest at the Γ point of the\ntwo-dimensional Brillouin zone corresponding to the x-y\nplane. This means that, due to the exponential depen-\ndence of the wave-function on κ, electron tunneling away\nfrom the Γ point will contribute very little to the trans-\nport. Assuch,intheanalysisthatwillfollow,wewillonly\nconsider transport through the Γ-point. We note that Γ-\npoint filtering is a highly desirable property for both tun-\nnel junctions and spin injection. As has been shown for\nthe Fe/MgO barrier, as the thickness of the MgO layer\nincreases so does the selectivity of the Γ-point. This in\nturn increases the tunneling magneto-resistance (TMR).\nAlthough the Γ-point filtering is not strictly necessary\nfor a large TMR, it significantly reduces the importance\nof the material choice for the electrodes.\nHaving established that the transport predominantly\noccursattheΓ-point, furtherinsightcanbegainedbyex-\nploringthe energydependence of κ(0,0;E). In particular\nit is important to establish the spin and orbital symme-\ntry of the complex bands corresponding to the smallest\nvalue of κ(0,0;E) for each energy, since incident waves\nwith that particular symmetry will dominate the tun-\nneling current. In Fig. 4 we present the complex band-\nstructuresof CoFe 2O4and NiFe 2O4, calculated alongthe\n[001] and [111] directions at the Γ-point in the transverse\n2D Brillouin zone for the Immaconfiguration. One can\neasily recognize that, for both CoFe 2O4and NiFe 2O4,\nthe main features which we discuss in the following are\nsimilar for the two different transport directions. We\nnote that the transport calculation along [111] requires\na larger unit cell, in order to obtain lattice vectors that\nare either perpendicular or parallel to the transport di-\nrection, which leads to a larger number of complex bands5\n(a)Imma-CoFe 2O4, transport along [001]\n(b)Imma-CoFe 2O4, transport along [111]\n(c)P4122-CoFe 2O4, transport along [001]\n(d)Imma-NiFe2O4, transport along [001]\n(e)Imma-NiFe2O4, transport along [111]\n(f)P4122-NiFe 2O4, transport along [001]\nFIG. 3. Minimal value of κat different energies (indicated at\nthe top left in each graph) within the gap for CoFe 2O4(a-c)\nand NiFe 2O4(d-f) along different transport directions, calcu-\nlated within the ASIC approach. Zero energy corresponds to\nthe middle of the band gap.00.20.4κ (Å-1)\n[001]NiFe2O4\n00.20.4κ (Å-1)\n[001]CoFe2O4-1.5-1-0.500.511.5200.20.4κ (Å-1)\n[111]\n-1.5-1-0.500.511.52\nE [eV]00.20.4κ (Å-1)\n[111]\nFIG. 4. The complex band-structure corresponding to kx=\nky= 0 for NiFe 2O4(upper two panels) and CoFe 2O4(lower\ntwo panels) along [001] and [111], calculated within ASIC\nfor theImmaionic configuration. The up and down arrows\nindicate the spin-character of the lowest lying complex ban ds.\nThe vertical dashed lines indicate the energies that were us ed\nfor thekx-kyplots in Fig. 3.\ncompared to the [001] case. In both materials the slow-\nest decay rate close to the valence band maximum cor-\nresponds to electrons with majority spin character (in\nagreement with the real band-structure shown in Fig. 2).\nThis remains the case for energies up to around 0.5 eV\nfrom the top ofthe valenceband, althoughthe decayrate\nincreases quickly with energy. In contrast, the lowest de-\ncay rate for energies taken in the upper part of the band\ngap is dominated by states with minority spin symme-\ntry. ForNiFe 2O4this decayrateremainsalmostconstant\nfor a wide energy window of about 1.5 eV, whereas for\nCoFe2O4the gap region is divided more symmetrically\nbetween the majority and minority spin-dominated re-\ngions. ThesmallerASICcalculatedbandgapofCoFe 2O4\ncomparedto that ofNiFe 2O4resultsin slightlyslowerde-\ncayswithinthegapregionforbothmajorityandminority\nspins.\nIn Fig. 5 we also present the complex band struc-\nture of CoFe 2O4and NiFe 2O4in theP4122 configura-6\n-1.5-1-0.500.511.5200.20.4κ (Å-1)NiFe2O4\n-1.5-1-0.500.511.52\nE [eV]00.20.4κ (Å-1)CoFe2O4\nFIG. 5. The complex band-structure corresponding to kx=\nky= 0 for NiFe 2O4(upper panel) and CoFe 2O4(lower panel)\nalong [001] for the P4122 configuration, calculated within\nASIC. The up and down arrows indicate the spin-character\nfor some of the lowest lying complex bands.\ntion for transport along [001]. One can recognize the\nslightly larger band-gap compared to the Immaconfig-\nuration, but for NiFe 2O4the complex bands look very\nsimilar compared to the Immacase. For CoFe 2O4one\ncan see that the bands in the mid-gap region connect in a\nsomewhat different way than in the Immaconfiguration.\nHowever, the spin-characters of the lowest complex band\nin the upper and lower gap region remain unaffected by\nthe different cation distribution, even though the energy\nrange dominated by the minority spin complex bands is\nsomewhat more extended in the P4122 case.\nFrom the complex bands it becomes clear that pos-\nitive as well as negative values for Psfare possible for\nboth NiFe 2O4and CoFe 2O4, depending on whether the\nFermi level of the electrode lies in the upper or lower gap\nregion ofthe spinel tunnel barrier, and on the availability\nof majority or minority spin carriers in the metal. If the\nFermi level of the metallic electrode lies within ∼0.5 eV\nfrom the top of the valence band, the slowest decay rate\nin both CoFe 2O4and NiFe 2O4will be for electrons with\nmajority spin. In contrast, if the Fermi level of the elec-\ntrode is more than 0.5 eV above the valence band edge\nof the spinel barrier, then the slowest decaying state is\nin the minority spin channel. The exact position of the\nFermi level of the metal depends on the band alignment\nbetween the two materials. Thus, the spin filter effi-\nciency of the spinel ferrite barrier will depend strongly\non the band alignment and eventually also on the orbital\nsymmetry of the electrode states at the Fermi level. In\naddition, a good lattice match is of course required, oth-\nerwise translationalsymmetry is broken in the transverseplane and the complex band-structure argument breaks\ndown. Here, the possibility to grow good quality films\nof CoFe 2O4and NiFe 2O4with either [001] or [111] ori-\nentation (see e.g. Refs 35, 36, and 37) opens up a wide\nrange of possible electrode materials. In fact, high qual-\nity epitaxial junctions of CoFe 2O4or NiFe 2O4with var-\nious electrode materials, such as La 2/3Sr1/3MnO3, Au,\nFe3O4, Nb-doped SrTiO 3, Pt, Co, Al, and SrRuO 3, have\nalready been fabricated.5,7–12\nSo far we have only discussed the spin character of\nthe complex bands, whereas it is well known from the\nFe/MgO/Fesystem,thattheorbitalcharacteroftherele-\nvantbandscanalsohaveacrucialinfluenceonthetunnel-\ning properties. The determination of orbital character of\nthe complex bands in the inverse spinel ferrites CoFe 2O4\nand NiFe 2O4is complicated by the different symmetries\nof the specific cation configurations used in the calcula-\ntions. For example, the lowest lying state above the gap\nat Γ inImma-NiFe2O4, i.e. the one which connects to\nthe complex band with minority spin character that has\nthe smallest extinction coefficient over a rather large en-\nergy region within the gap, transforms according to the\nfully symmetric irreducible representation Agof the cor-\nrespondingorthorhombicpointgroup mmm. Thismeans\nthat, assuming an electrode with cubic bulk symmetry,\nthis state can in principle couple to ∆ 1and ∆ 2/∆′\n2bands\nfor transportalong the [001]direction (whether ∆ 2or ∆′\n2\ndepends on how exactly the electrode is oriented with re-\nspect to the spinel structure), or to Λ 1and Λ 3for trans-\nport along the [111] direction. However, these consider-\nation hold only for the case with Immasymmetry and\nit is unclear how different cation arrangement, in partic-\nular a completely disordered cation distribution, would\nchange these symmetry-based selection rules. Generally,\nthe lower symmetry of the various cation arrangements\nleads to fewer symmetry restrictions regarding the possi-\nble coupling with electrode bands. Since a full symmetry\nanalysisofall combinationsthat can possiblyoccur is be-\nyond the scope of this paper, we restrict our analysis to\nthe spin character of the decaying states within the bar-\nrier, which was discussed in the preceeding paragraphs.\nIV. SUMMARY AND CONCLUSIONS\nIn summary, we have calculated the electronic struc-\nture of both NiFe 2O4and CoFe 2O4using different ap-\nproaches to evaluate the exchange-correlation potential.\nTheseinclude GGA, GGA+ U, HSEand ASIC. We found\nthat, while there are certain characteristic differences in\nthe predicted band-structure, the densities ofstates ofall\nbeyond-GGA methods consistently suggest a lower tun-\nnel barrier for minority spin electrons. Due to the well-\nknown limitations of this simple density of states picture\noftunneling, wehavefurtheranalyzedthecomplexbands\nof the two materials at the ASIC level.\nWe have shown that the tunneling along the [001]\nand [111] directions is dominated by zone-center con-7\ntributions ( kx=ky= 0), and that for both NiFe 2O4\nand CoFe 2O4the spin character of the slowest decaying\nstate changes within the gap. Therefore, NiFe 2O4and\nCoFe2O4are both capable of acting as either positive or\nnegative spin filters, depending on the band alignment\nand wave-function symmetry of the electrodes. Given\nsuch a relatively sensitive dependence of the tunneling\ncurrent on the position of the electrode Fermi level, we\nenvision that gating may allow the spin filtering to be\nswitched from positive to negative.\nHowever, we also want to note that based on the com-\nplex band-structure of the barrier alone, it is not pos-\nsible to make a definite prediction about the transport\nproperties observed in a specific experiment. One may\nstill encounter a situation where incident wave-functions\nwith the desired symmetry, i.e. matching that of the\nsmallest κ(0,0;E) inside the barrier, are not available\nwithin the electrodes, simply because of the correspond-\ning real band-structure38,39. Furthermore, it has been\ndemonstrated recently for the case of an Fe-MgAl 2O4-Fe\ntunnel junction, i.e. containing a non-magnetic spinel\nas barrier material, that the different unit cell sizes of\nthe spinel barrier and the Fe electrodes can open up new\ntransport channels due to “backfolding” of bands from\nthe in-plane Brillouin zone boundary onto the Γ point.40This leads to a relatively low tunnel magneto-resistance\nfor the Fe-MgAl 2O4-Fe junction, even though the corre-\nsponding complex and real band structures would indi-\ncate a highly symmetry-selective barrier.40,41Therefore,\nin order to fully assess the spin-filter efficiency for a spe-\ncific combination of electrode and barrier materials, a\nfull transport calculation for the entire device needs to\nbe performed. Nevertheless, the analysis of the complex\nband-structure provides a powerful interpretative tool\nand offers a good indication on what are the dominant\ncontributions to the tunneling current. 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Lett.\n100, 222401 (2012)."    },    {        "title": "1509.04062v1.Ground_state_phases_of_rung_alternated_spin_1_2_Heisenberg_ladder.pdf",        "content": "arXiv:1509.04062v1  [cond-mat.str-el]  14 Sep 2015Ground-state phases of rung-alternated spin-1/2 Heisenbe rg ladder\nF. Amiri,1G. Sun,2H.-J. Mikeska,3and T. Vekua3\n1Department of Physics, University of Guilan, Rasht, Iran\n2Max Plank Institut f¨ ur Physik komplexer Systeme, Dresden, Germany\n3Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at H annover, Germany\nThe ground-state phase diagram of Heisenberg spin-1/2 syst em on a two-leg ladder with rung\nalternation is studied by combining analytical approaches with numerical simulations. For the case\nof ferromagnetic leg exchanges a unique ferrimagnetic grou nd state emerges, whereas for the case of\nantiferromagnetic leg exchanges several different ground s tates are stabilized depending on the ratio\nbetween exchanges along legs and rungs. For the more general case of a honeycomb-ladder model\nfor the case of ferromagnetic leg exchanges besides usual ru ng-singlet and saturated ferromagnetic\nstates we obtain a ferrimagnetic Luttinger liquid phase wit h both linear and quadratic low energy\ndispersions and ground state magnetization continuously c hanging with system parameters. For\nthe case of antiferromagnetic exchanges along legs, differe nt dimerized states including states with\nadditional topological order are suggested to be realized.\nINTRODUCTION\nSpin-1/2 Heisenberg two-leg ladder systems have at-\ntracted a great deal of interest both from experiment\nand from theory[1]. Ladders with antiferromagnetic ex-\nchange along rungs and antiferromagnetic [2–4] as well\nas ferromagnetic [5, 6] exchanges along legs have been\nrealized experimentally.\nFor the case of a ladder with ferromagnetic legs and a\nuniform inter-leg (rung) exchange, so called rung-singlet\nor saturated ferromagnetic phases are realized depend-\ning whether the inter-leg coupling is antiferromagnetic\nor ferromagnetic.\nForthecaseofaladderwithantiferromagneticlegsand\na uniform rung coupling it has been established that for\nantiferromagnetic inter-leg coupling a rung-singlet phase\nis realized, whereas for ferromagnetic rung coupling a\nHaldane phase is stabilized. Both phases are stable both\nin weak rung-coupling and in strong rung-coupling lim-\nits. Weak rung-coupling limit is a proper limit for effec-\ntive field theory bosonization analyses [7], where in the\ncase of uniform rung exchanges, in the lowest (first) or-\nder of the inter-chain coupling the relevant operators (in\nthe renormalization group sense) are present that at low\nenergies drive system towards the strong-coupling fixed\npoints of the rung-singlet and Haldane states for positive\nand negative rung exchanges respectively.\nStability of a unique ground state from arbitrary weak\n(non-zero) up to arbitrary strong inter-leg exchanges, is\ndue to the fact that the ladder system is non-frustrated.\nThe question that we are going to address in our work\nis what happens when the ladder system is frustrated\nby rung exchange alternating in sign from rung to rung.\nFrustration in this case, for both signsof exchangesalong\nthe ladder legs, will be caused by the presence of an odd\nnumberofantiferromagneticexchangesin theelementary\nclosed path that is a ladder plaquette in our case. This\nproblem has not been addressed before and we will try\nto fill this gap in the following.J\nJS\n1,2jS2,2j 2,2j+1S\nS1,2j+1−JJ\nFIG. 1. Geometry of the two-leg ladder with alternated\nrung exchanges. Antiferromagnetic couplings along are cho -\nsen along the even rungs, J⊥>0.\nWe will use different complementary analytical ap-\nproaches: strong-rung coupling expansion for strongly\ncoupled legs and bosonization for weakly coupled antifer-\nromagnetic legs. To cover the intermediate regimes we\nwill use numerical techniques. We will as well consider\nthe more generalized case of a rung-alternated model\nwhere we will relax the constraint of equal absolute value\nof exchanges along the even and odd rungs.\nMODEL OF FRUSTRATED SPIN LADDER\nIn this work we study a Heisenberg spin-1/2 model\ndefined on a two leg ladder with Lrungs and with alter-\nnating rung exchanges, depicted in Fig. 1,\nH=JL/summationdisplay\nj=1,lSl,jSl,j+1+J⊥L/summationdisplay\nj=1(−1)jS1,jS2,j,(1)\nwhereSl,jare spin-1\n2operators acting on spins on the\nj-th rung of the l= 1,2 leg. For definiteness we will put\nJ⊥≥0, the case of J⊥≤0 will be recovered by one-site\ntranslation of the ladder along the legs.\nThe anisotropic XYcase of Eq. (1) was studied\nrecently in the context of single-component hard-core\nbosons on a two-leg ladder at half-filling with a flux π\nper plaquette [8, 9]. It was shown that depending on2\nthe ratio of strengths of exchanges along ladder legs and\nrungs there are two different ground-state phases. For\n|J| ≥Jc\nxy, whereJc\nxy≃2/3J⊥and for the XYcase\nthe sign of Jis irrelevant the ground state was shown\nto be a vortex-liquid Mott insulator, a state with gapped\nmagneticexcitation(excitationschangingtotal Sz), how-\never having a gapless mode to non-magnetic excitation.\nFor|J| ≤Jc\nxythe ground state was shown to be a fully\ngapped non-degenerate state, adiabatically connected to\ntheJ= 0 case where the ground state is of simple prod-\nuct form and is composed of alternating Sz= 0 com-\nponents of triplets and singlets from rung to rung. The\npresenceofmultiplegroundstateswhenchangingasingle\nparameter (here ratio of exchanges along legs and rungs)\nis an indicator of frustration present in the system.\nSTRONG RUNG-COUPLING LIMIT\nLet us start from the limit |J| ≪J⊥. ForJ= 0,\neven rungs in the ground state form rung-singlet states,\nwhereas odd rungs form rung-triplet states. Hence for\nJ= 0thegroundstatemanifoldis3L/2timesdegenerate.\nTreatingJperturbatively, integrating out singlets that\noccupy even rungs, we obtain an effective Hamiltonian\ndescribing a collection of odd rungs,\nH1\neff=J3\n16J2\n⊥L/2/summationdisplay\nj=1TjTj+1+O(J4),(2)\nwhereTj=S1,2j+1+S2,2j+1are effective S= 1 spins\nformedalongtheoddrungs. Becausetheexchangesalong\nboth legs have equal strengths the lowest, second order\ninJcontribution to the effective spin-1 chain formed\non odd rungs vanishes. The ground state of our model\nEq.(1) for |J| ≪J⊥will be hence a direct product of\nsinglets formed on even rungs and the ground state of a\nHeisenberg spin-1 chain formed on odd rungs. Depend-\ning on the sign of Jthe ground state of this Heisenberg\nspin-1 chain is either ferromagnetic state ( J <0), or\nHaldane state [10] ( J >0). For the original Hamiltonian\nEq.(1) ferromagnetic state of the effective S= 1 chain\nEq.(2) is the half-ferromagnetic state with the ground\nstate total spin equal to half of the maximum possible\nvalue,ST=L/2. This state will be called half-Ferro\nstate. The effect of anisotropies on the Haldane state\nrealized for 0 < J≪J⊥has been studied recently [11].\nFor the rung-alternated ladder with J >0 it was shown\nthat the application of an external magnetic field induces\nthe half-magnetization plateau state [12].WEAK RUNG-COUPLING LIMIT,\nBOSONIZATION\nIn the other limit J≫J⊥we use bosonization\napproach[7]. We represent spin operators with the help\nof bosonic operators:\nSl,j→a[Jl,L(x)+Jl,R(x)+(−1)jNl(x)].(3)\nDecoupled chains have N´ eel-like quasi long-range or-\nder and the above representation captures the impor-\ntant low energy fluctuations by smooth bosonic fields,\nat wave-vector 0 and π. Uniform spin magnetiza-\ntion is represented in terms of chiral currents Jl,L/R\nofalevel-1SU(2) Wess-Zumino-Witten model per-\nturbed by marginally irrelevant current-current interac-\ntions, describing an isolated antiferromagnetic Heisen-\nberg chain[13].\nWe will need the following important operatorproduct\nexpansion (OPE) rules[14],\nJa\nl,L/R(x,τ)Nb\nl(x′,τ′) =±iδabǫl(x′,τ′)+iǫabcNc\nl(x′,τ′)\n4π[v(τ−τ′)±i(x−x′)],\n(4)\nwherev=πJ/2 is the spin-wave velocity of the\nHeisenbergspin-1/2chain, knownfrom the Bethe ansatz\nsolution, and on the right hand side the dimerization\noperatorǫl, thatisthecontinuumlimitof( −1)jSl,jSl,j+1\nhas appeared.\nTreatingthe inter-chaincoupling J⊥perturbativelyfor\nJ⊥/J≪1 in the continuum limit the staggered inter-\nchain coupling, H⊥=J⊥/summationtextL\nj=1(−1)jS1,jS2,j, has the\nfollowing form in terms of smooth bosonic fields,\nH⊥=/integraldisplay\ndxH⊥(x) =J⊥/integraldisplay\ndx(J1,L(x)+J1,R(x))N2(x).\n+J⊥/integraldisplay\ndx(J2,L(x)+J2,R(x))N1(x). (5)\nTheH⊥perturbation has non-zero conformal spin and\ndoes not open a gap in the first order of J⊥. The relevant\nscalar operator from inter-chain exchange comes in the\nsecond order of J⊥coupling.\nUsing OPE for the same-leg operators at short-\ndistances Eq.(4) and integrating with the relative coor-\ndinates we obtain in the order J2\n⊥the following relevant\ncontributions that should be added to effective Hamilto-\nnian describing the long wavelength properties of decou-\npled bosonic chains,\n∼ −J2\n⊥/integraldisplay\ndx[3ǫ1(x)ǫ2(x)−2N1(x)N2(x)].(6)\nAt this point it is convenient to introduce 4 Majorana\nfermions [15]. The perturbation that we identified in\nEq.(6) is translated as mass term of triplet and singlet\nMajoranafermions and the complete ladder Hamiltonian3\nS =L/2\n0J<0 J Rung−SingletT\nHalf−Ferro H1\nJ>>J?\nJ<<J\nFIG. 2. Analytically conjectured ground-state phase di-\nagram of rung-alternated Heisenberg ladder as function of\nJ. H1denotes a Haldane phase, however only half of the\nrungs (odd numberedrungs) providethe effective S= 1 spins,\nwhereas the other half (even numbered rungs) are in the ap-\nproximate rung-singlet states and mediate antiferromagne tic\nexchange among effective S= 1 spins.\nin Majorana basis looks,\nHM=/integraldisplay\ndx[3/summationdisplay\nγ=1ivt\n2(ψγ\nL∂xψγ\nL−ψγ\nR∂xψγ\nR)+imtψγ\nLψγ\nR]\n+/integraldisplay\ndx[ivs\n2(ψ0\nL∂xψ0\nL−ψ0\nR∂xψ0\nR)+imsψ0\nLψ0\nR],(7)\nwhere the masses of Majoranas are: mt∼5J2\n⊥and\nms∼3J2\n⊥. Followingthe analysesof the groundstates of\nthe original ladder system from Majoranas [7, 14] we can\nconclude that the ladder system for J⊥/J→0 is in the\nphase that is adiabatically connected to the rung-singlet\nphase of the uniform ladder with antiferromagnetic ex-\nchanges. This is surprising, because the spin exchanges\nare ferromagnetic along every other rung.\nComparing the effective Hamiltonian Eq. (2) and the\noriginal one Eq. (1) reveals a very important compe-\ntition forJ >0. Namely, for small 0 < J≪J⊥the\nnext nearest neighbour spins on the same chain belong-\ning to the odd rungs show antiferomagnatictendencies,\n/angbracketleftSl,2j+1Sl,2j+3/angbracketright<0, whereas for the single-chain domi-\nnated regime J≫J⊥, it is clear that /angbracketleftSl,jSl,j+2/angbracketright>0 for\nanylandj. There is no such competition on J <0 side\nand hence the case of ferromagnetic legs is simpler and\nfor the entire region of J <0 the ferromagnetic phase of\nthe effective spin-1 chain is expected to be realized (for\nthe original ladder it is the half-Ferro state with ground\nstate total spin ST=L/2).\nSo far, using analytical approaches, we have estab-\nlished ground-state phases of rung-alternated Heisenberg\nspin-1/2 ladder in the limiting cases of: 0 <J≪J⊥and\nJ≫J⊥, where a Haldane phase of effective S= 1 spins\nformed along the odd rungs and a rung-singlet phases\nare stabilized respectively. The conjectured sequence of\nphases with changing Jis depicted in Fig. 2.\nTo check whether there are additional phases for in-\ntermediate values of J∼J⊥or there is a direct phase\ntransition between H1and RS states, we will use numer-\nical approaches.\nα−1\n1,2jSJJS2,2jα+1\nS1,2j+1S2,2j+1\nFIG. 3. Geometry of honeycomb-ladder model. For α= 0\nrung-alternated ladder presenting in Fig. 1 is recovered.\nHONEYCOMB-LADDER MODEL\nIn this section we will study a slightly generalized case\nof the rung-alternated ladder, introducing the additional\nparameterαthat relaxes the condition of equal absolute\nvalues of exchanges along even and odd rungs. This way\nwe obtain a SU(2) symmetric version of the so called\nhoneycomb-ladder model [16],\nH=J⊥L/summationdisplay\ni=j[(−1)j+α]S1,jS2,j+JL/summationdisplay\nj=1,lSl,jSl,j+1.(8)\nTo simplify notations in the following we will set J⊥= 1.\nForα∈(−1,1) the spin system is frustrated, because\nthe number of antiferromagnetic bonds per ladder pla-\nquette is odd. We will study the ground-state phases for\narbitary values of αandJ(both negative and positive).\nThe simpler case to start from is J <0.\nHoneycomb-ladder with ferromagnetic legs\nForJ <0 we can estimate the boundary of the fer-\nromagnetic phase. We derive the single-particle disper-\nsions, composed of 4 branches, since the unit cell con-\ntains 4 spins,\nε1(k) =α\n2+Jcosk, ε2(k) =α\n2−Jcosk\nε3(k) =−α\n2−√\n2+J2+J2cos2k√\n2\nε4(k) =−α\n2+√\n2+J2+J2cos2k√\n2. (9)\nthe chemical potential is attached to the minimum of\nε1(k) band, realized at k= 0,µ=α/2+J. By looking\nat the single magnon instability we estimate the bound-\nary of the fully polarized state, equating the chemical\npotential to the minimum of ε3(k) band, which is given\nfor any system size by\nαFM=−/radicalbig\n1+J2−J≃ −1\n2|J|for|J| ≫1.(10)\nNext, for |J| ≪1 the single-particle dispersion bands\nbecome flat and we can estimate the transition from the4\nferrimagnetic state into the half-Ferro state with total\nS=L/2 by equating the chemical potential with the\nmaximum of ε3(k) band realized at k=±π/2,\nαc=−J−1+O(J2). (11)\nTransitions induced by changing α<0 can be under-\nstood in a simple way for |J| ≪1. In the half-Ferro\nstate spins on even sites form rung-singlets, whereas on\nodd sites spins form effective spins S= 1. The effective\nspin-1 chain, formed by spins on odd sites, is in the fully\npolarized phase as follows for J <0 from Eq. 2. The in-\nteraction between the spins belonging to even sites along\nthe leg direction is mediated by intermediate S= 1 spins\n(all of which are ponting in the same, spontaneously cho-\nsen, direction due to the fact that they are in fully po-\nlarized ground state) and this interaction is ferro since\nJ <0. Hence, thesystemisequivalenttothedirectprod-\nuct of the ferromagneticstate ofthe spin-1 chain (formed\nby spins belonging to odd sites) and a 2-leg spin-ladder\nwith antiferromagnetic exhcange along rung and ferro-\nmagnetic exchange along legs[17] (the ladder is formed\nby spins belonging to even sites). Decreasing αweak-\nens the antiferromagnetic coupling along the rung and\nthe effective magnetic field (produced by the fully polar-\nized neighbouring spin-1 chain) induces two consequtive\nsecond ordercommensurate-incommensuratetransitions,\nfirst one from rung-singlet state of the two-leg ladder\n(with ferromagnetic legs and antiferromagnetic rungs) to\nan intermediate Luttinger liquid state with finite polar-\nization and then to fully polarized state[18].\nSingle-magnon (equivalently hard-core boson) disper-\nsions as presented in Fig. 4 shed light on the evolution\nof the low-energy excitation spectrum of model Eq.(8)\nas function of αforα <0 andJ <0 and fully con-\nfirm the above picture anticipated from the interpene-\ntrating spin-1 chain and the two-leg ladder with ferro-\nmagneticlegsand antiferromagneticrungexchanges. For\nα<−√\n1+J2−Jsystem is in the ferromagnetic state\nand the low energy excitation is a conventional fer-\nromagnetic magnon, with ∼cosk-like gapless quadratic\ndispersion shown in Fig. 4 (a). At the boundary be-\ntween ferromagnetic and ferrimagnetic phases there are\ntwo quadratic dispersions, two kinds of magnons with\nthe mass ratio of√\n2 presented in Fig. 4 (b). Inside the\nferrimagnetic state the ground state total spin changes\ncontinuously L/2≤ST≤Land there are two differ-\nent kinds of dispersions: one quadratic at low momenta\nand a second one that is linear shown at ±kFit Fig. 4\n(c), givingLuttingerliquid likeproperties. Atthebound-\narybetweenferrimagneticandhalf-Ferrostatesagaintwo\nquadratic in momenta low energy dispersions are present\nshown in Fig. 4 (d) and inside the half-Ferro state with\nST=L/2 only one magnon branch with ∼cosk-like\nquadratic low energy dispersion remains as shown in Fig.\n4(e). Bothphasetransitions,fromferromagnetictoferri-\nmagnetic and from ferrimagnetic to half-Ferro states arek\n(a)\n−π/2ε(  )\n0kπ/2k\n(b)\n−π/2 0 π/2kε(  )\nk\n(c)ε(  )\n0 −π/2 π/2εεεε1\n2\n3\n4\nkkF −kFε(  )k\n−π/2 π/2 0k(d)\n(e)\n−π/2 π/2kε(  )\n0k\nFIG. 4. (Color online) Single particle dispersion relation s\n(a) deep in ferro state, (b) at the boundary of ferro and ferri\nstates, (c) inside ferri state, (d) at the boundary of ferri a nd\nhalf-ferro states and (e) inside the half-Ferro state. the\ndashed line indicates the chemical potential.\nsecond order commensurate-incommensuratephase tran-\nsitions, where the linear mode disappears in favor of a\nquadraticdispersion. Theoverallgaplessquadraticmode\nremains in the background, due to the spontaneously\nbrokenSU(2) symmetry in all phases where the ground\nstate is not a global spin singlet. This makes SU(2) sym-\nmetric honeycomb-ladder model Eq.(8) for J <0 a very\nattractive and simple case to study the behavior known\nas unsaturated ferromagnetism, an effect that has been\nnoticedtooccurinotherfrustratedsystems[19–23]. With\nfurther increase of αto positive values there is a phase\ntransition from half-Ferro to rung-singlet state, not cap-\ntured by the single particle picture.\nEffective model for α≃1and|J| ≪1\nForα≃1 and|J| ≪1 we can derive an effective\nmodel for spins on odd rungs by integrating out spins\nbelonging to even rungs that are in rung-singlet states.\nTo capture the difference between J <0 andJ >0 cases\nwe have to go beyond the lowest (second) order in inter-\nrung coupling Jthat we treat in perturbation theory. To\nthe third order in Jthe effective ladder model formed by\nspins belonging to odd rungs is givenby the the following5\nS\nSS\nSJJ\nJxr2,2j−1\n1,2j+12,2j+1\n1,2j−1\nFIG. 5. Effective model for spins belonging to odd rungs\nvalid for α≃1 andJ≪1, where spins belonging to even\nrungsformapproximaterung-singletstates. Phasetransit ions\nare expected for Jr<0.\nhamiltonian (see Fig. 5),\nHeff=J||L/2/summationdisplay\nj=1[S1,2j−1S1,2j+1+S2,2j−1S2,2j+1]\n+JxL/2/summationdisplay\nj=1[S1,2j−1S2,2j+1+S1,2j+1S2,2j−1]\n+JrL/2/summationdisplay\nj=1S1,2j+1S2,2j+1 (12)\nwhere\nJ||=−J2\n2(α+1)−J3\n4(α+1)2\nJx=J2\n2(α+1)+3J3\n4(α+1)2\nJr=α2+J2−1\nα+1+3J3\n2(α+1)2. (13)\nInterchanging legs with diagonals of the effective\nmodel presented in Fig. 5 by interchanging spins on ev-\nery other rung, the model Eq.(12) for parameters given\nin Eq. (13) is equivalent to a two-leg spin ladder with\nantiferromagnetic legs, ferromagnetic diagonals and rung\nexchange that changes from antiferromagnetic to ferro-\nmagnetic with decreasing α. For the two-leg antiferro-\nmagnetic ladder weakly coupled by competing diagonal\nand rung exchanges (where bosonization is applicable)\nthreephasesareexpectedtobestabilizedwithdecreasing\nα: rung-singlet, dimer and Haldane phase [14]. Parame-\nters of our effective model are outside the weak-coupling\nlimit, but later with the help of numerical simulations\nwe will show that the same sequence of phases are also\nrealized in our effective model Eq.(12). In particular, the\ndimerization pattern of the original ladder model will be\ndimers formed along next nearest neighbor diagonals, in-\nvolving spins belonging to odd rungs.Vicinity of α≃ −1,0< J≪1\nForJ >0 and 0<J≪1 we can as well estimate pos-\nsible ground states around α≃ −1, where we can borrow\nthe results from the mixed diamond chain [24, 25]. For\nJ= 0 andα=−1 the spins belonging to even rungs\nare disconnected and spins belonging to odd rungs in the\nground state are in the spin-triplet configuration, form-\ningS= 1 spins. For 0 <J≪1 there is a competition in\nthe nature of the exchange between the spins belonging\nto the even rungs: for α>−1 the direct exchange S= 1\nis antiferromagnetic, whereas the exchange mediated by\nnearbyS= 1 spins is ferromagnetic. For α<−1 there is\nnosuchcompetition andeffective spins S= 1areformed\non each rung.\nOne possibility that the above mentioned competition\nforα>−1 gets resolved is that some of the even rungs\nchooseto bein triplet state andothersinsinglet statepe-\nriodically alternating as happens in the mixed-diamond\nchain [24] where consecutive odd number of Mrung-\ntriplets (coupled antiferromagnetically with each other\nbyJ >0) will be sandwiched between the rung-singlets.\nCoupling an odd number MofS= 1 spins by antiferro-\nmagnetic exchange and assuming open boundary condi-\ntions, theM-rung segment will be in the triplet state in\nthe ground state, forming an effective S= 1 spin. The\napproximate rung singlets in the case of rung-alternated\nladder (as opposed to the exact rung singlets realized in\nmixed-diamond chain [24] that cut the chain) will medi-\nate an effective antiferromagnetic exchange among the\nabovementioned effective S= 1 spins formed by M-rung\nsegments, giving rise to generalized Haldane states with\nan enlarged unit cell composed of M+1 ladder plaque-\nttes. Such a state for M= 3 is depicted in Fig. 6 and\ncalled Haldane-dimer.\nThere are in total 5 different ground states in the\nmixed-diamond chain when changing the equaivalent of\nαfromα>−1 toα<−1 withM= 1,3,5,7 andM=\n∞. TheM= 1 case is equivalent to the H 1state,\na Haldane state of S= 1 spins formed on odd rungs,\nand theM=∞case is equivalent to the Haldane state\nof the effective S= 1 spins formed on every rung. For\nthe spin-1/2 ladder with alternated rungs M= 1,3,5\nandM=∞are suggested to have finite extent in the\npresence of exchange anisotropy [25].\nNUMERICAL RESULTS\nRung-alternated ladder\nInitially we will present our numerical data for α= 0\ncase corresponding to the rung-alternated ladder model\nEq. (1). We use both Lanczossimulation and the density\nmatrix renormalizationgroup(DMRG) approach[26, 27]\nin order to access large system sizes. For the case of6α+1S=1 S=1\nS=0 S=0α−1magneticferroantiexchange\nFIG. 6. Cartoon of one of the possible ground state config-\nurations of the Haldane-Dimer phase that can be realized\nforα≃ −1, 0< J≪1. Spins encircled by open rectangles\nform approximate rung-singlet state, whereas those encirc led\nby shaded rectangles form rung-triplet states. 6 spins enci r-\ncled by the dotted rectangle form effective S= 1 spins which\nare connected via intermediate singlets to produce an effec-\ntive Haldane chain. In the mixed-diamond chain the singlets\ndepicted above are exact eigenstates and they do not mediate\nany exchange among the effective S= 1 spins.\nferromagnetic legs we systematically obtain (both using\nlarge scale DMRG as well as Lanczos algorithm for both\nperiodic and open boundary conditions) that the ground\nstate belongs to the multiplet with total spin half of the\nmaximal possible value, ST=L/2 for anyJ <0.\nIn the following we will discuss J >0, where we pre-\ndicted at least two different phases in the limiting cases\nJ≪1 andJ≫1 respectively. In Fig. 7 we plot the fi-\ndelity susceptibility[28–31] with changing control param-\neterJfor different system sizes.\nχL=−2\nLlim\nδJ→0ln|/angbracketleftψ0(J)|ψ0(J+δJ)/angbracketright|\n(δJ)2,(14)\nwhere|ψ0(J)/angbracketrightis the (non-degenerate)groundstatewave-\nfunction for the corresponding parameter J. We see that\nthereisawellpronouncedpeakin thefidelitysusceptibil-\nity and the height of the peak increaseswith system size,\nwhereas the width decreases. We extrapolate the loca-\ntion ot the peak to J=Jc1≃0.45 in the thermodynamic\nlimit. Thus, we can estimate the extent of the H 1phase\nforα= 0 as 0< J≤Jc1. Note, the rather similar esti-\nmate ofJc1follows from the position of the level-crossing\nof the lowest excited states, which are triplet with mo-\nmentumk=πin H1phase (J < J c1) and triplet with\nk= 0 in its neighboring phase ( J >Jc1).\nHoneycomb-ladder with ferromagnetic legs: J <0\nWe start by presenting our numerical data with the\ncase of ferromagnetic legs. In Fig. 8 we plot for the\nhoneycomb-ladder with ferromagnetic legs the ground\nstate multiplicity as function of αforJ=−1, which is a\ntypical behavior in the whole J <0 region. The bound-\nary of the ferromagnetic phase is captured exactly from\nthe spin-wave instability (indicated by dashed verticalJ0.3 0.35 0.4 0.45 0.5 0.55 0.6χL\n05101520\nL=12\nL=16\nL=24\nFIG. 7. (Color online) Fidelity susceptibility of the rung-\nalternated ladder with antiferromagnetic legs for periodi c\nboundary conditions and 3 different system sizes. In DMRG\ncalculations periodic boundary conditions are assumed (re -\nstricting considerably the available ladder lengths) to av oid\ndegeneracies of the Haldane-like states due to edge spins fo r\nthe open boundaries.\nα -0.5 0 0.5 1∆E\n00.020.040.060.080.1\nE0(Sz=1) -E0(Sz=0)\nE0(Sz=2) -E0(Sz=0)\nE0(Sz=48)-E0(Sz=0)\nE0(Sz=49)-E0(Sz=0)\nFIG. 8. (Color online) DMRG results for the lowest energy\nlevels in different total Szsubspaces relative to the lowest\nenergy in the Sz= 0 subspace as function of αforJ=−1\nandL= 96 rungs for open boundary conditions. The dashed\nline indicates the boundary of the ferromagnetic state.\nline in Fig. 8). Here we only present the data from which\nwe determine the boundaries of the half-Ferro state.\nWe see from this plot that for J=−1 the half-Ferro\nstateST=L/2 is sandwiched between α≃ −0.2 and\nα≃0.36. Note, for 0 .36<α<0.4 the excitations from\nthe singlet ground state to low total spin states e.g. to\nstateswith ST= 1and2becomepracticallygapless,thus\nwe cannot rule out the existence of an intermediate thin\nphase between half-Ferro and rung-singlet states based\non our numerical data.\nFor values of α <−0.2 energies of the lowest states\nwithSz> L/2 merge gradually with the ground state\n(only one state Sz=L/2 + 1 is indicated in Fig. 8)\nuntil the energy of the fully polarized state with Sz=7\n0.94 0.95 0.96 0.97 0.98 0.99 100.00050.001\nk=0  triplet\nk=    triplet\nk=    singlet\nα∆E\nππJ=0.1 (a)\n0.9 0.92 0.94 0.96 0.9800.0020.0040.006\nk=0  triplet\nk=    triplet\nk=    singlet\nα∆E(b) J=0.2\nπ\nπ\nFIG. 9. (Color online) Lanczos results of the lowest excited\nstates relative to the ground state obtained for L= 12 rungs\nof the effective model Eq.(12) for (a) J= 0.1 and (b) J=0.2.\nA similar picture is expected to hold for the Honeycomb-\nladder model with L= 24 rungs for the same extent of α\nandJ. Periodic boundary conditions are used that allows to\nassign a definite lattice momentum to each level.\nLbecomes degenerate with the ground state energy for\nα≤αFM,αFM(J=−1) = 1−√\n2≃ −0.414.\nHoneycomb-ladder with antiferromagnetic legs:\nJ >0\nForJ >0 case we start presenting numerical data near\nthe pointα≃1 for small Jwith changing α. To distin-\nguish different phases it is usefull to start from looking\nat the gap between the ground state and lowest excited\nstates. In Fig. 9 we depict the lowest excited states as\nfunction of α. We use the effective model Eq. (12) to\nreach system sizes of L= 12 rungs, that is equivalent\ntoL= 24 rungs for the Honeycomb-ladder model Eq.\n(8). We have checked that for available system size (up\ntoL= 12 for Honeycomb-ladder model) agreement be-\ntween the low energy levels of effective and full models is\nperfect for small Jvalues. In Fig.9 we present the level\nspectroscopy results for J= 0.1 (a) andJ= 0.2 (b) for\nthe effective model Eq. (8). In an antiferromagnetic lad-\nderwith auniformantiferromagneticexchangethelowest\nexcited state in the rung-singlet phase is a triplet state\nwith wavevector k=πin units of the ladder lattice con-stant. Since in our model the unit cell is made oftwo pla-\nquettes, in the rung-singlet phase of Honeycomb-ladder\n(in the phase that is adiabatically connected with rung-\nsinglet phase of the uniform ladder, but with a unit cell\nhalf of the Honeycomb-ladder model) the lowest excited\ntriplet should have momentum k= 2πin the units of the\nHoneycomb-ladder unit cell that is equivalent to k= 0\nmomentum.\nWe see that with decreasing αbelowα= 1 the gap to\nthe lowest excitation (triplet state with k= 0 momen-\ntum) shows a minimum and then with reducing αthis\nlowest triplet excitation level crosses with the lowest ex-\ncited singlet state that has momentum k=π. Note, for\nanyJ >0 and anyαthe ground state is a spin singlet\nstate withk= 0 momentum. There is a finite extent in\nαwhere the lowest excited state is a singlet state with\nk=π. With further reducing αthere is a level-crossing\nbetween the lowest spin singlet excitation with k=π\nand spin triplet excitation with k=π. The spin triplet\nexcitation with k=πin units of Honeycomb-ladder unit\ncell is the lowest excitation on top of the Haldane state\nthat is defined on the effective spin-1chain with the same\nunit cell as the original microscopic model. One can use\nthe abovementioned two level crossings in excited states\nto estimate the stability region of the intermediate dimer\nphase. In fact with increasing system size the singlet ex-\ncitation atk=πshould get degenerate with the ground\nstate singlet in the dimerized phase. The boundary be-\ntween the rung-singlet and dimer states can be estimated\nfrom the position of the minimum of the gap of the k= 0\ntriplet state.\nComparing the energy levels of the effective ladder\nmodel Eq.(12) for different system sizes with L≤12\nrungs we see that the energy of singlet state with k=π\nmomentum decreases faster, with increasing the system\nsize, than energies of the triplet states in the parameter\nregion where the dimerized phase is expected.\nThe numerical ground-state phase diagram of the\nhoneycomb-ladder model obtained with the help of\nDMRG simulations is presented in Fig. 10. For J <0\nthere are four different phases realized with decreasing α:\nrung-singlet, half-ferro, ferrimagnetic and ferromagnetic.\nFor the case J >0 the rung-singlet state, the H 1and\nthe conventional Haldane state and different dimerized\nstates: NNND and Haldane-Dimer are realized. Between\nthe NNND-Dimer and Haldane-Dimer states we can not\nlocate numerically the phase transition line, neither can\nwe exclude emergence of an intermediate (gapless) state\nlocated around α= 0.\nIt is worth noting that the topology of the H 1phase\nrealized for J≥0 can be captured by studying one pla-\nquette of the ladder, L= 2. Consider e.g. the α= 0\ncase. For this case for J <1/√\n2 a triplet state is realized\nas ground state, whereas for J >1/√\n2 the ground state\nbecomes a singlet, a direct product of the singlet states\non the first chain (2-site chain) and on the second chain8\nJ −1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5\nα\nFerroHalf−FerroRung−singlet\nHaldane1HNNND−Dimer\nHaldane−DimerFerri\nFIG. 10. Numerical ground-state phase diagram of\nthe honeycomb-ladder model in the parameter plane ( J,α).\nNNND-Dimerstandsfornextnearestneighbordiagonal dimer\nphase where dimers are formed along next nearest neighbor\ndiagonals involving spins of odd rungs. In the vicinity of\nα=−1 and for 0 < J≪1 before the transition from\nHaldane-DimertoHaldanephaseadditional phasesmayoccur\n(e.g. with M= 5 and M= 7 as discussed in previous sec-\ntions). In dimer phases ground states are doubly degenerate\nin the thermodynamic\nlimit.\n(that is an exact eigenstate for any Jin the case of a\nsingle plaquette). Hence at J= 1/√\n2 there is a triplet-\nsinglet levelcrossingin the groundstate ofoneplaquette.\nThe (threefold) degeneracy of the ground state for small\nvalues ofJis a particular case and omit:it stems from\nthe fact that there is only one effective spin 1 (formed\non one of the two rungs). As soon as the number of\nladder plaquettes is increased and more than one effec-\ntive spin 1 is formed on odd rungs the ground state\nbecomes a singlet (for periodic boundary conditions) for\nthe whole range of J >0 and there is no level crossing in\nthe ground state any more. However, when one assem-\nbles many plaquettes into the ladder geometry, instead\nof the level-crossing, one can identify the avoided level\ncrossing in the lowest energy singlet states of the finite\nladder (for system sizes L≤12 rungs), that is located at\nJ∼0.5 (data not shown).\nThe phase transition points indicated in Fig. 10 for\nJ >0 were obtained by studying the behavior of the fi-\ndelity susceptibilityasfunction of αfordifferentvaluesof\nJas presented in Fig. 11. For small values of J(roughly\nJ <0.5) the fidelity susceptibility shows typically well\npronounced four peaks, whereas for J >0.5 only two\npeaks are visible, one for positive and a second one for\nnegativeα. The peak for the α >0 side becomes less\nand less pronounced with increasing J >1.\nTo describe the regime corresponding to J≫1 in\nFig.12 we present the behavior of the lowest excitation\ngapasfunctionof αusingDMRGforlargevalueof J= 5.\nIn order to access large system sizes we use open bound-\nary conditions. One can see that with decreasing αfirstα -0.5 0 0.5 1χL\n051015L=12\nL=16\nL=24(a)\nα -1 -0.5 0 0.5 1χL\n00.20.40.60.81\nL=12\nL=16\nL=24(b)\nFIG. 11. (Color online) Ground state fidelity susceptibilit y\nper site as function of αfor (a)J= 0.4 and for J= 1 showing\ntwo peaks. In DMRG calculations periodic boundary condi-\ntions are assumed to avoid degeneracies of the Haldane-like\nstates due to edge spins for open boundaries.\nthere is a minimum in gap and then there is a cusp-\nlike behavior. Using the finite system size data for sys-\ntems withL= 48,96 and 144 rungs the position of the\ngap minimum in the thermodynamic limit extrapolates\nclearly to negative values of α. Starting from the rung-\nsinglet phase, the gap decreases linearly with decreasing\nαand the position of the minimum of the gap we inter-\npret as a boundary of the rung-singlet phase.\nOn the other hand, for J≫1, extending the bosoniza-\ntion analyses to α/negationslash= 0 gives that for α >0 the\nrung-singlet phase smoothly evolves into the rung-singlet\nphaseoftheuniformantiferromagneticladderrealizedfor\nα≫1. Forα<0 interestingly bosonization suggests the\nsequence of two consecutive second order phase transi-\ntions, first from rung-singlet to an intermediate dimer\nphase and then from dimer to Haldane phase with de-\ncreasingα. Hence we expect to see two values of α <0\nwhere gap should close in the thermodynamic limit. In-\nstead wesee only one minimum in the finite-size gap data\npresented in Fig. 12. The reason why we do not see the\nsecond minimum may be the fact that finite-size effects\nare still large (even for L= 144 rungs). In addition,9\nα−0.5−0.4 −0.3−0.2 −0.100.1 0.2 0.3 0.4 0.5E\n00.10.20.30.40.5\nL=48\nL=96\nL=144\nJ=5∆\nFIG. 12. Gap between the ground state and the first excited\n(triplet) state as function of αforJ= 5 obtained by DMRG\nusing open boundary conditions.\nsince we use open boundary conditions, it is difficult to\nseparate true bulk gap from the boundary gap of the\nHaldane phase (realized on the left side from the kink\nin Fig. 12). It is desirable to study the gap for peri-\nodic boundary conditions, however DMRG calculations\nbecome less accurate and only much smaller system sizes\ncan be addressed.\nCONCLUSIONS\nWe have studied the ground-state phase diagram of\nthe rung-alternated SU(2) symmetric spin −1/2 ladder.\nBoth cases with ferromagnetic as well as antiferromag-\nnetic leg exchanges have been considered. For the case of\nferromagneticlegsweshowedthata unique ferrimagnetic\nground state emerges, with ground state magnetization\nequaltohalfofthemaximumpossiblevalue, forarbitrary\nstrength of the leg exchanges. The case of antiferromag-\nnetic leg exchange is much richer and depending on the\nratio of leg to rung couplings several different ground\nstates can emerge starting from the Haldane phase H 1\nfor small leg couplings and ending with the rung-singlet\nphase for strong leg couplings. Based on Fig. 10 it is\ntempting to speculate that dimer order extends to α= 0\nand hence the intermediate phase of rung-alternated lad-\nder can be dimerized, even though we havenot succeeded\nin either directly measuring dimerization order, or find-\ning a second singlet state as the lowest excited state of\nthe finite chain or even resolving a finite excitation gap\nnumerically.\nWe have as well studied a generalization of rung-\nalternated ladder: the spin −1/2 Heisenberg system on\nhoneycomb-ladder lattice. For the case of ferromag-\nnetic legs we have identified a peculiar Luttinger liquidferrimagnetic state, where the ground state magnetiza-\ntion changes continuously as function of system param-\neters and low energy gapless excitations consist of two\nbranches one of which is linear and another quadratic\nin momentum. For the case of antiferromagnetic leg\ncouplings different short-range ground states, including\nthose with possible Haldane-like topological order have\nbeen suggested to occur.\nThis work has been supported by DFG Research\nTraining Group (Graduiertenkolleg) 1729 and center for\nquantum engeneeringand space-time research(QUEST).\nWork of F. A. was done while visiting Institute of The-\noretical Physics, Leibniz University of Hanover. F. A.\nacknowledges grant from the ministry of science and\ntechnology of Iran and support from deputy of research\nand technology of university of Guilan. We thank S.\nGreschner for numerical assistance.\n[1] H.-J. Mikeska and A. K. Kolezhuk, Lecture Notes in\nPhysics645, 1 (Springer-Verlag Berlin Heidelberg 2004).\n[2] M. Takano, Z. Hiroi, M. Azuma, and Y. Takeda, Jpn. J.\nAppl. Phys. Ser. 7, 3 (1992); Z. Hiroi et al., J. Solid State\nChem.95, 230 (1991).\n[3] T.M. Rice, S. Gopalan, and M. Sigrist, Europhys. Lett.\n23, 445 (1993).\n[4] R. S. Eccleston, T. Barnes, J. Brody, and J. W. Johnson,\nPhys. Rev. Lett. 73, 2626 (1994).\n[5] H. Yamaguchi, et al. Phys. Rev. Lett. 110, 157205\n(2013).\n[6] H. Yamaguchi, et al. Phys. Rev. B 89, 220402 (2014).\n[7] A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik,\nBosonization and Strongly Correlated Systems , Cam-\nbridge University Press (1998); T. Giamarchi, Quan-\ntum Physics in One Dimension , Oxford University Press\n(2003).\n[8] M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, S.\nGreschner, T. Vekua, and U. Schollw¨ ock, Phys. Rev. B\n91, 140406(R) (2015).\n[9] M. Di Dio, S. De Palo, E. Orignac, R. Citro, M. L. Chio-\nfalo, Phys. Rev. B 92, 060506(R) (2015).\n[10] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n[11] T. Tonegawa et al, in preparation. We thank Dr. Tone-\ngawa for sending us a so far unpublished account of this\nwork.\n[12] G. I. Japaridze and E. Pogosyan, J. Phys.: Condens.\nMatter18, 9297 (2006); G. I. Japaridze and S. Mah-\ndavifar, Bulletin of the Georgian National Academy of\nSciences, vol. 2, no. 4, (2008).\n[13] I. Affleck, Phys. Rev. Lett. 55, 1355 (1985).\n[14] O. A. Starykh and L. Balents, Phys. Rev. Lett. 93,\n127202 (2004).\n[15] D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, Phys.\nRev. B53, 8521 (1996).\n[16] G. Karakonstantakis, L. Liu, R. Thomale, and S. A.\nKivelson, Phys. Rev. B 88, 224512 (2013).\n[17] T. Vekua, G.I. Japaridze, and H.-J. Mikeska, Phys. Rev.\nB67, 064419 (2003).\n[18] T. Vekua, G.I. Japaridze, and H.-J. Mikeska, Phys. Rev.10\nB70, 014425 (2004).\n[19] K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Con-\ndens. Matter 8, 6405 (1996).\n[20] H. Nakano and M. Takahashi, J. Phys. Soc. Jpn. 66, 228\n(1997).\n[21] T. Shimokawa and H. Nakano, J. Phys. Soc. Jpn. 80,\n043703 (2011).\n[22] G. Sun, A. K. Kolezhuk, L. Santos, and T. Vekua, Phys.\nRev. B 89, 134420 (2014).\n[23] S. C. Furuya and T. Giamarchi, Phys. Rev. B 89, 205131\n(2014).\n[24] K. Takano, H, Suzuki, and K. Hida, Phys. Rev. B 80,104410 (2009).\n[25] K. Hida, J. Phys. Soc. Jpn. 83, 114711 (2014).\n[26] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys.\nRev. B48, 10345 (1993).\n[27] U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005).\n[28] L. Campos Venuti and P. Zanardi, Phys. Rev. Lett. 99,\n095701 (2007).\n[29] W.-L. You, Y.-W. Li, and S.-J. Gu, Phys. Rev. E 76,\n022101 (2007).\n[30] S.-J. Gu, Int. J. Mod. Phys. B 24, 4371 (2010).\n[31] G. Sun, A. K. Kolezhuk, and T. Vekua, Phys. Rev. B 91,\n0114418 (2015)."    },    {        "title": "1905.00595v1.Strong_magnetoelectric_coupling_in_mixed_ferrimagnetic_multiferroic_phases_of_a_double_perovskite.pdf",        "content": "Strong magnetoelectric coupling in  mixed ferrimagnetic-multifer roic \nphases of a double perovskite \n \nM. K. Kim, J. Y. Moon, S. H . Oh, D. G. Oh, Y. J. Choi*, and N. Lee* \nDepartment of Physics, Yonsei  University, Seoul 03722, Korea \n \nExploring new magnetic materials is essential for finding advan tageous functional properties such as \nmagnetoresistance, magnetocaloric effect, spintronic functional ity, and multiferroicity. Versatile \nclasses of double perovskite compounds have been recently inves tigated because of intriguing physical \nproperties arising from the proper combination of several magne tic ions. In this study, it is observed \nthat the dominant ferrimagnetic phase is coexisted with a minor  multiferroic phase in single-crystalline \ndouble-perovskite Er 2CoMnO 6. The majority portion of the ferrimagnetic order is activated by the long-\nrange order of Er3+ moments below TEr = 10 K in addition to the ferromagnetic order of Co2+ and Mn4+ \nmoments arising at TC = 67 K, characterized by compensated magnetization at TComp = 3.15 K. The \ninverted magnetic hysteresis loop observed below TComp can be described by an extended Stoner–\nWohlfarth model. The additional multiferroic phase is identifie d by the ferroelectric polarization of ~0.9 \nμC/m2 at 2 K. The coexisting ferri magnetic and multiferroic phases a ppear to be strongly correlated in \nthat metamagnetic and ferroelectric transitions occur simultane ously. The results based on intricate \nmagnetic correlations and phases in Er 2CoMnO 6 enrich fundamental and a pplied research on magnetic \nmaterials through the scope of di stinct magnetic characteristic s in double perovskites. \n \n \nCorrespondence and requests for materials should be addressed t o Y. J. C. \n(phylove@yonsei.ac.kr) or N . L. (eland@yonsei.ac.kr).  \n  Introduction \n \nOne of the ideas behind examining magnetic materials aims to de velop desired functional \nproperties utilized in a wide range of technologies, for exampl e, energy storage, 1 memory \ndevices 2, medical appliances 3, and environmental monitoring sensors 4. In particular, magnetic \noxides comprising metal cations  and oxygen anions have been ext ensively investigated owing \nto the abundance of the constituents and stability of the compo unds. A prominent example can \nbe found in perovskite rare-earth m anganites that have been the  focus of research on magnetic \nmaterials over the last few decades. In mixed-valence manganite s, the subtle balance between \nhopping and localization of charge carriers leads to the phase coexistence of ferromagnetic \n(FM) metallic and antiferromagnetic (AFM) insulating states via  the kinetic arrest of the phase \ntransition. The formation of mixed magnetic glass, which is sus ceptible to an external magnetic \nfield ( H), is essential to the origin of colossal magnetoresistance 5,6. The variation of rare-earth \nions in perovskite manganites also generates several types of m ultiferroic (MF) phases 7-10. In \nmedium-sized rare-earth ions, the spiral spin modulation can be  stabilized, inducing \nferroelectricity via antisymmetric exchange strictions with str ong controllability of \nferroelectric properties by external magnetic fields 7. With a smaller radius, the crystallographic \nstructure changes into a hexagonal structure, which represents a unique improper \nferroelectricity due to structural trimerization 9. However, the perovskite structure remains \nintact under high pressure and accompanies the E-type AFM phase  that results in another type \nof the MF phase driven by sy mmetric exchange strictions 10.  \n As an extension of studies on perovskite manganites, double per ovskites of R\n2CoMnO 6 (R = \nLa, …, Lu, and Y) have recently been explored owing to their fa scinating magnetic and \nfunctional properties, such as metamagnetism 11-13, spin-glass state 14-16, exchange bias effect \n17-19, magnetocaloric effect 20-22, and multiferrocity 23-27. By replacing half the Mn ions with Co \nions in perovskite manganites, a  double perovskite structure is  formed with Co2+ (S = 3/2) and \nMn4+ (S = 3/2) ions, alternatingly locate d in corner-shared octahedral environments. As the size \nof rare-earth ions decreases, the magnetic transition temperatu re (T) arising from the dominant \nCo2+ and Mn4+ superexchange interactions decreases from 204 K for La 2CoMnO 6 to 48 K for \nLu2CoMnO 6 28. In these compounds, the difficulty in attaining the impeccabl e alteration of \nCo2+ and Mn4+ ions naturally entails additional AFM clusters which involve an other valence \nstate of Co3+- Mn3+, and anti-sites of ionic disord ers and/or antiphase boundaries  leading to Co2+- Co2+ or Mn4+- Mn4+ pairs. The formation of anti-sites in addition to the dominant  FM \norder 27,29,30 of Co2+ and Mn4+ moments is known as the mechan ism for the observed magnetic \nexchange bias in polycrystalline Y 2CoMnO 6 19. In Tm 2CoMnO 6 and Er 2CoMnO 6 (ECMO), the \nneutron diffraction studies c onfirm that the order of Co2+ and Mn4+ moments is FM and the \norder of Er3+/Tm3+ moments at lower temperature activates the additional ferrimag netic (FIM) \norder between Er3+/Tm3+ and ferromagnetic Co2+/Mn4+ sublattices 31-33. The FIM order exhibits \nan inversion of the magnetic hysteresis loop in polycrystalline  ECMO 34. In Yb 2CoMnO 6 and \nLu2CoMnO 6, the Co2+ and Mn4+ ions display the up-up-down-down (↑↑↓↓) spin configuration \nin which the ferroelectricity emerges perpendicular to the c-axis from the cooperative O2- \ndisplacements through the sym metric exchange striction 23-25. Evidently, a scientific \nunderstanding of diverse magnetic  phases and interactions is cr ucial for finding novel \nfunctional properties in  double perovskites. \n In this work, the magnetic and magnetoelectric properties of si ngle crystals of double-\nperovskite ECMO were studied to reveal the characteristics corr esponding to the mixed FIM \nand MF phases. The dominant FIM order between Er\n3+ and FM Co2+/Mn4+ sublattices was \nidentified by compensated magnetization ( M) occurring at TComp = 3.15 K. From our precise \nmeasurement of isothermal M in the low T regime, the inversion of the magnetic hysteresis \nloop was observed below TComp, which can be explained by the delicate balance between \ndifferent magnetic moments, and qualitatively by an extended St oner–Wohlfarth model 35-38. \nThe ferroelectric polarization ( P) and dielectric constant ( ') measurements demonstrated an \nadditional inclusion of the MF phase as found in Yb 2CoMnO 6 and Lu 2CoMnO 6 23,24. Associated \nwith the coexistence of FIM and  MF phases, the disappearance of  MF phase by an external H \noccurs simultaneously with the m etamagnetic transition, reveali ng exclusive characteristics of \nthe double perovskite.  \nResults and Discussion \n \nFigure 1(a) shows the X-ray powder diffraction pattern for the ground single crystals of double \nperovskite ECMO at room T. The crystallographic structur e was refined as a monoclinic \nstructure with the P2\n1/n space group. The lattice constants were found to be a = 5.228 Å, b = \n5.594 Å, and c = 7.477 Å with  = 90.244º with good agreement factors, χ2 = 1.74, Rp = 7.97, \nRwp = 6.23, and Rexp = 4.72. The crystal structures viewed from the a- and c-axes are depicted in Figs. 1(b) and (c), respectively. Co2+ and Mn4+ ions are alternatingly located in corner-shared \noctahedral environments. The oxygen octahedral cages are strong ly distorted due to the small \nradius of the Er3+ ion 28.  \n To investigate intricate magnetic properties as anticipated in the double perovskite \nincorporating three different magnetic ions, the T-dependence of magnetic susceptibility ( χ = \nM/H) was obtained. The anisotropic χ in H = 0.05 kOe along ( H//c) and perpendicular ( Hc) \nto the c-axis was measured upon warming in H after zero-field cooling (ZFC) and upon cooling \nat the same H (FC), as shown in Fig. 2(a). The overall T-dependence of χ’s for two different \norientations exhibits strong magnetic anisotropy, which indicat es that the spins are mainly \naligned along the c-axis. The FM order relevant to the dominant Co\n2+ and Mn4+ superexchange \ninteractions sets in at TC = 67 K, which can be determined by the sharp anomaly in the T \nderivative of χ in H//c. The T-dependence of heat capacity divided by the temperature ( C/T) \nmeasured upon warming in zero H also exhibits the anomaly starting from TC, shown in Fig. \n2(b). Upon further cooling, C/T shows an abrupt increase below TEr ≈ 10 K, which corresponds \nto the ordering of Er3+ moments. Below TEr, the reversal of χ was observed in both ZFC and \nFC measurements (Fig. 2(a)) as a c haracteristic signature of a ferrimagnet 39-46.  \n A ferrimagnet is a substance that involves a portion of opposin g magnetic moments as in \nantiferromagnetism, but generates a net M from unequal magnetic moments in the opposite \ndirections, thus exhibiting dis tinct characteristics of magneti sm. The FIM interaction between \nEr\n3+ and FM Co2+/Mn4+ sublattices generates the intriguing T-dependence of χ following the \ndifferent sequence of measurement. In the FC measurement  in H//c, χ increases smoothly below \nTC with the parallel alignment of Co2+ and Mn4+ moments. Upon cooling further below TEr, the \nEr3+ moments begin to align oppositely to the Co2+/Mn4+ moments, which leads to a gradual \ndecrease in χ. At lower T, χ intersects the zero point owing to the large moment of Er3+ spin. \nOn the other hand, ZFC χ shows a positive value at 2 K since the Er3+ moments tend to orient \nalong the H direction. The decrease in the effective Er3+ moments upon increasing T results in \nthe sign change of χ. Above TEr, the negatively magnetized Co2+/Mn4+ spins begin to flip along \nthe applied H due to thermal fluctuation, which causes another sign change o f χ at 48 K. To \nfind the compensation T precisely, the thermoremanent magnetization ( Mrem) 47 was measured \nin H//c (Fig. 2(c)). At 2 K, H = 50 kOe was applied in H//c and then turned off, and Mrem was recorded in the absence of H upon warming from 2 K. The sign reversal of Mrem occurs at TComp \n= 3.15 K, which manifests the FIM  feature of this double perovs kite compound. \n \nThe anisotropic M in H//c and Hc was measured up to ±90 kOe at T = 2 K, shown in Fig. 3(a). \nFor the hysteresis loop in H//c, solid and dashed lines denote sweeping H from +90 to −90 kOe \nand from −90 to +90 kOe, respectively. M in H//c is not saturated at +90 kOe with the magnetic \nmoment of 17.2 B/f.u., but it is much larger than the moment in Hc (8.71 B/f.u.), indicating \nthe magnetic easy c-axis. Upon decreasing H from +90 kOe, M decreases smoothly until it \ndrops precipitously be low 15 kOe. At low H, M intersects the zero poin t at 1 kOe and exhibits \nthe negative remanent M (Mr) of −1 B/f.u. (inset of Fig. 3(a)). Further decrease in H in the \nnegative direction induces a sharp drop in M at HC = −26.5 kOe. The measurement of M in H//c \nin the opposite direction completes the inverted magnetic hyste resis loop. The inversion of the \nhysteresis loop in H//c can be analysed by an extended Stoner–Wohlfarth model within t he \nf r a m e  o f  t h e  F I M  o r d e r  b e t w e e n  E r3+ and Co2+/Mn4+ sublattices with a different magnetic \nanisotropy 35-38 (see Experimental section for detail). The experimental observ ation of inversed \nmagnetic hysteresis loop in ECMO s uggests the considerable diff erence of magnetic anisotropy \nenergies between Er3+ and Co2+/Mn4+ moments. In our calculation, we assumed that the \nmagnetocrystalline anisotropy energy of Co2+/Mn4+ moments is three times larger than that of \nEr3+ moments. With qualitative similarity, the magnetic hysteresis loop was attained from the \nmodel, as illustrated in Fig. 3(b). Based on the result, the ev olution of the spin configuration \nfor Er3+ and Co2+/Mn4+ ions during the sweeping of H from +90 to −90 kOe in H//c i s  \nschematically depicted in Fig. 3(b). The red and blue arrows in dicate the effective moments of \nEr3+ and Co2+/Mn4+ ions, respectively. At high H, the Er3+ and Co2+/Mn4+ moments tend to be \naligned in the same direction due to the dominant Zeeman energy . Upon decreasing H, the \nnegative exchange c oupling between Er3+ a n d  C o2+/Mn4+ spins accompanied by a smaller \nmagnetocrystalline anisotropy en ergy and larger moment of Er3+ ions leads to the progressive \ndecrease in the net Er3+ moments, followed by zero net M even at a positive H and negative Mr. \nDecreasing H further in the negative direction induces an abrupt drop in M, where the \nCo2+/Mn4+ spins are fully reversed because the Zeeman energy of Co2+/Mn4+ sublattices \novercomes the anisotropy energy.  Since the change in magnitude of M caused by the reversal \nof Co2+/Mn4+ moment at the metamagnetic transition is found to be ~9 B/f.u. (Fig. 3(a)), the \nnet magnetic moment of Co2+/Mn4+ spins should be ~ 4.5 μB/f.u., which is smaller than the \nsummation of Co2+ and Mn4+ moments (6 B/f.u.). The smaller net magnetic moment of Co2+/Mn4+ spins is acceptable because a small portion of Co2+/Mn4+ spins is naturally reversed \nduring the magnetization process from +90 kOe to −26.5 kOe and antiferromagnetic exchange \ncouplings of Co2+- Co2+ or Mn4+- Mn4+ pairs are originally include d from the presence of anti-\nsites of ionic disorders an d/or antiphase boundaries.  \n The close relevance of M\nr to TComp was cautiously examined by the T dependent evolution of \nMr. The full hysteresis curves up to ±90 kOe were recorded in H//c at various T’s. The \nhysteresis loops below and above TComp are shown within the range of  ±5 kOe in Fig. 3(c) and \nd, respectively. Below TComp, all the curves present the i nverted magnetic hysteresis. Upon  \nincreasing T, the inverted loop becomes narrow and the magnitude of negativ e Mr decreases \nlinearly, resulting from the reduced net Er3+ moments by thermal fluctuation. By crossing TComp, \nthe sign of Mr changes and it increases gr adually with an increasing T.  \n Recently, new magnetism-driven ferroelectrics, i.e. type-II mul tiferroics, were found in \ndouble-perovskite Yb\n2CoMnO 6 and Lu 2CoMnO 6 23,24. The initial polycrystalline analysis of \nneutron diffraction and bulk electric properties for Lu 2CoMnO 6 suggested that the \nferroelectricity arises from the symmetric exchange striction o f the ↑↑↓↓ spin chains with \nalternating Co2+ and Mn4+ charge valences 48, consistent with the Ising spin chain magnet of \nCa3CoMnO 6 49. However, studies on the single crystals of Yb 2CoMnO 6 and Lu 2CoMnO 6 \nrevealed that the ferroelectricity emerges perpendicular to the  c-axis below TC = 52 and 48 K, \nrespectively. Several theoretical  works provided a plausible ex planation for the ferroelecticity, \nin which the symmetric exchange strictions along the ↑↑↓↓ spin chain with alternatingly shifted \nO2- ions generate cooperative O2- displacements perpendicular to the c-axis 50-52. \n The possible formation of an additional MF phase in ECMO was ex amined by the H-\ndependence of P obtained by integrating magnetoelectric current density ( J), measured \nperpendicular to the c-axis ( Ec) at 2 K, shown in Figs. 4(a) a nd (b). After poling from 100 K \nto 2 K in H = 0 kOe and E = 5.7 kV/cm, the J in H//c exhibits a very sharp peak with peak \nheight of ~0.76 μA/m\n2 at the metamagnetic transition, HC = 26.5 kOe. The corresponding P \nvalue at H = 0 kOe and 2 K was estimated as ~0.9 μC/m2, which is only two orders of magnitude \nsmaller than the P observed in Lu 2CoMnO 6 and signifies the presence of a small amount of the \nMF phase. The tiny magnitude of P at 2 K implies that the exact  magnetic configuration of MF \nphase could hardly be identified by the neutron diffraction exp eriment. Upon increasing H, the P shows the sharp step at HC and disappears above HC. The simultaneous transitions at HC for \nthe suppression of the ferroel ectricity and the reversal of Co2+/Mn4+ spins in the FIM state \nsuggest that the small amount of the additional MF phase is str ongly influenced by the \ndominant FIM phase. In analogy w ith the ferroelectricity in Lu 2CoMnO 6, the P e m e r g e d  \nperpendicular to the c axis at H=0 kOe in ECMO suggests that  the most plausible spin \nconfiguration of the minor MF phase would be ↑↑↓↓. The disappea rance of the P by applying \nH along the c axis can be explained by the cha nge of spin configuration from  the ↑↑↓↓ to ↑↑↑↑. \n \nIn Fig. 4(c), the H-dependence of ' in Ec is plotted, measured in H//c up to ±90 kOe at f = \n100 kHz and T = 2 K. By sweeping H between +90 to −90 kOe, the whole variation of ' is \nonly about 1 % with strong hysteretic behaviour. The maximum va lues occur at H = ±6 kOe, \nfollowed by the sharp transitions at HC. The ' shows the rather complicated H dependence in \ncomparison with the H dependence of P. In addition to the small portion of MF phase, the \nadditional AFM clusters formed by anti-site disorders and antip hase boundaries in the \nferromagnetic Co2+/Mn4+ sublattices would also affect the isothermal '. The complicated but \ntiny magnitude variation of isothermal ' may result from the intricate contributions from the \nsmall portions of MF phase and AFM clusters. For a comparison w ith '(H), the H derivative \nof isothermal  M, dM/dH at 2 K is also plotted in Fig. 4(d). The d M/dH reveals the similar \nhysteretic variation of '. The isothermal  M mainly reflects the response of the FIM order \nbetween the Er3+ and Co2+/Mn4+ moments to the external H, as illustrated in Fig. 3(b), but also \naffects strongly on the hysteretic behaviour of '(H) .    \n \nThe T-dependence of the d ielectric constant ( ') and tangential loss (tan ) is displayed in Figs. \n5(a) and (b), respectively, m easured perpendicular to the  c-axis ( Ec) at f = 100 kHz in H//c \nwith H = 0, 10, 20, and 30 kOe. At zero H, a small and broad peak of ' at TC = 67 K  was \nobserved in Fig. 5(a), which signifies the emergence of a small  amount of MF phase. Compared \nto the peak height of ~15 %,  normalized by the value at TC = 48 K in Lu 2CoMnO 6,23 it can be \nestimated as only about 1 % in ECMO. Despite a small portion of  the MF phase in ECMO, TC \nis fairly enhanced. The broad peak of ' is gradually suppressed by applying H along the c-axis, \nascribed to the change in the spin configuration from ↑↑↓↓ to ↑ ↑↑↑, similar to that in \nLu2CoMnO 6.23 Upon decreasing T, ' decreases linearly until it declines faster below 20 K. The overall T-dependence of ' and tan  (Fig. 5(b)) below TC appears similar to those of \nLu2CoMnO 6. \n While the intrinsic coupling phenomena between magnetic and fer roelectric states in single-\nphase type-II multiferroics were extensively explored, detailed  properties of an MF phase \nmixed with another magnetic phase have scarcely been revealed. The T evolution of \nmagnetoelectric effect in the mixed FIM and MF phases was exami ned by comparison between \nisothermal P and M at T’s below T\ncomp. Figures 6(a) and (b) show the H-dependence of P’s and \nM’s, respectively, in Ec and H//c at T = 2, 2.25, 2.5, 2.75, and 3 K, indicating that both of P \nand M vary delicately to the change of T. The estimated P’s at 2.25 and 2.5 K were 0.79 and \n0.47 μC/m2, respectively. As H is increased, the P’s are suppressed with steep steps at H = 28.0 \nand 28.7 kOe. The i nitial curve of M at 2, 2.25, and 2.5 K also show s the step at the same H as \nP, suggesting the strong intercorr elation between FIM and MF pha ses. At 2.75 and 3 K, P \nmagnitudes at 0 kOe are reduced as 0.43 and 0.32 μC/m2, respectively. Upon increasing H, the \nP’s are gradually reduced and vanish above ~37 kOe, correspondin g to the overall broad feature \nof M’s. Note that P above Tcomp could not hardly be obtained because of the almost suppressed \nmagnitude of J with a broadened feature. Figur es 6(c) and (d) display isother mal ' in Ec at f \n= 100 kHz and J in H//c, respectively,  at T = 2, 2.25, 2.5, 2.75, and 3 K. The initial curve of ' \nat 2 and 2.25 K indicates both a sharp peak and step-like featu re at the metamagnetic transition \nbut the ' at 2.5 K shows only a step. The sharp peak of the J at 2 K shifts to higher H and the \npeak height is reduced upon slightly increasing the T. The weak anomaly was observed in the \n' at 2.75 and 3 K, corresponding to the disappearance of P. As shown in the inset of Fig. 6(d), \nJ’s at 2.75 and 3 K exhibit wide  and small peaks around 35 kOe. \n The T evolution of the magnetodielect ric effect in a wide range of T’s in the mixed FIM and \nMF phases was also investigated by comparison between isotherma l \n' and M. Figure 7 displays \nthe isothermal ' in Ec at f = 100 kHz and M in H//c and Hc, at T = 5, 10, 20, 35, 50, and 65 \nK. At 5 K, a butterfly-like shape of ' was observed with a strong magnetic hysteresis, with the \nabsence of the step-like metama gnetic transition (Fig. 7(a)). T he broadened feature of ' is \ncompatible with the modulation of M in H//c with the narrow magnetic hysteresis described as \nsmall values of Mr = 1.22 B/f.u. and the coercive field of Hc = 2.10 kOe (Fig. 7(g)). At 10 K, \nthe butterfly-like shape of ' is maintained (Fig. 7(b)), but the magnetic hysteresis is considerably reduced. The central  part of the hysteresis loop i n H//c is extended as Mr = 2.73 \nB/f.u. and Hc = 7.24 kOe (Fig. 7(h)), indicativ e of the reduced strength of t he Er3+ spin order. \nIn addition, the slight and elonga ted hysteretic behaviour of M in Hc emerges. As T increases \nfurther, the magnetic  hysteresis in both ' and M is progressively reduc ed. At 65 K, just below \nTC, the sharp cusp of ' occurs at zero H with the hysteresis loop in M vanishing. \n \nIn summary, we explor ed the magnetic and magnetoelectric proper ties of mixed ferrimagnetic \nand multiferroic phases of single-crystalline double-perovskite  E r 2CoMnO 6. The dominant \nCo2+ and Mn4+ superexchange interactions lead t o the ferromagnetic order bel ow TC = 67 K, \naligned mainly along the c-axis. The long-range order of Er3+ m o m e n t s  b e l o w  TEr = 10 K \ninduces the ferrimagnetic order and magnetization compensation at TComp = 3.15 K, delicately \nbalanced with the ferromagnetic Co2+/Mn4+ sublattice. The extende d Stoner–Wohlfarth model \ndepicts qualitatively the invert ed magnetic hysteresis loop obs erved below TComp. The \nobservation of electric polarizat ion at low temperature is indi cative of the presence of a small \nportion of a multiferroic phase simultaneously with the ferrima gnetic phase. The strong \nmagnetoelectric correlation at the metamagnetic transition in t he phase coexistence reveals the \nunique characteristic of the double perovskite compound, which o f f e r s  c r u c i a l  c l u e s  f o r  \nexploring suitable materials for magnetoelectric functional app lications. \n \nMethods  \n \nRod-shaped single crystals of ECMO with a typical size of 2  2  5 mm3 were grown by the \nconventional flux method with Bi 2O3 flux in air. Er 2O3, Co 3O4, and MnO 2 powders were mixed \nin the stoichiometric ratio for  ECMO and ground in a mortar, fo llowed by pelletizing and \ncalcining at 1000 °C for 12 h in a box furnace. The calcined pe llet was delicately reground and \nsintered at 1100 °C for 24 h. Th e same sintering procedure afte r regrinding was carried out at \n1200 °C for 48 h. A mixture of pre -sintered polycrystalline pow der and Bi 2O3 flux with a ratio \nof 1:12 ratio was heated to 1300 °C in a Pt crucible. It was me lted at the soaking T for 5 h, \nslowly cooled to 985 °C at a rate of 2 °C/h, and cooled to room  T at a rate of 250 °C/h. The \ncrystallographic structure and absence of a second phase were c hecked by the Rietveld \nrefinement 53 using the FullProf program 54 for the power X-ray diffraction data. The data were \nobtained with a Rigaku D/Ma x 2500 powder X-ray diffractometer u sing Cu-K α radiation.  \n The T and H dependences of DC M were examined by using a VSM magnetometer in a \nQuantum Design PPMS (Physical Properties Measurement System). T he specific heat ( C) was \nmeasured with the standard relaxation method in PPMS. The T and H dependences of ' were \nobserved at f = 100 kHz using an LCR meter (E4980, Agilent). The H dependence of electric \npolarization ( P) was obtained by the integrati on of magnetoelectric current me asured with the \nH variation of 0.1 kOe/s after polin g in a static electric field  of E = 5.7 kV/cm. \n In our extended Stoner–Wohlfarth model \n35-38, the magnetic energy density can be expressed \nas  \nܧൌെ ܯ ாܪcosߠ ாെܯ/ெܪcosߠ /ெെܬcos൫ߠ ாെߠ/ெ൯ \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t \tܭ ாsinଶሺߠாെ߮ாሻܭ/ெsinଶ൫ߠ/ெെ߮/ெ൯          ( 1 )  \nThe first two terms represent the Zeeman energy density, where ܯா and ܯ/ெ  are the \neffective magnetic moments of Er3+ and Co2+/Mn4+ ions, respectively, and ߠ is the angle \nbetween the corresponding M and applied H. The third term denotes the exchange energy \ndensity, where the moments in the Er3+ and Co2+/Mn4+ sublattices tend to be ordered oppositely, \nand thus, the exchange coupling constant ( ܬ )is negative. The last two terms signify the densities \nof magnetocrystalline anisotropy energy ( ܭா and ܭ/ெ ) for Er3+ and Co2+/Mn4+ ions, \nrespectively, where ߮ is the angle between H and the magnetic easy axis . The energy density \ncan be minimized to determine the direction of net M at an applied H by solving both பா\nபఏಶൌ0 \nand பா\nபఏ/ಾൌ0. T h e  e s t i m a t e d  n e t  M can be written as follows: ܯൌܯ ாcosߠா\nܯ/ெcosߠ/ெ . The calculated hysteresis loop in Figure 3(b) was obtained wi th ܬൌ\nെ4.241ൈ10ହ\tJ / mଷ, ܭாൌ 8.482ൈ10ସ\tJ / mଷ, and ܭ/ெൌ 2.545ൈ10ହ\tJ / mଷ. \n \nAcknowledgements \n \nThis work was supported by the NRF Grant ( NRF-2016R1C1B2013709, NRF-\n2017K2A9A2A08000278, 2017R1A5A1014862 (SRC program: vdWMRC center), and NRF-\n2018R1C1B6006859 ). \n \nAuthor contributions \n Y.J.C. and N.L. designed the experiments. M.K.K. calculated Sto ner–Wohlfarth model, and \nS.H.O. and J.Y.M. synthesised th e single crystals. M.K.K., J.Y. M. and D.G.O. performed \nmagnetization, heat capacity, diel ectric constant, and magnetoe lectric current measurements. \nM.K.K., Y.J.C., and N.L. analysed the data and prepared the man uscript. All the authors have \nread and approved the final ve rsion of the manuscript. \n \nAdditional information \n \nThe authors declare no competing interests . \n \n \n \n \n \n \n \n  \n \nFigure 1. Crystallographic structure of Er 2CoMnO 6. (a) Observed (open circles) and \ncalculated (solid line) powder X-ray diffraction patterns for g round Er 2CoMnO 6 (ECMO) \nsingle crystals. Y obs, Y cal, and Y obs−Y cal r e p r e s e n t  t h e  i n t e n s i t i e s  of the observed patterns, \ncalculated patterns, and their difference, respectively. The gr een short lines denote the Bragg \npositions. (b) and (c) Views of the crystal structure of double  perovskite ECMO from the a- \nand c-axes, respectively. The purple,  pink, blue, and yellow spheres represent Er3+, Co2+, Mn4+, \nand O2 ions, respectively. \n \n \n \nFigure 2. Temperature-dependent magnetic properties of Er 2CoMnO 6. (a) Temperature \n(T) dependence of the magnetic susceptibility ( χ = M/H , 1 emu = 4π × 10-6 m3) of a double-\nperovskite ECMO single crystal along ( H//c) and perpendicular ( Hc) to the c-axis, measured \nupon warming in H = 0.05 kOe after zero-magnetic-fie ld cooling (ZFC) and upon coo ling in \nthe same field (FC), shown up to 80 K. (b) T-dependence of specific heat divided by \ntemperature ( C/T) measured without magnetic field ( H). (c) T-dependence of the \nthermoremanent magnetization ( Mrem) of the ECMO crystal, measured in H//c warming from \n2 K in the absence of H after cooling in 50 kOe. The v ertical dashed lines indicate th e \nferromagnetic transition temperature ( TC), the Er3+ spin ordering temperature ( TEr), and the \ncompensation temperature ( TComp), respectively. \n  \n \n \nFigure 3. Observed and calculated isothermal magnetization at 2  K and temperature \nevolution of inverted magnetic hysteresis.  (a) Isothermal magnetization ( M) of the ECMO \ncrystal in H//c and Hc, measured at T = 2 K up to 90 kOe after ZFC. The inset shows the \nmagnified view in the range of H = ±10 kOe of the hysteresis loop in H//c. For the hysteresis \nloop in H//c, the solid and dashed curves indicate the data obtained by swe eping H from +90 \nkOe to −90 kOe, and by sweeping H from −90 kOe to +90 kOe, respectively. (b) Calculated \nhysteresis loop in H//c by adopting the extended Stoner-W ohlfarth model.  The schematic  spin \nconfigurations depicted as net moments of Er3+ (light red arrows) and Co2+/Mn4+ (light blue \narrows) spins are illustrated for the curve of sweeping H from +90 kOe to −90 kOe. (c) and (d) \nMagnified views in the range of H = ±5 kOe of isothermal M’s in H//c, measured at various \nT’s below TComp (T = 2, 2.25, 2.5, 2.75, and 3 K) and above TComp (T = 3.25, 3.5, 4, 4.5, and 5 \nK), respectively. \n  \n \n \n Figure 4.  Isothermal ferroelectric polarization and dielectric constant a t 2 K.  ( a )  H-\ndependence of ferroelectric polarization ( P) at 2 K, obtained by integrating the magnetoelectric \ncurrent in b). (b) H-dependence of current density ( J), measured with the H variation of 0.1 \nkOe/s in H//c after poling from 100 K to 2 K in E = 5.7 kV/cm perpendicular to the c axis. (c) \nH-dependence of \n' i n  Ec, measured up to ±90 kOe in H//c at 2 K. (d) H-derivative of \nisothermal M (dM/dH) at 2 K, taken from the  data in Figure 3a).  \n \n  \n \n  \n \n \n \nFigure 5. Temperature dependence of  the dielectric properties o f Er 2CoMnO 6. (a) and (b) \nT-dependences of dielectric constant ( ') and dielectric tangential loss (tan ), respectively, \nmeasured upon warming from 2 K to 100 K in an applied AC voltag e of V = 1 V at f = 100 kHz \nperpendicular to the  c-axis  (Ec), and H = 0, 10, 20, and 30 kOe along the c-axis  (H//c).  \n \n    \n \n \n \nFigure 6.  Temperature evolution of ferroelectric polarization below Tcomp in comparison \nwith that of magnetization and dielectric constant.  (a) H-dependence of P at T = 2, 2.25, \n2.5, 2.75 and 3 K s hown in the range of 25-40 kOe. (b) Initial curves of isothermal M at T = 2, \n2.25, 2.5, 2.75 and 3 K. (c) H-dependence of ' a t  T = 2, 2.25, 2.5, 2.75 and 3 K. (d) H-\ndependence of J, measured with the H variation of 0.1 kOe/s in H//c at T = 2, 2.25, 2.5, 2.75 \nand 3 K after poling in Ec. The inset shows the magnified view of J.  \n  \n \n \n \nFigure 7. Temperature evolution of the isothermal dielectric co nstant.  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Yáñez-Vilar, S.  et al.  Multiferroic behavior in the double-perovskite Lu 2MnCoO 6. Phys. \nRev. B  84, 134427, doi:10.1103/PhysRevB.84.134427 (2011). \n49. Choi, Y. J.  et al.  Ferroelectricity in an Ising Chain Magnet. Phys. Rev. Lett.  100, 047601 \n(2008). \n50. Xin, C.  et al.  Spin rotation driven f erroelectric polari zation with a 180° fl op in double-\nperovskite Lu 2CoMnO 6. RSC Adv.  5, 43432-43439, doi:10.1039/c5ra03727a (2015). \n51. Zhang, J. T., Lu, X. M., Yang, X. Q., Wang, J. L. & Zhu, J.  S. Origins of ↑↑↓↓ magnetic \nstructure and ferroelectricity in multiferroic Lu 2CoMnO 6. Phys. Rev. B  93, \ndoi:10.1103/PhysRevB.93.075140 (2016). \n52. Jia, T., Zeng, Z. & Lin, H. Q. The collinear ↑↑↓↓ magnetism  driven ferroelectricity in \ndouble-perovskite multiferroics. J. Phys.: Conf. Ser.  827, 012005, doi:10.1088/1742-\n6596/827/1/012005 (2017). 53. Rietveld, H. A profile refinement method for nuclear and ma gnetic structures. J. Appl. \nCrystallogr.  2, 65-71, doi:10.1107/S0021889869006558 (1969). \n54. Rodríguez-Carvajal, J. Recent advances in magnetic structur e determination by neutron \npowder diffraction. Phys. B (Amsterdam, Neth.)  192, 55-69, doi:10.1016/0921-\n4526(93)90108-I (1993). \n "    },    {        "title": "1201.2144v1.Direct_observation_of_magnetic_phase_coexistence_and_magnetization_reversal_in_a_Gd___0_67__Ca___0_33__MnO___3___thin_film.pdf",        "content": "arXiv:1201.2144v1  [cond-mat.str-el]  10 Jan 2012Direct observation of magnetic phase coexistence and magne tization reversal in a\nGd0.67Ca0.33MnO 3thin film\nJeehoon Kim, Nestor Haberkorn, Leonardo Civale, Paul Dowden,\nAvadh Saxena, J. D. Thompson, and Roman Movshovich\nLos Alamos National Laboratory, Los Alamos, NM 87545∗\nEvgeny Nazaretski\nBrookhaven National Laboratory, Upton, NY 11973\nWe have investigated the ferrimagnetic domain structure in a Gd0.67Ca0.33MnO3thin film using\nmagnetic force microscopy. We observe clear signs of phase s eparation, with magnetic islands\nembedded in a non-magnetic matrix. We also directly visuali ze the reversal of magnetization of\nferrimagnetic domains as a function of temperature and attr ibute it to a change in the balance of\nmagnetization of anti-aligned Mn and Gd sublattices.\nMixed-valent perovskite manganites A 1−xBxMnO3(A\nand B are rare-earth and divalent alkaline elements, re-\nspectively), such as La-based manganites, have been\nstudied extensively in recent years.[1–4] These materials\nexhibit a colossal magnetoresistance (CMR) effect for a\nwide rangeof dopingscentered at x = 1/3where the dou-\nble exchange mechanism is maximized.[5] The resulting\ncombinationoffascinatingphysicalphenomena anda po-\ntential fortechnologicalapplicationshasbeen the driving\nforce in sustaining high interest in these compounds.[1–4]\nElectronic inhomogeneity and phase separation are ubiq-\nuitous in doped manganites, and the resulting CMR ef-\nfect is driven by percolative transport.[6] CMR manifests\nitself by a dramatic drop in resistivity and a discon-\ntinuous decrease in the equilibrium Mn-O bond length\nat a first order phase transition in an applied magnetic\nfield.[7, 8] Their complex electronic structure and a vari-\nety of competing interactions lead to a rich ensemble of\nground states in this family of compounds.\nIn this Letter we report a low temperature\nmagnetic force microscopy (MFM) investigation of\nGd0.67Ca0.33MnO3(GCMO), a compound with an in-\nsulating ferrimagnetic (FIM) ground state. Compared\nto other ferromagnetic (FM) perovskite manganites,\nGCMO exhibits arelativelylowCurie temperature( TC),\nand its small structural tolerance factor t<0.89[9, 10]\nleadstoarobustinsulating groundstate. Magneticprop-\nerties of the system reflect those of the two sublattices of\nMn and Gd ions (see below). The different temperature\ndependence of magnetization of each of the two sublat-\ntices results in a changeofsign ofthe totalmagnetization\nas a function of temperature at a characteristic compen-\nsation temperature Tcomp, where the Mn and Gd sub-\nlattices have magnetic moments of the same magnitude\nand opposite direction.[9–11] A small tolerance factor,\na structural distortion, and the antiferromagnetic inter-\naction between Gd and Mn sublattices yield remarkable\nproperties, such as a giant magnetostrictive effect in a\n∗Electronic mail: jeehoon@lanl.govwide temperature range[12] and inhomogeneousFIM-like\nbehavior with an exchange bias effect close to Tcomp.[13]\nLow values of the saturation magnetization ( MS) sug-\ngest phase coexistence.[12, 13] MFM studies described\nbelow, with the wide range of field and temperature em-\nployed, allow us to visualize the magnetic structure of\nGCMO and provide direct evidence of phase separation.\nThe magnetization reversal at Tcompofeach individual\ndomain provides strong support for the scenario of anti-\naligned Mn and Gd sublattices with the Gd (Mn) mag-\nnetization dominating below (above) Tcomp.\nThe Gd 0.67Ca0.33MnO3thin film was grownby pulsed-\nlaser deposition (PLD) on a SrTiO 3(100) substrate from\nacommercialtargetwith the samechemicalcomposition.\nThe substrate temperature was kept at 790◦C in an oxy-\ngen atmosphere at a pressure of 200 mTorr. After depo-\nsition, the O 2pressure was increased to 200 Torr, and\nthe temperature was decreased to room temperature at\na rate of 30◦C/min. Bulk GCMO is an orthorhombic\nperovskite (Pbnm (no. 62); a= 5.52˚A,b= 5.34˚A,\nc= 7.50˚A).[13, 14] The GCMO film was examined by\nx-ray diffractometry, and was found to be single phase\nwith a (00 l) orientation. The lattice parameters ( a=\n5.55(2)˚A,b= 5.36(2) ˚A, andc= 7.50(1) ˚A) were de-\ntermined using (00 l), (200), and (020) reflections from\na four-circle diffractometer/goniometer. No additional\npeaks due to secondaryphasesor different crystallineori-\nentationswereobserved. Therockingcurvewidtharound\nthe(004)peakofthefilmwas ∼0.27◦. Thefilmthickness\nof 45 nm was determined by a low-anglex-rayreflectivity\nmeasurement with an angular resolution of 0.005◦.\nA Quantum Design SQUID magnetometer was used\nfor measurements of the global magnetization with the\nmagnetic field oriented perpendicular to the film surface.\nAlllocalizedMFM measurementsdescribedinthisLetter\nwere performed in a home-built low-temperature MFM\napparatus.[15] MFM images were taken in a frequency-\nmodulated mode, with the tip-lift height of 100 nm\nabove the sample surface. High resolution SSS-QMFMR\ncantilevers,[16] magnetized along the tip axis in a field\nof 3 T, were used for MFM measurements; the external\nmagnetic field was always applied perpendicular to the2\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s50/s48/s52/s48/s54/s48/s56/s48/s48/s72\n/s99/s32/s91/s84/s93\n/s84 /s32 /s91/s75 /s93/s84\n/s99/s111/s109/s112\n/s77/s110/s32/s71/s100\n/s40/s98/s41/s32/s48/s46/s48/s48/s49/s32/s84\n/s32/s48/s46/s48/s49/s32/s84\n/s32/s48/s46/s49/s32/s84\n/s32/s48/s46/s53/s32/s84\n/s32/s49/s32/s84/s77/s32 /s91 /s101/s109/s117/s47/s99/s109/s51\n/s93 \n/s40/s97/s41\nFIG. 1. (Color online) (a) Field-cooled M(T) curves in differ-\nent magnetic fields ( H). (b) Coercive field ( Hc) as a function\nof temperature obtained from magnetic hysteresis loops.\nfilm surface and parallel to the MFM tip.\nFig. 1(a) shows the field-cooled (FC) magnetization M\nas a function oftemperature at different values ofapplied\nmagnetic field H. The temperature dependence of mag-\nnetization was discussed previously by Snyder et al.[10]\nGCMO undergoes a phase transition from paramagnetic\ninsulating to ferromagnetic insulating states, associated\nwith the ferromagnetic ordering of Mn cations, at TC≈\n80 K. The local field due to FM order in the Mn sub-\nlattice and the negative f-dexchange interaction on the\nGd spins force the moments on the Gd sublattice to be\nanti-aligned to those in the Mn sublattice. The Mn sub-\nlatticedominatesthemagnetizationathightemperature,\nbut the absolute magnitude of magnetization of the Gd\nsublattice grows faster when the temperature is reduced.\nConsequently, the total magnetization Mreaches a max-\nimum value close to 50 K (see Fig. 1), starts to decrease\nwith decreasing temperature, and goes toward zero at\nTcomp≈15 K in low fields ( Tcompdepends on H), where\nmagnetizations of the Mn and Gd sublattices compen-\nsate each other. Below Tcompthe local magnetization of\nGd is larger than that of Mn, |MGd|>|MMn|, and the\nsign of the total magnetization is determined by the di-\nrection of magnetization of the Gd sublattice. When the\napplied magnetic field His below the coercive field Hcof\nthe system at Tcomp, the magnetization of the Gd sublat-\ntice is locked in a direction opposite to the applied field,\nand the total magnetizationis negativebelow Tcomp. For\n10 K (b) \n(d) \n4 K (a) Δf (Hz) \n1\n15 K (c) \n0Correlation  (a.u.) 0\n-80 T< Tcomp (e) \nnon-magnetic Matrix : Gd \n: Mn T> Tcomp (f) : Gd \n: Mn \nnon-magnetic Matrix \nFIG. 2. (Color online) (a)-(c) MFM images acquired at dif-\nferent temperatures. Solid and dashed circles represent th e\nsame sample area. (d) Cross-correlation map between images\nshown in panels (a) and (c). The large negative value at the\ncenterof the mapsignifies the anticorrelation between imag es.\n(e) and (f) Schematical illustration of the temperature evo -\nlution of phase-separated magnetic regions above and below\nTcomp≈12 K in 1 mT. The field of view in the images ((a)-\n(d)) is 6 µm×6µm. Features on the left side are broader\nthan those on the right side because the scan plane is not\nperfectly parallel to the sample surface.\nH > H cthe magnetization is reversed immediately be-\nlowTcomp, producing a characteristic sharp kink and a\nV-shape in the data. This sharp reversal of the change\nin magnetization (from decreasing to increasing with de-\ncreasing temperature) is facilitated by a strong decrease\nofHcatTcomp, as shown in Fig. 1(b), which is deter-\nmined on the basis of an analysis of full hysteresis curves\nat different temperatures (data not shown). The positive\noffsetofMatthe kink at0.5Tand 1Tin Fig.1(a) isdue\nto a paramagnetic background. All magnetic transition\ntemperatures observed in the film are in good agreement\nwith the values previously reported for bulk polycrystal\nand single crystal samples.[10, 13, 14] The data at 0.1\nT has a clear kink as it crosses M= 0 and Tcomp, indi-\ncating that some small number of the magnetic domains\nflip their orientationat Tcomp. This is consistentwith the\nbulk measurementsof Hc≈0.1T atTcomp, and points to\ncoercivefieldin thissystembeingalocalproperty, proba-\nbly dependent on the magnetic domain’s size, shape, and\nenvironment.\nThe MFM images depicted in Figs. 2(a)-(c) were taken\nsequentially at 4 K ( T<Tcomp), 10 K (T≈Tcomp), and 15\nK (T>Tcomp), respectively, in a magnetic field of 1 mT\n(belowHc) applied above TC(field-cooled data). The\ndashed circles in Figs. 2(a)-(c) show the same sample re-\ngion (thermal drifts are negligible for images taken below\n15 K, see below). Regions of a non-zero magnetic sig-\nnal, either blue or red, change color as the temperature3\n(e) \n(c) 0.5 T \n1 T (d) \n (f) \n0 T (a) \n 0.1 T (b) Δf (Hz) 1\n0Correlation  (a.u.) 0220 \nFIG. 3. (Color online) (a)-(d) Field-cooled MFM images\ntaken at 4 K in different magnetic fields. (e) and (f) Cross-\ncorrelation maps between (a)-(b), and (b)-(c), respective ly;\nno correlation is observed. The field of view in the images\n((a)-(f)) is 6 µm×6µm. Dashed circles correspond to the\nsame sample area.\nchanges from 4 K to 15 K, but the green areas remain\ngreen in all images (a)-(c) in Fig. 2. The sample, there-\nfore, is phase separatedinto FIM (blue and red) and non-\nmagnetic (green) regions.[17] At 4 K (Fig. 2(a)) Gd dom-\ninates the magnetization of FIM domains, as depicted\nschematicallyinFig.2(e). Redislandsin Fig.2(a), there-\nfore, represent those parts of the sample where Gd mag-\nnetic moments point “down”, and the blue regions are\nthosewithGdmagneticmomentspointing“up”. At15K\n(Fig. 2(c)) all of the red regions switch to blue, signaling\na reversal in their magnetization, as Mn magnetization is\ndominant above Tcomp≈12 K. This situation is depicted\nschematically in Fig. 2(f). The small magnetic contrast\nacross the sample at 10 K (see Fig. 2(b)) indicates al-\nmost equal magnetic contributionsofthe anti-alignedGd\nand Mn sublattices in FIM regions near Tcomp. In addi-\ntion, Fig. 2(b) demonstrates the domains’ breakup and\na reduction in their size close to Tcomp(at 10 K). This\nphenomenon is consistent with the exchange bias effect\npreviously observed in single crystals.[13] The reduction\nof the size of FIM domains close to Tcompalso leads to a\ndecrease of the coercive field (see Fig. 1(b)).[14, 18]\nFig. 2(d) shows a cross-correlation map between im-\nages (a) and (c) and allows us to investigate qualitatively\nthe temperature evolution of magnetic domains in the\nsample. The large negative value in the center of the\ncross-correlation map demonstrates the anti-correlation\nbetween 4 K and 15 K magnetization data in Fig. 1(a),\nindicating that red and blue islands reverse their magne-\ntization (and colors) upon the temperature change from\n4 K to 15 K. The central location of the cross-correlation\nminimum also demonstrates the small thermal drift in\nour MFM apparatus.Figs. 3(a)-(d) show MFM images obtained at 4 K after\nfield-coolingtheGCMOsamplethrough TCin 0T, 0.1T,\n0.5 T, and 1 T applied fields. In order to understand the\nthermalandfield evolutionofthesample’smagnetization\nwe evaluated cross-correlationmaps for these images. No\ncorrelation was observed between data sets obtained at 0\nT and 0.1 T (panels (a) and (b)), as shown in Fig. 3(e),\nand 0.1 T and 0.5 T (panels (b) and (c)), as shown in\nFig. 3(f). The lack of cross-correlation indicates signifi-\ncant evolution of the spin magnetization due to the re-\nversal process inside FIM clusters in a field up to 0.5 T.\nOn the other hand, magnetic domains imaged in 0.5 T\nand 1 T FC experiments show a similar pattern, suggest-\ning saturation of the magnetization reversal process as\nwell as a clear phase separation between ferrimagnetic\nclusters and the paramagnetic matrix. (Data taken at\n3 T, not shown, are similar to those at 1 T.) The lack\nof correlation between the images (a)-(c) cannot be the\nresult of thermal drift of the tip position over the sam-\nple, as this was observed repeatedly to be under 1 µm for\nour system (e.g., see panels (c) and (d)). The magnetic-\nnonmagnetic phase coexistence could be attributed to lo-\ncalized disorder or a localized strain distribution, similar\nto observations in Y- and Pr-based manganites with a\ncomparably low tolerance factor [Y 2/3Ca1/3MnO3(t∼\n0.884) and Pr 2/3Ca1/3MnO3(t∼0.91)].[19–22] Results\nof x-ray diffraction measurements on our thin-film sam-\nple, however, are close to those on bulk samples and tend\nto rule out a strain mechanism of phase separation.\nIn conclusion, we have performed MFM experiments\nonaferrimagneticGCMOthin filmanddirectlyobserved\nphase separation in the sample, with magnetic (FIM)\nregions of characteristic dimensions between 0.1 to 0.5\nµmembeddedinanon-magneticmatrix. Thebehaviorof\nmagnetic regions is consistent with the presence of anti-\naligned Mn and Gd magnetic sublattices, forming a FIM\nstate. The observed magnetization reversal in the FIM\ndomains as a function of temperature, for small external\nmagnetic field, is consistent with the Mn sublattice being\ndominant at T > T comp≈15 K, but the Gd sublattice\n(with magnetization locked to be antiparallel to a small\napplied field) is dominant for T < T comp. We attribute\nthe phase separation to localized disorder rather than\na strained state of the sample. These results will have\nsignificant bearing on the potential utilization of GCMO\nand otherrelated compoundsin magneticmemorydevice\napplications.\nWork at LANL (sample fabrication, SQUID measure-\nments, MFM, data analysis, and manuscript prepara-\ntion) was supported by the US Department of Energy,\nOffice of Basic Energy Sciences, Division of Materials\nSciences and Engineering. Work at Brookhaven (data\nanalysis and manuscript preparation) was supported by\nthe US Department of Energy under Contract No. DE-\nAC02-98CH10886. N.H. is a member of CONICET (Ar-\ngentina).4\n[1] S. Jin, T. H. Tiefel, M McCormack, R. A. Fastnacht, R.\nRamesh, and L. H. Chen, Science 264, 413 (1994).\n[2] Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen,\nRev. Mod. Phys. 77, 721 (2005).\n[3] Weida Wu, Casey Israel, Namjung Hur, Soonyong Park,\nSang-Wook Cheong, and Alex De Lozanne, Nat. Mater.\n5, 881 (2006).\n[4] J. P. Zhou, J. T. McDevitt, J. S. Zhou, H. Q. Yin, J. B.\nGoodenough, Y. Gim, and Q. X. Jia, Appl. Phys. Lett.\n751146 (1999).\n[5] Lev P. Gor’kov and Vladimir Z. Kresinc, Phys. Rep. 400\n149 (2004).\n[6] E. Dagotto, T. Hotta and A. Moreo, Phys. Rep. 344, 1\n(2001), and references therein.\n[7] Liuwan Zhang, Casey Israel, Amlan Biswas, R. L.\nGreene, and Alex de Lozanne, Science 298, 805 (2002).\n[8] John B. Goodenough, J. Appl. Phys. 81, 5330 (1997).\n[9] H. Y. Hwang, S-W. Cheong, P. G. Radaelli, M. Marezio,\nand B. Batlogg, Phys. Rev. Lett. 75, 914 (1995).\n[10] G. Jeffrey Snyder,C. H. Booth, F. Bridges, Ron Hiskes,\nSteve DiCarolis, M. R. Beasley, and T. H. Geballe, Phys.\nRev. B.55, 6453 (1997).\n[11] Octavio Pe˜ na, Mona Bahouta, Karim Ghanimia, Pe-\ndro Duranb, Dionisio Gutierrezb, and Carlos Moureb,\nJ. Mater. Chem. 12, 2480 (2002).[12] V. F. Correa, N. Haberkorn, G. Nieva, D. J. Garcia, and\nB. Alascio, arXiv:1109.0259v1.\n[13] N. Haberkorn, S. Larr´ egola, D. Franco, and G. Nieva, J.\nMagn. Magn. Mater. 321, 1133 (2009).\n[14] Yanwei Ma, M. Guilloux-Viry, P. Barahona, O. Pe˜ na,\nand C. Moure, Appl. Phys. Lett. 86, 062506 (2005).\n[15] E. Nazaretski, K. S. Graham, J. D. Thompson, J.\nA. Wright, D. V. Pelekhov, P. C. Hammel, and R.\nMovshovich, Rev. Sci. Instrum. 80, 083074 (2009).\n[16] A SSS-QMFMR cantilever, Nanosensors, Inc.\n[17] Y. H. Sun, Y. G. Zhao, H. F. Tian, C. M. Xiong, B. T.\nXie, M. H. Zhu, S. Park, Weida Wu, J. Q. Li, and Qi Li,\nPhys. Rev. B. 78, 024412 (2008).\n[18] B. D. Cullity and C. D. Graham, Introduction To Mag-\nnetic Materials , TheInstituteofElectrical andElectronic\nEngineers, Inc. (2009).\n[19] R. Mathieu, P. Nordblad, D. N. H. Nam, N. X. Phuc,\nand N. V. Khiem, Phys. Rev. B 63, 174405 (2001).\n[20] P. G. Radaelli, R. M. Ibberson, D. N. Argyriou, H.\nCasalta, K. H. Andersen, S. W. Cheong, and J. F.\nMitchell, Phys. Rev. B 63, 172419 (2001).\n[21] V. N. Smolyaninova, A. Biswas, P. Fournier, S. Lofland,\nX. Zhang, G. Zhao, and R. L. Greene, Phys. Rev. B 65,\n104419 (2002).\n[22] D. Saurel, A. Brˆ ulet, A. Heinemann, C. Martin, S. Mer-\ncone, and C. Simon, Phys. Rev. B 73, 094438 (2006)."    },    {        "title": "1308.0203v1.Exchange_relaxation_as_the_mechanism_of_ultrafast_spin_reorientation_in_two_sublattice_ferrimagnets.pdf",        "content": "arXiv:1308.0203v1  [cond-mat.str-el]  1 Aug 2013Exchange relaxation as the mechanism of ultrafast spin reor ientation in two-sublattice\nferrimagnets.\nV. G. Baryakhtar,1V. I. Butrim,2and B. A. Ivanov1,3,∗\n1Institute of Magnetism, 03142 Kiev, Ukraine\n2Taurida National V.I. Vernadsky University, 95007 Simfero pol, Ukraine\n3National Taras Shevchenko University of Kiev, 03127 Kiev, U kraine\nIn the exchange approximation, an exact solution is obtaine d for the sublattice magnetizations\nevolution in a two-sublattice ferrimagnet. Nonlinear regi mes of spin dynamics are found that include\nboth the longitudinal and precessional evolution of the sub lattice magnetizations, with the account\ntaken of the exchange relaxation. In particular, those regi mes describe the spin switching observed\nin the GdFeCo alloy under the influence of a femtosecond laser pulse.\nPACS numbers: 75.10.Hk, 78.47.J-, 05.45.-a\nMagnetic materials have various applications in\nmodern electronics and informatics, but probably the\nmost important research direction is still the creation\nof information storage and processing systems. The\nchallenge of designing magnetic devices with ever\nincreasing information density and recording speed\nrequires solving certain fundamental problems of the\nmagnetism dynamics. The possibility to manipulate the\nmagnetization by means of femtosecond laser pulses\nopens wide opportunities in this direction. This field\nhas been incepted by the work [1], where a fast (within\na time shorter than a picosecond) reduction of nickel\nmagnetization after the exposure to a 100 femtosecond\nlaser pulse has been observed, as well as the subsequent\nrelaxation of the magnetization with a characteristic\ntime of the order of picoseconds. The authors explained\nthe initial drop in the magnetization either by an\nextremely rapid heating of the sample above the Curie\npoint, see review [2], or by spin-dependent super-diffusive\nelectron transfer in the laser-excited metal [3]. Further\nwork in this area followed for various materials, and\nunexpected and rather unusual effects were discovered.\nIn the ferrimagnetic rare earth and transition metal\nalloy GdFeCo, a femtosecond pulse lead, in the first\nstage, to a similar spin reduction (i.e., the reduction of\nthe magnetization of sublattices) as for nickel, but the\nsubsequent evolution turned out to be fundamentally\ndifferent. Instead of a simple relaxation to the initial\nvalue, within about the same time (a few picoseconds),\nboth sublattice magnetizations changed their signs, i.e.,\na switching of the net magnetic moment took place [4],\nand during this picosecond-scale evolution there occurred\nana priori energetically unfavorable state with parallel\nsublattice moments. Such a magnetization switching\neffect is of a threshold type, and is observed only for\nsufficiently strong pulses. It has been detected in films\nas well as in microparticles [5] and nanoparticles [6],\nboth for ferromagnets with and without a compensation\npoint [5]. There is also a way of “selective” switching:\ndue to the magnetic dichroism, the absorbed energy ofa circularly polarized pulse depends on the direction of\nthe magnetic moment of the particles, and a pulse of\ncertain polarization would only switch the moments of\nthe particles which are in a matching state [7]. All that\nmakes possible to create a purely optically-controlled\nmagnetic memory with a picosecond recording speed.\nAlthough an analytical explanation of this effect is\nhighly desirable, The theoretical description has been\nperformed only by means of numerical simulation [4, 5].\nIt has been found that the change of the sublattice\nmagnetization lengths S1=|S1|иS2=|S2|, i.e. a\nlongitudinal spin evolution, is crucially important for th is\nphenomenon [5, 8]. The magnetization length is formed\nby the exchange interaction, and all the salient features\n(particularly, picosecond-scale characteristic evoluti on\ntimes, and the fact that the effect persists even in\nmagnetic fields up to 300 KOe) point out to the\nimportance of the exchange-dominated evolution [5].\nThe Landau-Lifshitz (LL) equation [9], with the\nstandard relaxation terms [9, 10] preserves the\nmagnetization length. The problem of the correct\nstructure of the relaxation terms in the LL equation,\nincluding the question of a purely exchange relaxation,\nwas previously considered by one of the present authors\n[11, 12]. It was shown that the longitudinal spin evolution\narises naturally when the general equations describing\nthe magnetization dynamics of ferromagnets [11] and\nantiferromagnets [12] are constructed, but has certain\nlimitations. Because of the obvious symmetry of the\nexchange interaction, it can not lead to a change (in\nparticular, relaxation) of the total spin of the system.\nTherefore, the evolution of the magnetization length of\na simple ferromagnet is reduced to a diffusion process\n(that is generally nonlinear), and is absent in the\nhomogeneous case which we are interested in, see the\ndetailed analysis in [11]-[13]. However, for a magnet with\ntwo sublattices, the situation is different, and a purely\nexchange relaxation is possible even for a homogeneous\ndynamics [12].\nThese ideas were used in [8] for a qualitative2\ndescription of the experimental data. Since the duration\nof the laser pulse used (less than 100 fs) is much shorter\nthan the characteristic evolution time, the analysis\ncan be performed by considering the dynamics of the\nmagnetization outside of the time interval of the pulse.\nIn doing so, a highly non-equilibrium state created by\nthe pulse plays the role of the initial condition for the\nequations describing the magnetization dynamics. The\nfollowing scheme has been proposed: the light pulse\ntransfers the system into a non-equilibrium state, in\nwhich, however, the direction of the spin sublattices is the\nsame as in the initial state. The system evolves further\nunder the influence of a faster exchange relaxation,\nfollowing along the straight line S1+S2=S1(0) +\nS2(0) = const in the(S1,S2)plane, see Fig. 1 of Ref.\n[8]. The analysis showed that the evolution of the system\nquickly leads to a state of partial equilibrium, which\ncorresponds to the spin values differing from the initial\nonesS1(0),S2(0)not only in the magnitude, but in the\nsign as well. The further evolution is due to the slower\nrelativistic relaxation, and the system goes to one of\nthe two equivalent states of complete equilibrium. In\na wide range of the initial values, consistent with the\nexperiment, the final state after the two-step process\ndiffers from the initial one only by the signs of S1and\nS2, which explains the effect of spin switching. However,\nRef.[8] studied a purely longitudinal dynamics, that is, it\nwas assumed that the vectors S1andS2remain collinear\nto their initial values.\nIn the present work, the exchange evolution of\nsublattice spin vectors of a ferrite is investigated in\na general way, without the assumption of collinearity.\nWe have found nonlinear regimes of spin dynamics,\nincluding both longitudinal and precessional evolution\nof the sublattice spins. It is shown that in the case of\na strong deviation from equilibrium an instability of the\nlongitudinal dynamics is possible, in which the amplitude\nof the precession increases due to the transfer of the\nenergy associated with the nonequilibrium character of\nlength of the antiferromagnetism vector L=S1−S2into\nthe deviation of Lfrom its equilibrium direction that is\ncollinear to the total magnetization M=S1+S2.\nThe LL equations for a two-sublattice magnet, with\npurely exchange relaxation terms can be written as\n/planckover2pi1∂S1\n∂t= [S1,H1]+λ(H1−H2)−λ1∇2H1,\n/planckover2pi1∂S2\n∂t= [S2,H2]−λ(H1−H2)−λ2∇2H2,(1)\nwhereS1,S2are the sublattice spins, H1,2=\n−δw/δS1,2are the effective fields for the sublattices, and\nw=w{S1,S2)}is the non-equilibrium thermodynamic\npotential per elementary cell, written as a functional of\nthe sublattice spin density. In what follows we set the\nPlanck constant to unity, and it will only be recovered\nin some final results. The relaxation terms can bewritten in the form δQ/δH1,2, whereQis the dissipative\nfunction, dw/dt−= 2Q, whose density in the exchange\napproximation is given by the following expression [11]:\n2Q=λ(H1−H2)2+λ1(∇H1)2+λ2(∇H2)2.\nHereafter, we will discuss only the homogeneous\ndynamics, and the terms containing λ1,2, which\ndetermine the spin diffusion, will be neglected.\nHere a general remark is in order, regarding the\nequations of motion of the magnetization. For the LL\nequation, both dynamic and dissipative terms (including\nthe standard relaxation term of the relativistic nature as\nwell as the exchange terms such as those in Eq. (1)) are\nchosen to be linear in the components of the effective\nfield. This approach is consistent with the Onsager\nprinciple, see [11]. However, the linearity of equations\nin the effective field does not limit the applicability\nof these equations to the linear approximation . For a\nmagnetically ordered state, a significant nonlinearity\nis present in the expression for the nonequilibrium\nthermodynamic potential, which determines well-known\nnon-linear properties of the LL equation. The presence\nof this non-linearity, reflecting the properties of the\nsystem, makes this approach very natural and reasonable.\nOf course, it is possible to consider generalizations of\nthese equations including the terms nonlinear in the\ncomponents of the effective field, but we do not know\nany examples where such a generalization would lead to\nnew physical effects.\nIn the homogeneous case and in the exchange\napproximation, the relaxation for two-sublattice magnets\nis actually determined by a single parameter λ. This is\neasy to understand by noticing that Eqs. (1) preserve\nthe total spin M, which is the consequence of the\nexchange approximation. We remark that the SU(2)\nexchange symmetry does not exclude the change of\nlength as well as the direction of the antiferromagnetism\nvectorL. Thus, we come to the conclusion that the\ninter-sublattice exchange plays the dominating role in\nrelaxation (in contrast to the independent relaxation of\nevery sublattice, as it comes out when the relaxation\nterm is taken in the Gilbert form), which is supported\nby recent experiments [14].\nNaturally, (1) describes only the relaxation to a\npartially equilibrium state, which corresponds to a\nminimum of the thermodynamic potential at fixed (and,\ngenerally, non-equilibrium) M. The value of λcan\nbe found from the damping decrement γlinof small-\namplitude Loscillations, which in the framework of\n(1) is determined by the formula γlin=λJ12(¯S1−\n|¯S2|)2/¯S1|¯S2|, where ¯S1,¯S2are the equilibrium spin\nvalues. It is important that the damping of optical spin\nwaves, connected to the transversal oscillations of the\nantiferromagnetism vector L, and the relaxation of the\nlength of Lare both determined by the same constant\nλ. First, this allows one to establish the value of λ3\nfrom independent measurements, and second, one can use\nthe known results of microscopic calculations of magnon\ndamping [15] to estimate it, which yields λ∝T4.\nIn what follows, our starting point will be the following\nexpression for the thermodynamic potential of a two-\nsublattice ferrite with purely exchange symmetry, written\nfor the homogeneous case as a function of the sublattice\nspins:\nw(S1,S2) =f1(S2\n1)+f2(S2\n2)+J12S1S2, (2)\nwhereS2\n1,2=S2\n1,2, and the exact form of the functions\nf1andf2is not yet specified. It is clear that the terms\ncontaining f1,f2do not contribute to the dynamical part\nof (1), and [S1,2,H1,2]→ ±J12[S1,S2]. It is convenient\nto pass to the equations for irreducible vectors Mand\nL. The equation for Myields∂M/∂t= 0, and for L\none obtains the closed-form vector equation ∂L/∂t=\nJ12[M,L] + 2λeHL,HL=−∂w/∂L. Let us choose\nthezaxis along the constant vector M=Mez. In the\nconvenient notation L=Ll,l2= 1those equations take\nthe form\n∂L\n∂t= 2λe(lHL)−2λe∂w\n∂L,\n∂l\n∂t=J12[M,l]+2λe\nL[HL−l(lHL)], (3)\nwhere the dissipative term in the equation for lresembles\nthe Landau-Lifshitz one. Equations for Landl, with the\naccount taken of the specific form of the thermodynamic\npotential can be also cast in the following convenient\nform:\n∂L\n∂t=−2λe(J12L−∂f1\n∂S1+∂f2\n∂S2),\n∂l\n∂t=J12[M,l]+λe\nL/parenleftbigg∂f1\n∂S2\n1−∂f2\n∂S2\n2/parenrightbigg\n[M−l(lM)].(4)\nHaving written lx+ily= sinθexp(iϕ), lz= cosθ, it is\neasy to show that ϕ=ωt,/planckover2pi1ω=J12M, and at θ∝ne}ationslash= 0,π\nvectorlprecesses with a constant frequency ω, and the\nprecession amplitude Lsinθchanges with time because\nof the dissipation. It is interesting that for small Ma\n“slowdown” of this precession takes place. Thus, nonlinear\noscillations of arbitrary (not small) amplitude have the\nform of a precession of Laround the constant vector M,\nwith the frequency /planckover2pi1ω=J12M:\nL=Lzez+L⊥(excosωt+eysinωt),\nLz=Lcosθ, L⊥=Lsinθ,\nwhere the quantities Lz(t),L⊥(t)exhibit a dissipative\nevolution. It is convenient to write down the equations in\nL,θvariables:\n∂L\n∂t= 2λL(J12−∂f1\n∂S2\n1−∂f2\n∂S2\n2)+2λ(∂f2\n∂S2\n2−∂f1\n∂S2\n1)Mcosθ,\n∂θ\n∂t=−M2λ\nL(∂f2\n∂S2\n2−∂f1\n∂S2\n1)sinθ. (5)For the sake of simplicity and physical clarity let us\ntakef1,2in the form of the Landau expansion of the\nform\nf1=J1\n4(S2\n1−S2\n0)2, f2=J2S2\n2\n2. (6)\nHere we assume that the second sublattice consists\nof paramagnetic rare-earth ions, f2is determined by\nthe spin entropy, and J2is of the order of the\ntemperature T. The parameter S0=S0(T)formally\ncoincides with the equilibrium value of the iron sublattice\nmagnetization if one neglects its interaction with the\nrare-earth sublattice. Using (5), one obtains simple close d\nformulae for the equilibrium values of the sublattice\nspins,¯S1=/radicalBig\nS2\n1,0+J2\n12/J1J2and¯S2=−J12¯S1/J2,\nwhile the equations can be written in the form\nt0∂L\n∂t=f(L,θ), t0∂θ\n∂t=g(L,θ), t0=4/planckover2pi1\nλJ1,(7)\nгдеf(L,θ) =−L3−3L2Mcosθ+AL+B,\ng(L,θ) =−Msinθ\nL(4J2\nJ1+4S2\n0,1−L2−2LMcosθ−M2)\nA=−M2(1+2cos2θ)−4J2\nJ1+4S2\n0,1+8J12\nJ1,\nB=Mcosθ(4J2\nJ1+4S2\n0,1−M2).\nIt is worth noting that the evolution of Lz(t),L⊥(t)\noccurs on a naturally emerging universal time scale t0=\n4/planckover2pi1/λJ1, which is larger than the “purely exchange” time\ntex∼/planckover2pi1/J1∼t0/λsince the relaxation constant λis\nsmall. For not very small values of M∼1and not too\nweak inter-sublattice interaction J12∼J1, this time scale\nis also larger than the precession period of vector L.\nProceed further to the analysis of the evolution of Lz\nandL⊥. It is clear that all singular points occur at θ=\n0,π, and their positions are determined by zeros of the\nfunction f(L,θ)atsinθ= 0. The condition f(L,sinθ=\n0) = 0 can be represented as a cubic equation in Lcosθ=\n±L(it is convenient to assume that L >0, andθvaries\nin the range 0≤θ≤π). AtM= 0the three roots\nareL1,3cosθ=±√\nAandL2= 0, so it is clear that at\nsufficiently small M≤Mcthere will also be three real\nroots. A simple analysis shows that L=L1, θ= 0(or\nLz=L1>0,L1=√\nAatM= 0)corresponds to the\nequilibrium position (a stable node), L=L3, θ=π,\ni.e.,Lz=−L3<0corresponds to a saddle point, and\nthe unstable node lies at L=L2. ForM≤Mcone has\nL1< L3, and the unstable node will correspond to a\nnegative value of Lz=L2cosθ <0. AtM=MctheL2\nandL3roots merge, and for M > M cthe system has\nonly one singular point at Lcosθ=L1>0.\nIt is important to note that for all values of Mthe\nsystem (7) has another solution Lz=Lz(t),L⊥(t) = 0,4\n/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s76 \n/s122 /s76 /s32\nРис. 1: The evolution of L⊥andLz, calculated numerically\nforJ1=J2= 2J12andS0= 1atM= 0.4and shown as\na phase portrait. Singular points are shown with circles, an d\nthe separatrix is shown as a dotted line.\nwhich corresponds to a purely longitudinal evolution,\nbut the physical sense of this solution is very different.\nAtM > M c, for all initial conditions ( L(0)< L1or\nL(0)> L1), the value of Lztends to its equilibrium\nvalueL1for this class of solutions. Numerical analysis\nshows that in his case the evolution remains close to the\npurely longitudinal one even if the direction of Ldeviates\nfrom the equilibrium. The only exception is for large\ndeviations, when L(0)∼ −L1; in this case the length of\nLis already close to the equilibrium, and a rotation of L\nbecomes favorable. At extremely small Mthe evolution\nis degenerate: θchanges much slower than L, and the\nphase portrait in the (L⊥,Lz)plane consists of radial\nstraight line intervals θ= const and of parts of the circle\nL=L1≈√\nA. For finite M < M cthe situation is much\nmore interesting: in this case one also has a solution of\nthe form Lz=Lz(t), L⊥(t) = 0 , but with the initial\nconditions −L3< Lz(0)<−L2, i.e., between the saddle\npoint and the unstable node, the longitudinal evolution\ntakes the system away from the equilibrium. This is\nillustrated in Fig. 1, which shows the phase portrait of\nthe system in (L⊥,Lz)plane, calculated numerically for\nJ1=J2= 2J12andS0= 1atM= 0.4< Mc(at those\nparameter values one has Mc= 4/(3√\n3)≃0.77).\nThus, the exact solution of the full system of\nequations of motion for sublattice spins in the exchange\napproximation shows the existence of two qualitatively\ndifferent regimes. The character of the evolution is mainly\ndetermined by the initial value of the magnetization M,\nwhich is conserved in the exchange approximation. For\nlargeM > M c, as well as for all Mand the initial\nvalueLz(0)>0, a longitudinal relaxation occurs, as\nstudied previously in [8]. For M < M c, approximately\nthe same behavior is also retained for negative Lz(0),provided that Lz(0)>−L2. In all those cases, there is\na special solution of the form Lz=Lz(t)which leads to\nthe equilibrium, and even for a nonzero (but small) value\nof the transversal initial deviation L⊥(0)the value of\nL⊥remains small in the process of relaxation. However,\nthe situation is changed dramatically, if the initial value\nenters the region of the unstable node situated around\nLz≃ −L2(Lz≃0.83in Fig. 1). As seen in Fig. 1,\nin the vicinity of this point and to its left, even small\ninitial values of L⊥increase with time. In this case, in\na wide range of the initial conditions all trajectories in\nthe(Lz,L⊥)plane tend to the separatrix which connects\nthe saddle point and the unstable node; the values of\nL⊥are not small at the separatrix. In this way, strongly\nnonlinear evolution regimes with L⊥∼Lz∼1become\npossible. The solution with L⊥∝ne}ationslash= 0atM∝ne}ationslash= 0is of\nthe precession type, i.e., for the initial condition with\nLz<−L2approaching equilibrium is accompanied by\nthe growth of the precession amplitude of L, at the\nconstant precession frequency /planckover2pi1ω=J12M, so that L=\nLzez+L⊥(excosωt+eysinωt). It should be remarked\nthat the experimentally observed time dependence of the\nsublattice magnetizations in the time interval between\n0.5and3ps shows some non-monotonic behavior at the\nbackground of a smooth magnetization change, which\nresembles oscillations with the period of about 0.3ps, see\nFig. 2 of Ref. [4]. The results of numerical simulation of\nthis process, reported in the same work [4], did not show\nsuch a behavior, but oscillations were found in recent\nnumerical studies [16].\nTaking into account the transversal spin deviations\nin the process of evolution may be important for\nunderstanding the recent experiment on TbFeCo alloy\n[17]. An obvious difference between this material and\nGdFeCo is the presence of a strong easy-axis anisotropy,\nbut it is clear that such anisotropy should not affect\na purely longitudinal evolution. Despite that, spin\nswitching characteristic for GdFeCo was not observed in\nTbFeCo, although the initial reduction of the sublattice\nmagnetizations was roughly the same as in the GdFeCo\nexperiment. Of course, there could be other reasons\nfor such a different behavior, e.g., the presence of\nunquenched orbital moment of Tb, but the detailed\nanalysis of this problem is beyond the scope of the present\npaper.\nThis work is partly supported by the joint Grant\n0113U001823 of the Russian Foundation for Fundamental\nResearch and the Presidium of the National Academy of\nScience of Ukraine, and by the joint Grant Φ53.2/045 of\nthe Russian and Ukrainian Foundations for Fundamental\nResearch.\n∗Electronic address: bivanov@i.com.ua5\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot ,\nPhys. Rev. Lett. 76, 4250 (1996).\n[2] A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod.\nPhys.82, 2731 (2010).\n[3] M. Battiato, K. Carva, P. M. Oppeneer, Phys. Rev.\nLett.105, 027203 (2010); D. Rudolf, C. La-O-Vorakiat,\nM. Battiato et. al , Nature Comm. 3, 1037 (2012); M.\nBattiato, K. Carva, and P. M. Oppeneer, Phys. Rev. B\n86, 024404 (2012)\n[4] I. Radu, K. Vahaplar, C. Stamm et. al , Nature (London)\n472, 205 (2011).\n[5] T. A. Ostler, J. Barker, R. F. L. Evans et. al, Nature\nCommun. 3, 666 (2012).\n[6] L. Le Guyader, S. El Moussaoui, M. Buzzi, R.\nV.Chopdekar, L. J. Heyderman, A.Tsukamoto, A.Itoh,\nA.Kirilyuk, Th.Rasing, A. V.Kimel, and F.Nolting, Appl.\nPhys. Lett 101, 022410 (2012).\n[7] A. R. Khorsand, M. Savoini, A. Kirilyuk, A.V. Kimel, A.\nTsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett.\n108, 127205 (2012)\n[8] J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,\nA. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand Th. Rasing, Phys. Rev. Lett. 108, 057202 (2012).\n[9] L. D. Landau and E. M. Lifshitz, To the theory of\nmagnetic permeability of ferromagnetic bodies , In: L. D.\nLandau, The collection of works, Vol.1, p. 128 (Nauka,\nMoscow 1969) [in Russian].[10] T. L. Gilbert, Phys. Rev. 100, 1243 (1955).\n[11] V. G. Bar’yakhtar, Zh. Eksp. Theor. Fiz. 87, 1501 (1984);\nFiz. Tverd. Tela 29, 1317 (1987) [Sov. Phys. JETP 60,\n863, (1984); Sov. Phys. Solid State 29, 754 (1987)].\n[12] Fiz. Nizkikh Temp. 11, 1198 (1985); Zh. Eksp. Theor.\nFiz.94, 196 (1988) [Sov. J. Low Temp. Phys. 11, 662\n(1985); Sov. Phys. JETP 67, 757 (1988)].\n[13] V. G. Bar’yakhtar, B. A. Ivanov, T. K. Soboleva, and\nA. L. Sukstanskii, Zh. Eksp. Theor. Fiz. 91, 1454 (1986)\n[Sov. Phys. JETP 64, 857 (1986)]; V. G. Bar’yakhtar,\nB. A. Ivanov and K. A. Safaryan, Solid State Comm.\n72, 1117 (1989); E. G. Galkina, B. A. Ivanov and\nV. A. Stephanovich, JMMM 118, 373 (1993); V. G.\nBar’yakhtar, B. A. Ivanov, A. L. Sukstanskii and E. Yu.\nMelekhov, Phys. Rev. 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Lett.\n110, 107205 (2013)."    },    {        "title": "0704.3139v2.Element_resolved_x_ray_ferrimagnetic_and_ferromagnetic_resonance_spectroscopy.pdf",        "content": "Element-resolved x-ray ferrimagnetic and\nferromagnetic resonance spectroscopy\nG Boero, S Mouaziz, S Rusponi\nEcole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\nP Bencok\nEuropean Synchrotron Radiation Facility (ESRF), F-38043 Grenoble, France\nF Nolting\nSwiss Light Source (SLS), Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\nS Stepanow\nCentre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB Campus,\nE-08193 Bellaterra, Barcelona, Spain\nP Gambardella\nInstituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats (ICREA)\nand Centre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB\nCampus, E-08193 Bellaterra, Barcelona, Spain\nE-mail: pietro.gambardella@icrea.es\nAbstract. We report on the measurement of element-speci\fc magnetic resonance\nspectra at gigahertz frequencies using x-ray magnetic circular dichroism (XMCD). We\ninvestigate the ferrimagnetic precession of Gd and Fe ions in Gd-substituted Yttrium\nIron Garnet, showing that the resonant \feld and linewidth of Gd precisely coincide\nwith Fe up to the nonlinear regime of parametric excitations. The opposite sign of\nthe Gd x-ray magnetic resonance signal with respect to Fe is consistent with dynamic\nantiferromagnetic alignment of the two ionic species. Further, we investigate a bilayer\nmetal \flm, Ni 80Fe20(5 nm)/Ni(50 nm), where the coupled resonance modes of Ni and\nNi80Fe20are separately resolved, revealing shifts in the resonance \felds of individual\nlayers but no mutual driving e\u000bects. Energy-dependent dynamic XMCD measurements\nare introduced, combining x-ray absorption and magnetic resonance spectroscopies.\nPACS numbers: 76.50.+g, 78.70.Dm, 78.20.Ls, 76.30.DaarXiv:0704.3139v2  [cond-mat.mtrl-sci]  22 Jan 2008Element-resolved XFMR 2\n1. Introduction\nRecent interest in magnetization dynamics has been fostered by progress in fast\nmagnetic recording and microwave technologies [1, 2]. Despite considerable e\u000borts,\nhowever, the description of magnetodynamics remains essentially phenomenological.\nInductive, magnetoresistive, and magneto-optical techniques solely measure the\nintegrated magnetic response of complex heterogeneous materials, typically magnetic\nalloys and multilayer structures, whose functionality depends on the interplay of several\nelements. The development of methods capable of elemental analysis constitutes\nan obvious advantage for investigating fundamental problems related to time- or\nfrequency-dependent magnetization phenomena. Examples include the dynamic\ncoupling of elemental moments in ferrites [3, 4, 5, 6], metallic alloys [7], and spin-valve\nheterostructures [8, 9], as well as spin-orbit induced damping e\u000bects attributed to the\npresence of high [5, 10, 11] and low [12] Z elements. Advances in this direction are\nmostly based on stroboscopic pump-probe experiments exploiting the element-resolving\npower of x-ray magnetic circular dichroism (XMCD) and the sub-ns bunch structure of\nsynchrotron radiation beams. Pulsed magnetic \felds in synchrony with x-ray photon\nbunches are usually employed to excite the reversal [8, 13] or the precessional motion\n[7] of the magnetization. More recently, continuous wave rf \felds have been applied to\nexcite resonant modes in trilayer metal \flms [14, 15] and microstructures [16, 17].\nWith respect to time-resolved measurements, techniques such as ferromagnetic\nresonance spectroscopy (FMR) o\u000ber an alternative and powerful way to gain insight\ninto the energy scales that govern magnetization dynamics. Frequency-domain methods\nthat allow to detect magnetic resonance using the core level absorption of circularly\npolarized x-rays have been developed independently by our group in the soft x-ray\nenergy range [18] and by Goulon et al. in the hard x-ray regime [19, 20]. These methods\nexploit the XMCD dependence on the scalar product M\u0001Pof the magnetization vector\nMand photon helicity Pto measure the time-invariant changes of the longitudinal\nmagnetization component \u0001 Mzas a function of microwave (MW) \feld B1and bias\n\feldB0. Microstrip resonators [18] and tunable cavities [21] have been employed to\ngenerate MW excitations together with di\u000berent detection schemes. In the hard x-ray\nregime, XMCD at the Kedge of transition metals relates purely to orbital magnetization\ncomponents; measurements at the Fe K-edge and Y L2;3edges by Goulon et al. provided\nevidence for the precession of the Fe orbital moments as well as induced Y spin moments\nin yttrium iron garnet (YIG) [19, 20].\nIn this article, we report on di\u000berent applications of soft x-ray MCD to FMR\nmeasurements and on a novel way to combine FMR and XMCD spectroscopy. Element-\nspeci\fc magnetic resonance spectra are measured on both magnetic oxides and metallic\nmultilayers. We show that ferrimagnetic resonance measurements of Gd-substituted\nYIG are consistent with the antiferromagnetic (AFM) alignment of Gd and Fe ions in the\nferromagnetic resonance mode of YIG in the non-linear regime, above the threshold for\nparametric spin wave excitations. Further, FMR spectra of coupled thin metal bilayersElement-resolved XFMR 3\nFigure 1. (a) Diagram of the experimental setup. (b) Close-up view of the resonator\nand photodiode situated between the poles of the electromagnet. Note that one of the\nmagnet poles and the photodiode have an opening to allow for the passage of x-rays.\nare separately resolved, allowing the investigation of interlayer dynamics in stacks of\nmagnetic layers. Finally, we show that the x-ray FMR (XFMR) signal measured at\nresonance as a function of photon energy yields dynamic XMCD spectra, which relate\nto the magnetic state of the atoms undergoing microwave absorption. The latter can\nbe combined with static XMCD spectra to derive information on the dynamics of the\norbital and spin magnetization components.\n2. Experimental\nA schematic diagram of the experimental setup is given in Fig. 1. A coplanar waveguide\n\u0015=2-resonator is used to generate a MW \feld B1\u00190:01 to 0.5 mT parallel to the sample\nsurface with input power 0 to 34 dBm at frequency !=2\u0019= 2:21 GHz. The resonator-\nsample assembly is positioned between the pole expansions of an electromagnet, which\nproduces a \feld 0 \u0014B0\u00140:8 T aligned perpendicular to the sample surface and parallel\nto the photon propagation direction. In the absence of MW \feld, Maligns with B0\nparallel to P, which is the geometry commonly employed in static XMCD measurements.\nIfB1is turned on, as B0matches the resonance \feld of the sample ( Br) the precessional\nmotion of Minduces a reduction of the longitudinal magnetization component Mzthat\ncan be measured as a steady-state e\u000bect in the frequency domain, i.e., without requiring\nsub-ns time resolution. Here, x-ray absorption spectra (XAS) corresponding to positive\n(P+) and negative (P\u0000) helicity are measured by recording the dc \ruorescence yield\n(FY) of the sample as a function of photon energy using a Si photodiode (Eurisys-\nCanberra, Ref. [22]). XMCD is de\fned as the di\u000berence spectrum P+-P\u0000(Fig. 2). The\nXFMR signal, either P+or P\u0000, is obtained by square-modulating the MW power source\nat relatively low frequency ( <100 kHz) and by measuring the corresponding amplitude\nof the ac FY photocurrent by means of a lock-in ampli\fer, as shown in Fig. 1 (a). We\nintroduce two methods to measure magnetic resonance using XMCD: the \frst, in analogy\nwith FMR spectroscopy, consists in recording the XFMR intensity during a sweep of B0\nacrossBr, \fxing the photon energy in correspondence of a static XMCD peak [18]. We\ndenote this type of measurements as XFMR B-scan , which e\u000bectively generate element-Element-resolved XFMR 4\nFigure 2. (a) One octant portion of the unit cell of GdIG, showing the AFM\nspin alignment of octahedral Fe (black circles), tetrahedral Fe (gray circles), and\ndodecahedral Gd sites (empty blue circles), from Ref. [23]. Oxygen ions have been\nomitted. (b) FY XAS spectra and corresponding XMCD of Fe and (c) Gd sites\nmeasured at room-temperature with B0= 0:21 T.\nspeci\fc longitudinal magnetic resonance spectra. The second method consists in taking\nthe sample at resonance by setting B0=Brand recording the XFMR as a function\nof photon energy. This, denoted as XFMR E-scan , is analogous to recording XAS and\nXMCD spectra, but corresponding to the precessional motion of Mrather than to a\nstatic situation. Examples of either type of measurements will be given later.\nTwo di\u000berent type of samples are employed in the present study: a\nrare earth substituted iron oxide and a metallic heterostructure, which were\nchosen in order to highlight the broad spectrum of materials where new insight\ncan be obtained by XFMR. A polished 30 \u0016m-thick slab of polycristalline\nGd 1Y2Fe5O12(Gd:YIG) with lateral dimensions 1 \u00022 mm2was selected to investigate\nferrimagnetic resonance in garnet systems composed of di\u000berent magnetic ions. An\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer deposited on glass by e-\nbeam evaporation in high vacuum (1 \u000210\u00006mbar) was fabricated in order to address\nlayer-speci\fc resonance modes in metallic heterostructures. The x-ray spot size at the\nsample position was 0.1 mm long and 1 mm wide at full width half maximum, while\nthe coplanar resonator had a central conductor with a width of 1.5 mm and a length\nof 44 mm, thus ensuring that the MW excitation covers the whole area sampled by the\nx-ray beam. XAS and XFMR spectra were recorded at the L2;3edges of Fe and Ni,\nand at the M4;5edges of Gd. XAS spectra are normalized to the incident photon \rux\nmeasured by the photocurrent of an Au grid upstream from the sample, and are givenElement-resolved XFMR 5\nFigure 3. (a) Magnetization of a 30 \u0016m thick, 1\u00022 mm2wide Gd 1Y2Fe5O12slab\nmeasured by SQUID with applied \feld perpendicular to the sample plane at 300 K.\n(b) Magnetization vs. temperature of a 100 \u0016m thick Gd 1Y2Fe5O12slab \feld-cooled\nin a 3 mT \feld.\nin arbitrary units. Apart from normalization, the spectra are raw data; in particular, no\nenergy-dependent correction for self-absorption has been applied. As the signal-to-noise\nratio is proportional to the square root of the photocurrent [18], energy resolution has\nbeen sacri\fced to intensity by opening the exit slits of the beamline monochromator.\nThe e\u000bective energy resolution corresponds to about 1.2 and 3 eV at 700 and 1200 eV,\nrespectively, which results in signi\fcant broadening of the multiplet features of Fe and\nGd spectra in Gd:YIG, as shown in Fig. 2. This is not an essential problem for XFMR\nB-scans, but may limit the spectral resolution of E-scans; in the latter case, however,\nhigher resolution can be achieved simple by reducing the slit apertures while increasing\nthe averaging time to maintain a constant signal-to-noise ratio. Throughout the paper\nXFMRB-scans are given in pA, as measured by the FY photodiode. Simultaneously\nwith XFMR, the transverse part of the imaginary susceptibility \u001f00was measured, as in\nconventional FMR, by monitoring the power re\rected o\u000b the \u0015=2-resonator via a MW\nbridge and diode detector, as schematized in Fig. 1 (a). XFMR B-scans were measured\nat the ID08 beamline of the European Synchrotron Radiation Facility, while E-scans\nwere recorded at the SIM beamline of the Swiss Light Source; two undulators were\noperated in series with 99 \u00061 % circularly polarized beams in both type of measurements.\n3. Element-resolved XFMR spectra of Gd:YIG\nThe structure of Gd 1Y2Fe5O12(Gd:YIG) consists of three sublattices [Fig. 2 (a)]. Two\nof them, the octahedral and tetrahedral sites, contain Fe ions which are strongly AFM\ncoupled by superexchange. The third lattice, the dodecahedral sites, contains Gd and\ndiamagnetic Y ions [23]. While their mutual interaction is very weak, Gd ions couple\nAFM to tetrahedral Fe ions with a moderate exchange \feld of the order of 24 T (16 K)\n[24]. Such a system thus e\u000bectively behaves as a two-sublattice ferrimagnet, where\nthe Gd moments order spontaneously only at low temperature ( <50 K). Figure 3\n(a) shows the out-of-plane magnetization of Gd:YIG measured by superconducting\nquantum interference device magnetometry (SQUID) at room temperature. The curveElement-resolved XFMR 6\nFigure 4. FMR spectra of Gd:YIG measured by the re\rected power from the \u0015=2-\nresonator at 0 dBm using \feld and MW amplitude modulation (bottom and middle\ntraces, respectively). The top trace shows the high power (31 dBm) FMR for MW\namplitude modulation. B0is oriented perpendicular to the sample surface in all cases.\nis composed by a hard-axis ferromagnetic loop that saturates above 0.1 T, as expected\nfrom shape anisotropy considerations, and a linear term proportional to the applied \feld.\nThe latter is a common feature of rare-earth garnets and ascribed to the continuous\nrotation of MGdtowards MFewith increasing \feld, in accordance with N\u0012 eel's theory\nof ferrimagnetism. The temperature behavior of the magnetization, shown in Fig. 3\n(b), is characteristic of two AFM-coupled lattices with inequivalent magnetization.\nWhile for all rare-earth garnets the Curie temperature is associated to the pairing of Fe\nmoments and nearly independent on rare-earth composition [23, 25], the compensation\ntemperature depends sensibly on the rare-earth content. In Gd 3Fe5O12compensation\noccurs at 290 K [23]. Figure 3 (b) shows that the total magnetization of Gd 1Y2Fe5O12is\napproximately constant from 300 to 150 K; below this temperature magnetic order sets\nin throughout the Gd lattice, compensating the Fe magnetization at about 45 K. The\nXAS and XMCD spectra of Fe and Gd in Gd 1Y2Fe5O12recorded at room temperature\nwith applied \feld B0= 0:21 T are shown in Figs. 2 (b) and (c). The opposite sign of\ntheM5vsL3andM4vsL2intensity re\rects the static alignment of the resultant MGd\nagainst MFe.\nLinearization of the coupled equations of motion shows that two resonances can be\nexcited in a ferrimagnetic compound: the ferromagnetic mode, which is independent\nof the exchange \feld since the angle between MFeandMGddoes not vary during the\nprecession, and the high-frequency exchange mode, where the two sublattices precess\nout-of-phase but phase-locked to each other with non collinear magnetization vectors\n[3, 4, 26]. The \frst mode is the one accessible at relatively low \felds in usual FMR\nexperiments, as in our case, while the second one is situated at \felds of several\ntens of Teslas for frequencies in the MW range [27]. Neglecting magnetocrystalline\nanisotropy, the resonant \feld for uniform precession in the ferromagnetic mode is\ngiven byBr=!\n\r+\u00160Nz(MFe\u0000MGd) = 190 mT, where \ris the gyromagneticElement-resolved XFMR 7\nratio,Nz= 0:935 is the demagnetizing factor calculated for our geometry [28], and\n\u00160(MFe\u0000MGd) = 120\u00066 mT. Figure 4 shows the conventional FMR spectra of\nGd:YIG. Owing to the sample \fnite dimensions, the low power FMR shows a series of\nmagnetostatic modes with the principal one close to Br. The longer wavelength modes\nare resolved in the \feld-modulated spectrum (bottom trace) and appear as shoulders\nof the main peak in the MW-modulated spectrum (middle trace). For a sample 30 \u0016m\nthick with lateral dimensions of the order of 1 mm their separation corresponds to that\nexpected for magnetostatic forward volume wave modes with the excitation geometry\nof Fig. 1 [29, 30]. At high MW power (top trace) the FMR shifts to a lower \feld due to\nheating of the sample and related decrease of the resultant magnetization MFe\u0000MGd.\nMoreover, the FMR lineshape is signi\fcantly distorted due to e\u000bects such as foldover\nand nonlinear spin wave instabilities [31]. In such a regime, nonlinear terms in the\nLandau-Lifschitz equation of motion transfer energy from the uniform precession mode\ndriven by the external MW \feld to nonuniform magnon modes, which become unstable\nabove a critical \feld threshold [32]. These phenomena lead to saturation of the main\nresonance and precession angle together with excitation of spin waves above thermal\nvalues. Of relevance to the present discussion is the fact that nonlinear coupling terms\nescape conventional treatments of ferrimagnetic resonance, which reduce the dynamics\nof individual sublattices to that of a single macrospin (e.g., of amplitude MFe\u0000MGd\nfor Gd:YIG) [3, 4, 5, 6]. Moreover, the assumed equivalency of the equations of motion\nfor di\u000berent sublattices might not hold true when nonlinear phenomena are taken into\naccount. For example, substitution of foreign ions in a material where all equivalent\nlattice sites are occupied by identical ions, as in Gd:YIG, provides a site-dependent\nadditional scattering channel leading to spin wave excitations [33]. Element-resolved\nFMR spectra can thus put the macrospin concept to test, speci\fcally in the nonlinear\nregime where relatively large deviations \u0001 Mzmake the XFMR intensity easier to detect.\nFigure 5 compares the inductive FMR spectrum of Gd:YIG (a) with the XFMR\nP+-P\u0000intensity recorded at the Fe L2edge (b) and Gd M4edge (c) as a function of\nB0. Several comments are in order. First, we note that conventional FMR and XFMR\nspectra di\u000ber for obvious reasons, namely: (i) XFMR is a measure of \u0001 Mz, while\nFMR is proportional to the transverse dynamic magnetization component. Only if jMj\nis conserved the two measurements can be considered to be equivalent. (ii) XFMR is\nsurface-sensitive, with the same probing depth as FY XAS ( \u001820 nm at the Fe L2;3edges\n[34]) and probes a limited portion of the sample, while FMR averages over the whole\nsample volume. In Fig. 5 (a) the FMR lineshape is asymmetric and heavily saturated due\nto nonlinear e\u000bects that limit the FMR precession cone amplitude. The XFMR signal\nin (b), on the other hand, is composed of a broad resonant feature and a sharp peak\nlocated at about B0= 165 mT with linewidth \u0001 B= 1 mT. It may be observed that the\nintensity of both features is centered around the low-\feld rising edge of the FMR peak\nand does not follow the FMR intensity distribution. The origin of such di\u000berences lies\nin (i) and (ii); a detailed understanding of the XFMR vs. FMR lineshape, however, isElement-resolved XFMR 8\nFigure 5. (a) FMR spectrum of Gd:YIG measured simultaneously with the XFMR\ndata. (b) XFMR P+\u0000P\u0000intensity measured at the L2edge of Fe (723.8 eV) and (c)\nat theM4edge of Gd (1222 eV). The MW power is 31 dBm. The data are averaged\nover 40 sweeps of B0in the positive direction, with a sweep time of 80 s and lock-in\ntime constant of 100 ms.\npresently missing. To appreciate this point, we o\u000ber a number of consideration based on\nprevious FMR and XFMR studies of YIG. The sharp peak observed by XFMR denotes\na sudden increase of \u0001 Mz, whereMzis proportional to the total number of magnons\nin the system. De Loubens et al. , using magnetic resonance force microscopy on a\nsingle crystal YIG \flm, observed a dramatic increase of \u0001 Mzat the onset of the second\norder Suhl's instability threshold, which was attributed to the parametric excitation of\nlongitudinal spin waves with a low spin-lattice relaxation rate compared to the uniform\nmode [35, 36]. In this model, the total number of magnons is considered to be constant,\nwhile changes of Mzare attributed to a redistribution of their occupation number from\nmodes with relatively high to low relaxation rate, favoring larger precession angles [37].\nGoulon et al. , using XFMR on a single crystal Y 1:3La0:47Lu1:3Fe4:84O12\flm, also observed\na sharp decrease of Mzmeasured at the Fe K edge, taking place in correspondence with\nthe foldover critical \feld of the FMR spectrum [21]. They explained this e\u000bect by\nthe degeneracy of the uniform mode with long-wavelength longitudinal magnetostatic\nwaves caused by foldover in perpendicular FMR. In this regime, parametric excitation\nof coupled magnetostatic-magnetoelastic waves becomes possible [21], which may lead\nto an e\u000bective transfer of angular momentum to the lattice and therefore to a decrease\nofMz. This is substantially di\u000berent from the model proposed by De Loubens et al. ,Element-resolved XFMR 9\nFigure 6. Restricted range of (a) FMR and (b) XFMR spectra of Gd:YIG at the L2\nedge of Fe (723.8 eV) and M5edge of Gd (1191 eV) recorded with the parameters of\nFig. 5.\nas the total number of magnons needs not be conserved. The validity of either of these\nexplanations for the present measurements may be questioned due to the inhomogeneous\ncharacter of local magnetic \felds in polycrystalline samples, e.g., owing to magnetic\nanisotropy \ructuations or microstructure \raws, which results in broadened FMR lines.\nSpeci\fcally, if individual crystal grains went through resonance individually according\nto their orientation in the applied \feld and one would have to worry about strongly\ninhomogeneous resonance conditions; however, as the magnetocrystalline anisotropy\n\feld is more than a factor 10 smaller compared to the saturation magnetization in\nGd:YIG, dipolar coupling between di\u000berent grains predominates and resonance occurs\nas a collective phenomenon [38, 39]. The observation of di\u000berent magnetostatic modes\nin Fig. 4 supports this view, although a much smaller number of modes are resolved\ncompared to single crystal YIG \flms [21, 35]. The granular structure of the material\nand related local changes of the anisotropy \feld have also a well-known e\u000bect on the\ncritical \feld for parametric spin wave excitations, raising it up to 0.1-1 mT in YIG\n[40], and leading to a smooth onset of this e\u000bect rather than an abrupt threshold [41].\nThe saturation as well as the distorted shape of the FMR spectrum indicate that the\nconditions for foldover and parametric spin wave ampli\fcations are met at high power\nin Gd:YIG and likely contribute to the observed XFMR features. In general, however,\nwe cannot identify a unique origin for the XFMR peak nor exclude it to be related to\na mode localized at the vacuum-Gd:YIG interface, which would be selectively probed\nby XFMR and only weakly observed in the bulk FMR signal [see Fig. 6 (a)]. More\nmeasurements shall be performed to clarify this point.\nWe proceed now to compare the XFMR spectra of Fe and Gd, discussing whatElement-resolved XFMR 10\ntype of information may be derived on the relative motion and relaxation of dissimilar\nmagnetic moments in a bulk compound at resonance. Apart from the noise and a\nscaling factor, the Gd M4spectrum in Fig. 5 re\rects specularly the one measured at\nthe FeL2edge. The resonant \feld and linewidth derived from the Gd B-scan XFMR\nprecisely match those of Fe, but the XFMR intensity has opposite sign. This is even\nmore evident in the restricted range B-scan in Fig. 6 (b), where the Fe L2and GdM5\nspectra are reported; note that the relative sign of the Fe and Gd intensity depends on\nthe absorption edge, as for XMCD. Sign inversion of the XFMR at the Fe L2(L3) and Gd\nM4(M5) edges, consistent with that observed in the static XMCD [Figs. 2 (b) and (c)],\nreveals the coupled AFM dynamics of the Fe and Gd magnetic moments. Their relative\n\u0001Mz=Mdeviations can be quanti\fed in terms of the XFMR cross section, de\fned as\nthe ratio between the dynamic and static dichroism FY photocurrents \u001b=XFMR (E)\nXMCD (E),\nwhich depends on the x-ray photon energy Eas well as on the spin and orbital magnetic\nmoment precession in a way dictated by the XMCD sum rules [42]. At 31 dBm MW\npower, we have \u001bL2(Fe) = (2:0\u00060:2)\u000210\u00003and\u001bM4(Gd) = (1:7\u00060:2)\u000210\u00003. These\ndata, together with the above observations, are consistent with Fe and Gd maintaining\nrigid AFM alignment in nonlinear excitation modes (diagram in Fig. 6). We note that,\nin principle, the same result can be obtained for noncollinear MFeandMGdvectors\nprecessing on the cone shown in Fig. 6; however, in the noncollinear case, di\u000berent \rexing\nangles (\u001b) would be expected for Fe and Gd, given that the local exchange \felds acting\non the two ionic species are strongly dissimilar [3, 24, 27]. Full con\frmation of the type\nof AFM coupling would in any case require to measure the phase of the precessing Fe and\nGd moments, which may be retrieved only by time-resolved detection of the transverse\nmagnetization components [14, 15, 21]. Within the experimental error, XFMR data thus\nshow that the resonating longitudinal components of MFeandMGdhave opposite sign\nand equal relative deviations from static equilibrium up to the nonlinear regime of high-\npower MW excitations. This is consistent with collinear dynamic AFM alignment of\nMFeandMGdpredicted by the theory of ferrimagnetic resonance for uniform precession\nat low \felds, but extends into the nonlinear regime beyond the approximations usually\nmade in theoretical models [3, 4, 26] and at temperatures where thermal \ructuations\nstrongly a\u000bect magnetic order in the Gd lattice (Fig. 3). Further, the observation of\nequal Fe and Gd linewidths, within the experimental accuracy of the results reported\nin Fig. 6 (b), implies that the relaxation mechanisms of the Fe and Gd lattice can be\ndescribed by a common e\u000bective damping parameter, as also predicted by theory [4].\nEven though \u001b, and therefore \u0001 Mz, cannot be uniquely related to precessing\nmagnetic moments in the uniform mode due to the presence of nonlinear excitations,\nit is interesting to de\fne an e\u000bective precession angle related to \u0001 Mz=Mmeasured by\nXFMR. In doing so, one must take into account that \u001bis a photon energy-dependent\nparameter. In other words, considering that XAS involves 2 p!3d(3d!4f)\ntransitions for the Fe L2;3(GdM4;5) edges,\u001bdepends on the precession of both spin\nand orbital magnetic components of the d- (f-) projected density of states probed by\nphotons of energy E. This point has been discussed in detail by Goulon et al. inElement-resolved XFMR 11\nRef. [42], who have shown that the precession angles of the spin and orbital magnetic\ncomponents may be derived by combining \u001bL2and\u001bL3measurements and applying the\ndi\u000berential form of the XMCD sum rules. By assuming spin-only magnetic moments,\nthe relationship between \u001band the e\u000bective precession angle becomes extremely simple,\n\u001b= (1\u0000cos\u0012eff), yielding \u0012eff(Fe) = 3:6\u000e\u00060:2\u000eand\u0012eff(Gd) = 3:4\u000e\u00060:2\u000efor the\nmeasurements reported above. Even if the orbital magnetization of Gd and trivalent\nFe ions is usually very small, the extent to which orbital precession contributes to \u001b,\nin particular for Fe, remains to be determined. This matter touches on the interesting\nquestion of separately measuring the spin and orbital moment precession angles, which\nrequires either a comparison between Kedge andL2;3edges measurements recorded\nusing identical experimental conditions [42] or full XMFR E-scans over the entire L2;3\nregion. The latter possibility is further discussed in Sect. 5.\n4. Element-resolved XFMR spectra of metallic bilayers\nWe consider now the extension of XFMR to thin metallic \flms, and show that\nlayer-speci\fc magnetic resonance spectra of multilayer magnetic structures can be\nseparately resolved. This is of interest, e.g., to investigate interlayer coupling e\u000bects,\ndistinguish superposed spectra of layers with similar resonance \felds, and investigate\ncurrent induced precessional dynamics in spin-torque devices. Here we study a\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer, where the thickness of the\ntwo magnetic \flms was adjusted so as to reduce Brof Ni 80Fe20to within range of our\nelectromagnet for perpendicular FMR.\nFigure 7 (a) shows the inductive FMR of the magnetic bilayer, where two resonances\nare observed at 530 and 740 mT. These are close but not equal to the resonances\nof individual Ni and Ni 80Fe20\flms, respectively, that were prepared with the same\nprocedure. The high \feld resonance peak, in particular, appears to be shifted by an\namount \u0001B=\u0000170 mT with respect to the resonance of an individual Ni 80Fe20layer,\nwhich is indicative of ferromagnetic exchange coupling at the Ni - Ni 80Fe20interface.\nThe elemental components of the two resonance peaks are straightforwardly resolved by\nXFMR, as shown in Fig. 7 (b). We observe that the low-\feld resonance originates from\nthe Ni layer alone, while the high-\feld one comprises both Ni and Fe components. In\nthe high-\feld resonance, the scaled Ni and Fe XFMR intensities coincide, implying\na common g-value and relaxation channel for the two elements, as expected for a\nferromagnetic alloy such as Ni 80Fe20[7]. We therefore conclude that, despite the\npresence of exchange coupling at the interface, mutual resonance-driving e\u000bects between\nperpendicularly-magnetized Ni and Ni 80Fe20layers are not signi\fcant. This result can be\nrationalized within the theoretical model developed by Cochran et al. for a thin overlayer\ncoupled to a thick magnetic substrate [43]. The model assumes that two ferromagnetic\nlayersAandBdeposited on top of each other are exchange coupled at their interface by\na surface energy per unit area of the form Eexc=\u0000JMA\u0001MB, whereJis the interface\ncoupling constant [44, 45]. In the two extreme limits of strong and zero coupling, theElement-resolved XFMR 12\nFigure 7. (a) FMR of Ni 80Fe20(5 nm)/Ni(50 nm) measured simultaneoulsy with (b)\nL2XFMR spectra of Fe and Ni at E= 722:2 and 871.7 eV, respectively. The MW\npower is 34 dBm.\nmagnetizations of the two layers precess locked together or independently of each other,\nrespectively. For small but \fnite J, mutual driving terms in the equations of motion\nbecome unimportant, with the overlayer responding to the driving MW radiation as if\nit were an isolated \flm subject to an e\u000bective anisotropy \feld of magnitude JMB=tA,\nwheretAdenotes the overlayer thickness and MBthe thick \flm magnetization [43]. This\nbehavior corresponds to the data reported in Fig. 7. From the shift \u0001 Bwe estimate\nJ= 2:1\u000210\u000015Vs/A andEexc\u00196\u000210\u00004J/m2. According to theory [43, 45], also\nthe resonance position of the thicker Ni layer should be down-shifted in the presence of\nferromagnetic interface coupling, namely by the amount JMA=tB. Indeed, with respect\nto a single 50 nm thick Ni layer in a Al(10 nm)/Ni(50 nm)/Cr(5 nm) stack, a shift\n\u0001B=\u000030 mT is observed, which yields J= 1:9\u000210\u000015Vs/A, consistently with the\nvalue reported above.\nCompared to the exchange energy of ferromagnetic metals, Eexcestimated from the\nresonance shifts turns out to be rather small for metallic \flms in direct contact with\neach other. Although this explains the absence of Ni 80Fe20(Ni) response upon excitation\nof the Ni (Ni 80Fe20) resonance, its origin could not be uniquely determined during the\npresent study. The magnitude of Eexcis known to be extremely sensitive to the quality of\nthe interface between magnetic materials. Roughness, as well as adsorption of impurities\nsigni\fcantly diminish the coupling strength. In high vacuum, the few seconds intervened\nbetween evaporation of the Ni and Ni 80Fe20\flms are su\u000ecient to deposit a monolayer-\nlike quantity of contaminants, which may strongly decrease the magnetization of the\ninterface metal layers. In vacuum conditions similar to ours, Ho\u000bmann et al. found\nEexc= 1:2\u000210\u00003J/m2for a double Ni/Ni 80Fe20/Ni interface [44]. Fully oxidizedElement-resolved XFMR 13\nFigure 8. Static Fe XMCD (solid line) of Ni 80Fe20(5 nm) and Fe XFMR E-scan\nmeasured at B0= 0:74 T (squares) and 0.70 T (dashed line). The MW power is\n34 dBm.\nNiO/Ni 80Fe20interfaces, on the other hand, have interfacial coupling energies as small\nas 2\u000210\u00005J/m2[46].\nFinally, we note that the smallest XFMR cross-section measured for Ni 80Fe20(5 nm)\ncorresponds to \u001bFe= 5\u000210\u00004, representing a very remarkable dichroism sensitivity in\nthe soft x-ray range, still susceptible of further improvements.\n5. Dynamic XMCD spectra\nSo far we have dealt with the information contained in XFMR B-scans. One of the main\npoints of XFMR, however, is that the measured intensity contains all the information\nderived from the x-ray absorption process, in particular that related to the unoccupied\n\fnal density of states of a given chemical species together with its spin and orbital\nmagnetization components. In other words, two powerful spectroscopical methods, x-\nray absorption and magnetic resonance, are combined together in XFMR. Here we show\nhow the information related to the electronic state of the atoms whose magnetization is\nprecessing can be practically retrieved by XFMR E-scans, i.e., by recording the XFMR\nintensity as a function of photon energy at B0=Br. Figure 8 shows the XFMR\nenergy-dependent intensity of Fe in the Ni 80Fe20layer measured on- and o\u000b-resonance,\ncompared with the static XMCD signal measured at the same \feld value. One can\nsee that, while the on-resonance XFMR displays a strong energy dependent intensity,\nthe XFMR measured o\u000b-resonance is zero within the noise, emphasizing the dynamic\norigin of the XFMR E-scan. Indeed, the latter can be considered as a dynamic XMCD\nspectrum, where the probed magnetization corresponds to that resonantly excited by\nthe MW \feld into uniform precession or other resonant modes selected by the choice of\nB0. Here, although the signal-to-noise ratio needs to be improved to reach quantitative\nconclusions, the overall similarity between the static and dynamic XMCD lineshape\nsuggests a similar orbital-to-spin ratio for the static and precessing magnetic moments\nof Fe.\nThis method eliminates the need to resort to the di\u000berential form of the XMCD\nsum rules to extract information on the precession dynamics of the spin and orbitalElement-resolved XFMR 14\nmagnetization components of the d-density of states introduced in Ref. [42]. By\nintegrating XFMR E-scans and XMCD spectra simultaneously measured, the standard\nXMCD sum rules [47, 48] can be applied, deriving information on the dynamic vs. static\ntotal orbital and spin magnetic moments. Assumptions made in applying the XMCD\nsum rules regarding integration cut o\u000bs, magnitude of the spin dipole moment, and\nisotropic absorption intensity [47, 48, 49] shall hold equally well (or badly) for XFMR\nE-scans and XMCD spectra, thus making their relative comparison most relevant. Two\ncaveats should be mentioned concerning this type of measurements. The \frst is the\nquantitative accuracy of the XMCD sum rules for soft x-ray absorption spectra measured\nin the FY mode, as discussed, e.g., in Ref. [50]. The second is the presence of strong self-\nabsorption e\u000bects for thick \flms and bulk samples, which alter the measured intensity\nof the most prominent XAS and XMCD features. Di\u000berent methods may be used to\nretrieve the true XAS absorption coe\u000ecients from FY data [51, 52]; a relative, qualitative\ncomparison of static and dynamic XMCD measurements is nonetheless always possible\nsince self-absorption a\u000bects them in the same way. Moreover, such e\u000bects may be\nneglected in ultrathin \flms and dilute samples, and entirely bypassed by measuring\nXFMR in a transmission geometry, with a signi\fcant additional gain of XAS intensity.\nRecently, XAS and XMCD spectra have been measured also by time-resolved pump-\nprobe methods, addressing the transfer of angular momentum from the spin and orbital\nmagnetic moments to the lattice in Fe/Gd multilayers [53] and polycrystalline Ni \flms\n[54]. Ultrafast heat transients produced by fs-laser pulses are used to pump electronic\nexcitations, inducing strong demagnetization e\u000bects and consequent transfer of angular\nmomentum from the magnetic system to the lattice. XMCD spectra recorded at \fxed\ndelay times allow to monitor the spin and orbital magnetic moments during this process.\nTime resolution is achieved either by temporally dispersing the intensity of x-ray photon\nbunches transmitted by the sample using a streak camera [53] or by employing fs x-\nray probe pulses produced by femtoslicing techniques [54], achieving resolutions of the\norder of 2 ps and 100 fs, respectively. \"Slower\" time-resolved schemes based on pulsed\nmagnetic \felds [7, 13] or continuous wave excitations [14, 15] as pump and x-ray photon\nbunches of\u001850\u0000100 ps duration as probe may also be employed to measure full XMCD\nspectra, although this, to our knowledge, has not yet been reported. With respect to\ntime-resolved methods, XFMR E-scans appear particularly suited to study stationary\nprecessional dynamics. The averaging time required to measure the Fe spectrum in\nFig. 8 amounts to about 1 hour. Improving the detection e\u000eciency using transmission\nrather than FY is expected to reduce this time further while leading to a better XFMR\nsignal-to-noise.\n6. Conclusions\nIn summary, we have shown that time-invariant x-ray magnetic dichroism and magnetic\nresonance spectroscopy at GHz frequency can be combined to yield element-resolved\nmagnetic resonance spectra as well as dynamic XMCD spectra, depending on whetherElement-resolved XFMR 15\nthe photon energy is kept constant while the applied magnetic \feld is varied or\nviceversa. We reported two case studies concerning a Gd 1Y2Fe5O12garnet and an\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) metallic \flm. Antiferromagnetic\ncoupling at resonance between Fe and Gd sublattices in Gd:YIG has been resolved\nand shown to hold also in the nonlinear regime where the FMR response is heavily\nsaturated. The Fe and Gd XFMR linewidths coincide to within the experimental\naccuracy, supporting the notion of a common e\u000bective damping parameter for the two\nsublattices introduced in early theoretical treatments of ferrimagnetic resonance [4].\nThe Ni 80Fe20(5 nm)/Ni(50 nm) bilayer presents two resonance modes whose elemental\ncomponents have been separately identi\fed by XFMR. It was shown that while one\nlayer is excited the other is at rest, i.e., that interlayer driving e\u000bects are negligible\nfor moderate values of the interface exchange energy, as predicted by theory [43].\nFinally, the comparison between static and dynamic Fe XMCD lineshape in Ni 80Fe20\nsuggests a constant orbital-to-spin magnetic moment ratio for the steady and precessing\nmagnetization.\n7. Acknowledgments\nWe acknowledge the European Synchrotron Radiation Facility and Swiss Light Source,\nPaul Scherrer Institut, for provision of beamtime.\nReferences\n[1] Hillebrands B and Ounadjela K 2002 and 2003 Spin Dynamics in Con\fned Magnetic Structures I\nand II (Topics Appl. Phys. vol 83 and 87) (Springer)\n[2] Kiselev S I, Sankey J C, Krivotorov I N, Emley N C, Schoelkopf R J, Buhrman R A and Ralph\nD C 2003 Nature 425380\n[3] Wangsness R K 1953 Phys. Rev. 911085\n[4] Wangsness R K 1958 Phys. Rev. 111813\n[5] Kittel C 1959 Phys. Rev. 1151587\n[6] de Gennes P G, Kittel C and Portis A M 1959 Phys. Rev. 116323\n[7] Bailey W E, Cheng L, Keavney D J, Kao C C, Vescovo E and Arena D A 2004 Phys. Rev. B 70\n172403\n[8] Vogel J, Kuch W, Camarero J, Fukumoto K, Pennec Y, Pizzini S, Bon\fm M, Petro\u000b F, Fontaine\nA and Kirschner J 2005 Phys. Rev. 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Lett. 771508\n[51] Eisebitt S, B oske T, Rubensson J E and Eberhardt W 1993 Phys. Rev. B 4714103\n[52] Carboni R, Giovannini S, Antonioli G and Boscherini F 2005 Physica Scripta T115 986\n[53] Bartelt A F, Comin A, Feng J, Nasiatka J R, Einm uller T, Ludescher B, Sch utz G, Padmore H A,\nYoung A T and Scholl A 2007 Appl. Phys. Lett. 90162503\n[54] Stamm C, Kachel T, Pontius N, Mitzner R, Quast T, Holldack K, Khan S, Lupulescu C, Aziz E F,\nWietstruk M, D urr H A and Eberhardt W 2007 Nat. Mater. 6740"    },    {        "title": "1607.08689v1.A_rock_salt_type_Li_based_oxide__Li3Ni2RuO6__exhibiting_a_chaotic_ferrimagnetism_with_cluster_spin_glass_dynamics_and_thermally_frozen_charge_carriers.pdf",        "content": "1 \n (Accepted for publication in Scientific Reports)  \n \nA rock -salt-type Li-based oxide, Li 3Ni2RuO 6, exhibiting a chaotic ferrimagnetism with \ncluster spin-glass  dynamics  and  thermally frozen charge carriers  \n \nSanjay Kumar Upad hyay,1 Kartik K Iyer,1 S. Rayaprol2,  P.L. Paulose ,1  and E.V. \nSampathkumaran1,* \n1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India  \n2UGC -DAE Consortium for Scientific Research, Mumbai Centre, R -5 Shed, BARC Campus, Trombay, \nMumbai – 400085, India  \n \n*Corresponding author: sampath@mailhost.tifr.res.in  \nThe area of research  to discover new Li containing materials  and to understand their physical \nproperties has been of constant interest  due to applications potential for rechargeable batteries.  \nHere, we present the results of   magnetic investigations on a  Li compound, Li 3Ni2RuO 6, which was \nbelieved to be a ferrimagnet  below 80 K. While our neutron diffraction (ND) and isothermal \nmagnetization (M) data support ferrimagnetism, more detailed magnetic studies establish that this \nferri magnetic phase exhibits  some f eatures  similar to spin -glasses .  In addition,  we find another  broad  \nmagnetic anomaly  around 40-55 K  in magnetic susceptibility  (χ), attributable to  cluster  spin-glass  \nphenomenon . Gradual  dominance of cluster   spin-glass dynamics with  a decrease of temperature ( T) \nand the apparent spread  in  freezing temperature  suggest that the ferrimagnetism of this compound \nis a chaotic one.  The absence of a unique freezing temperature for a crystalline material is interesting. \nIn addition, p yroelectric current   (Ipyro)    data reveal s a featu re in the range 40 -50 K ,  attributable  \nto thermally s timulated depolarization current . We hope this finding motivates future work to \nexplore whether  there is any intriguing correlation of such a feature with  cluster  spin-glass \ndynamics . We attribute these magnetic and electric dipole anomalies t o the crystallographic disorder, \nintrinsic to this compound.  \n \nThe phenomenon of spin -glass ordering in which the magnetic moments are randomly frozen as the \ntemperature is lowered below   a characteristic temperature  (Tg) discovered  several decades ago  for magnetic \nimpurities in non -magnetic matrices , is commonly observed in many concentrated magnetic systems  as \nwell1,2. Such a type of magnetic ordering in compounds is usuall y facilitated b y crystallographic order and \ncan also be triggered by geometrical frustration [see, for instance, Refs. 3-5]. Some materials exhibit what \nhas been known as ‘re -entrant spin -glass behavior6,7; in such  materials ,  the one occurring at a higher \ntemperature is of  a ferro /antiferromagnetic type, which can enter into a spin -glass reg ime with a lowering \nof temperature  with a unique freezing temperature . A few ferromagnetic materials exhibiting spin -glass \ncharacteristics have also been labelle d ‘chaotic’ magnetic systems7.  Evidences for  multiple spin -glass \ntransitions are generally scarce  barring some exceptions8-10, and in some systems of this kind9,10, \nferromagnetic cl usters behave like spin -glasses . In this article, we provide evidence for an interesting \nsituation in which one sees a gradual  dominance of cluster  spin-glass features, as though there is no unique  \nfreezing temperature, with a  decrease of  temperature , for a crystalline  material, viz., Li 3Ni2RuO 6, which \nwas believed  to be a ferrimagnet  (TC  =80 K) ,  [Ref. 11 ]. Our results thus reveal that  the ferrimagnetis m \nof this compound is not that simple . This conclusion is based on  viewing together  the results of ac and dc \nmagnetization  and heat -capacity ( C) as well as neutron diffraction  studies . Interestingly,  pyroelectric \ncurrent reported here   also exhibits an anomaly, which appears to arise from thermally frozen -in electric \ndipoles12-14, in the same T-range in which spin -glass -like features appear . But it is not clear to us at present 2 \n whether these  electric dipole and magnetic phenomena are coupled.    It may be stated that, following Ref. \n11, this compound was not paid much attention in the literature.  \nThe  monoclinic  crystal structure (space group, C2/c)    in which the compound forms  is related to that \nof Li 2TiO 3-type rock-salt structure15.  In this structure, three distinct positions  for Li [Li1 ( 8f), Li2 ( 4d), \nand Li3 ( 4e)] and two different ( 4e) positions for Ti are possible. In the compound under investigation, it \nwas found that Li1 and Li2 positions are occupied by Li and Li3 is o ccupied by Ni. Ti1 site is  occupied by \nNi and almost all Ti2 site is occupied by Ru. A fraction (<10%) of Ru and Ni go to Li1 and Li2  sites \nrespectively and Li in turn occupies majority Ni and Ru sites. If one ignores this disorder, the structure \nessentially consists of LiO 6 octahedra alternating with (Ni 2/3Ru1/3)O6 octahedral planes, running along c-\ndirection. It is however clear from the above discussions that   there is a significant crystallographic  disorder \nin this material.  A  view of the crystal structure of this compound, however ignoring disorder, along a -axis \nis shown  in Supplementary  Information  (see Supplementary Fig. S1 online) .  \n \nResults  \nDc magnetization  \nThe results of dc magnetic susceptibility as a function of T obtained in a magnetic field of (H=) 5 kOe \nare shown  in Fig. 1 a. There is a gradual  increase of χ with decreasing T below 300  K, which is cut off by \nan upturn below 100 K , which becomes sharper below about 80 K,  due to the onset of magnetic ordering , \nfollowed by a peak  around 50 K and finally a fall. Inverse χ exhibits a linear region in a narrow temperature \ninterval ( 225-300 K), below which there is a deviation from this high -temperature Curie -Weiss behavior, \nattributable to short -range magnetic correlations; the value of the effective moment obtained from the linear \nregion is about 5.3  μB per formula unit which is very close to that expected (5.57 μ B)  for high -spin divalent \nNi (S= 1) and pentavalent Ru (S= 3/2). The value of the paramagnetic Curie -temperature is found to be  \nabout -305 K.  These findings are in agreement with those reported by Laha et al11 by measurements with 1  \nkOe. However,  a further study with low -fields ( e.g., H= 100 Oe,  see Fig.  1b), presented here, for zero-\nfield-cooled ( ZFC) and field -cooled (FC) conditions of specimen during measurements offer s an  insight .   \nWhile ZFC curve qualit atively resembles that obtained with 5 kOe, the FC curve deviates from this curve \nbelow about 100 K ,  without any downfall  even in ZFC curve. χ(FC) continues to increase  with a tendency \nto flatten , only  below about 30 K , but not at the onset of magnetic ordering . There is a weak peak at about \n50 K  in FC curve , coinciding with  the peak temperatu re in ZFC curve . It should also be noted that there is \na shoulder near 30 K (where FC curve flattens)  even in ZFC curve; the ZFC curve additionally shows a \nshould er near 80 K, which apparently gets smeared in the FC curve due to the steeper variation  in this \ntemperature range.  While irreversibility in ZFC -FC curves is a s ignature of spin -glass freezing, the \n“delayed” flattening of the FC curve  with multiple featu res as described above  already signals complex \nnature of  magnetic ordering . \nWe have also me asured hysteresis loops at low fields (Fig. 1 c) and isothermal magn etization up to 140 \nkOe  (Fig. 1 d) at selected temperatures. M(H)  plots  continue to  increase   without any evidence for saturation \nand thus the ferromagnetic state at high fields could not be obtained.  This finding emphasizes that the \nmagnetic ordering can not be of a ferromagnetic -type. The hysteresis loops  at 30 and 60 K to show  a weak \nhysteres is, which is a characteristic feature of spin -glasses  and ferrimagnetism .  \n \nNeutron diffraction  \nCrystallographic Structure : We have analyzed the crystallographic structure of this compound at \nroom temperature with  the ND pattern recorded at 300K. The ND pattern  was refined using a structural \nmodel given by L aha et  al11. The observed  pattern fits very well to this model . The occupancies for each \nsite obtained from  Rietveld refinement   is as follows: The Wyckoff site 8f is occupied by Li1 and Ru1 in \nthe ratio 92.5:7.5. There are three 4e sites, two of which are occupied by Li/Ni and Ni respectively, and the \nthird is occupied by Li/Ru. The first 4e site is occupied by Li3 and Ni1 in the ratio 87:13, and the second \n4e site is fully occupied by Ni2. The third  4e site is occupied by Ru:Li in 85:15 ratio. The 4d site is occupied \nby Li:Ni3 in the ratio 87:13. All the oxygen positions are fully occupied.   In general, good fits were obtained 3 \n between calculated and observed ND patterns recorded at different tempera tures (300K, 150K, 100K, 65K, \n30K, 10K, and 3K).  \nMagnetic features:  In Fig. 2 we have shown the ND data along with Rietveld refinement profile for \nthree selected temperatures, 3, 65 and 150K. The first observation one can make here is that, with decreasin g \ntemperature, there is no additional or un -indexed Bragg peak. There is a distinct  increase in peak int ensity \non entering magnetically ordered state  (see Supplementary Fig. S2  online ). Fig. 3  shows the variation of \ncell parameters  (including the monoclin ic angle  β (in degree )). A decrease in temperature decreases the \noverall unit cell volume.  \nSince the neutron diffraction patterns measured down to 3K  do not show  additional magnetic \nBragg peak, the magnetic ordering in this compound could be assumed to be ferrimagnetic with the \npropagation vector, k = (0 0 0), also taking note of the fact that  isothermal magnetization at high fields is \nnot ferromagnetic -like (that is, absence of saturation)  as mentioned earlier . Magnetic moments were refined \nfor temperatures well below TC only.   Using BasIreps progr am of the Fullprof suite16,17, irreducible \nrepresentations and basis vectors were obtained for all the magnetic ions at different crystallographic sites. \nFor each magnetic ion, the 1 representation was sufficient to correctly represent the magnetic structure \nwith reasonable values of the magnetic mo ment. As per the cationic distribution, Ru1 is found at the site 8f \n(shared with Li1). However, Ru1 does not seem to possess a magnetic moment  and hence was not \nconsidered for final refinement. Starting from the ND data measured at 3K, the magnetic moment s were \nrefined independently for Ni3 at site 4d, Ni1 and Ni2 at 4e sites and Ru2 at site 4e. This arrangement clearly \nshows that all moments at site 4e lie on the same plane and the moment on site 4d  lie above and below this \nlayer. The coupling of magneti c moments between sites 4d and 4e is anti -parallel, thereby giving rise to \nferrimagne tic structure as shown in Fig. 4 . The refined magnetic moments for  Ni and Ru at different sites \nare tabulated in Table 1  and shown pictorially in Fig. 5 . It can be clearly  seen that among the magnetic ions, \nNi3 (at 4d site) exhibits negative moments, indicating that these moments are  anti -parallel to the rest of the \nmagnetic ions (at site 4e) as clear ly seen in the Fig. 5 . It therefore appears that it is this antisite Ni w hich \nresults in net ferrimagne tism.  We believe that non -monotonic variation of magnetic moments with \ntemperature could be genuine, considering complex features in  the temperature dependence of dc magnetic \nsusceptibility.   \nAc  magnetic susceptibility  \nFig. 6a show s real (χ) and imaginary (χ ) parts of ac χ. It is obvious that, following the upturn below \n100 K with lowering temperature,  there is a peak in both these parts at 82 K for the frequency ( ν) = 1 Hz \nand this peak shifts towards  higher T range wi th increasing ν, for instance by 2 K for 1333 Hz; apart from \nthis, the intensity of the peak also decreases w ith increasing ν. χ also exhibits a ν -depe ndent peak near 80 \nK, which is a characteristic feature of  spin -glass freezing2. This implies that the ferrimagnetism  could be a \nchaotic one, as proposed for a nother  re-entrant ferromagnet long ago7.  With a further lowering of \ntemp erature, a broad peak appears around  53 K in χ for ν= 1.3 Hz,  which varies with  frequency, with  this \npeak -temperature increasing by about 1 K for 1333 Hz.  A careful look at the left side of this  χ  peak \nsuggest s a weak change of slope near 40 K, as though there is a superposition o f at least two peaks below \n70 K, as though there is more than one characterist ic freezing  temperature.  In fact, this is more clearly \nreflected in  χ (T), which peak s near 40 K for ν= 1.3 Hz. This peak  shows an apparent upward shift by a \nfew degrees when measured with 1333 Hz. We also measured ac χ in the presence of a dc magnetic field \n(5 kOe) and the above -described features are completely suppressed  with a dramatic reduction in the values  \nand with overlapping -curves for different ν  (see Fig. 6 a). This is a key support for  spin-glass -like dynamics . \n \nHeat -capacity  \nIn the inset of Fig.  6c, we show the plot of C(T) below  100 K and there is no evidence for any feature \ndown to 1.8 K that can be attributed to long range magnetic ordering.   The absence of a feature at the onset \nof magnetic ordering (near 80 K) may be either due to the fact that rapidly varying large lattice contribution 4 \n around this temperature obscures the expected λ -anomaly.  Crystallographic disorder also can contribute to \nsmearing the feature in the entire temperature range.   Therefore, it is difficult to delineate the contributions \ndue to  magnetic frustration, though this phenomenon also  must play a role for the lack of C(T)anomaly.  \n \nIsothermal remnant magnetization  (MIRM) \nWe measured MIRM at three se lected temperatures, 1.8, 30,  65 and 125 K. The specimen was zero -field-\ncooled to desired temperature, and then a field of 5 kOe was applied. After waiting for some time, the field \nwas switched off, and then MIRM was measu red as a function of time ( t). We find that MIRM drops to \nnegligibly  small values within seconds of reducing the field to zero  at 125 K ; however, it decays slowly \nwith t at other te mperatures, as shown in Fig. 6b. These offer support to spin -glass dynamics. The curves \ncould be fitted to a stretch ed exponential form of the type18 MIRM (t) = M IRM(0)[1+Aexp( -t/τ)1-n), where A \nand n are constants and τ here is the relaxation time.  It is found that the value of n falls in the range 0.5 – \n0.7. The values of relaxation times  are rather large (e.g., 100 mins at 2 K and about 28 mins for 30 and 65 \nK). These values are in fact in agreement with th at reported for cluster spin-glasses18.   \n \nWaiting time dependence of dc magnetization  \nWe looked for aging effects   [see, for instance, Ref. 19 ] in dc magnetization at two temperatures \n(30 K and 65 K) characterizing spin -glass phase. For this purpose we have followed ZFC and FC protocols \nas des cribed, for instance, in Ref. 20 . In ZFC proto col, we cooled the sample to the desired temperature, \nwaited for certain period of time, switched on a dc field of 100 Oe and measured the increase of M as a \nfunction of time. In FC protocol, the specimen was cooled in 100 Oe , and after waiting  for certain period, \nthe decay of M was measured as a function of time  after the field was switched off. Th e curves thus obtained \nfor two  waiting times are s hown in Fig.  7. It is obvious from this figure that the curves for 3000 s are \ndisplace d with respe ct to those for a lower waiting time of 300 s. This is very distinct at 30 K for both ZFC \nand FC protocols, establishing spin -glass -like spin dynamics at such low temperatures. For 65 K,  this \ndisplacement of curves is visible for ZFC protocol, but it  is feeble for FC protocol, as though spin -glass -\nlike behavior tends to weaken with increasing temperature in the magnetically ordered state. Clearly aging \nphenomenon is present in this compound  and demonstrates gradual nature of the variations in spin -glass \nfreezing with changing temperature . \n \n‘Memory effect’ in dc magnetization  \nIn order to look for memory ef fect, we obtained χ(T) curves in different ways. In addition to ZFC curve \nin the presence of 100 Oe  without a long wait at any temperature (which is a reference curve),  we have \nobtained a ZFC curve after waiting at two temperatures 25 and 60 K for 3 hours each (and also for 6 hours \neach in another independent experiment). We obtained the difference bet ween these two curve s and plotted \nthe same as  ΔM versus T in Fig.  6c. It is distinctly clear that there are clear ‘dips’ at these two temperatures \nin this plot. It is found that the intensity of the ‘dip’ is increase d for a wait of 6 hours, with respect t o tha t \nfor 3 hours. This is a signature of frustrated magn etic behavior, even in the ferri magnetic phase (just below \n80 K), as discussed for assemblies of nanoparticles with ferromagnetic core  and antiferromagnetic shell21.   \n \nComplex permittivity and pyroelectric current behavior  \nComplex permittivity  and  pyroelectric current studies also reveal  interesting behavior  well below TC. \nDielectric constants ( ɛ) and the loss factor (tan ) are shown in Fig.  8a below 100 K  for  two selected \nfrequencies (1 and 100 kHz). Beyond 100 K, tan  increases dramatically and therefore extrinsic \ncontributions tend to dom inate . The observation we would like to stress is that both ɛ and tan undergo a \ngradual increase with T from 1.8 K, with out any apparent peak or any other feature. Therefore, we rule out \nthe presence of any ferro electricity below 150 K in this compound. We have also measured \nmagnetocapacitance  at various temperatures and the changes observed in ɛ for H= 140 kOe are 0.01 a nd \n0.1%  at 2 and 25 K respectively . Therefore , magnetodi electric coupling is rather weak .  5 \n However, Ipyro as a function of T exhibits a distinct feature (measured with two poling electric fields \n100 and 200  V corresponding to 2.08 kV/cm and 4.16 kV/cm for the sample used ). That is, the plot (Fig.  \n8b) shows a peak at about 40 K for a rate of warming of temperature (dT/dt ) of 2K /min for the poling by -\n2.08 kV/cm at 100  K. The peak gets reversed in sign when pole d by +100 V. The intensity of the peak \nincreases  for 200 V, as shown in Fig.  8b. This finding mimics that expected for ferroelectricity. However, \nsince we do not find any anomaly in ɛ(T) (Fig.  8a), these peaks can n ot be attributed  to ferroelectricity . To \ngather further support for this conclusion, we have performed pyroelectric current measurements  for \ndifferent dT/dt . The results (see  Fig. 8c) reveal that the peak in fact shifts to  higher temperatures with \nincreasing  dT/dt , for instance, to ~43 K an d ~46  K for 5 K/min and 8K/min respectively. Such a str ong \nvariation is not expected13 for ferroelectric transitions . We have also obtained the behavior of Ipyro in the \npresence of a dc magnetic field of 10 kOe and we find (se e Fig. 8 b) that the intensity of the peak in the plot \nis dra matically suppressed, as in the case of ac χ.  \n \nDiscussion  \nFrom the results presented above, it is clear that there appears to be a contradiction between  the \nconclusion from  neutron dif fraction results (suggesting well -defined  magnetic structure, say, \nferrimagnetism ) and that from other bulk measurements  (in particular,  frequency dependence in  ac \nsusceptibility, and aging,  memory and ZFC -FC curves bifurcation behavior in dc magnetization, revealing \nspin-glass features ). Clear ly, the magnetism of this compound is very complex. The fact that the peaks in \nχ in the range 40 to 60 K are not cusp -like suggests that there is a spread in the freezing temperatures in \nthis T-range.   This spread is consistent with various features noted aro und 30 - 50 K and 75 K in Fig.  1b. \nFor this reason, it is tempting to  claim that this compound could be one of the rare examples for multiple \nspin-glass freezing phenomenon .  It is not clear whether a relaxation phenomenon of ferrimagnetic structu re \nis operative.  \n  In order to understand the nature of magnetism better, w e have  analyzed the ac χ results in terms of \nthe conventional power law, associated with the critical slowdown of relaxation time, τ/τ0 = (Tf/Tg − 1)−zν. \nHere, τ  represents the observation time (1/2πν), τ0 is the microscopic relaxation time, Tg is the spin -glass \ntransition temperature, T f corresponds to freezing temperature for a given observation time and zv is the \ncritical exponent.  For the feature around 40 K, we  obtained  Tg ≈28 K, zν ≈ 7.12 and τ0 ≈ 1.8×10−4 s. For \nthe one around 80 K, corresponding values are: ~80 K, ~2.6 and ~1.6 ×10−7 s.  For  a conventional spin \nglasses2, the zν value falls in the range ~4 -13, and τ 0 value ranges between 10−10 and 10−13 s.  It is clear that \nthe values of τ 0 obtained are in general higher than that for the conventional spin glasses. But the deviation \nis highly pronounced for the feature around  20-40 K. Judged by t hese values, one can interpret22,23 that the \nferrimagnetic  regions form cluster s  exhibiting spin -glass -like inter-cluster dynamics, with the  cluster -glass \nbehavior gradual ly strengthening  with decreasing  temperature.   The parameters derived from isothermal \nremnant behavior also offer s support for  cluster -glass behavior, as described earlier. Therefore, long-range \nmagnetic ordering  is labelled  ‘chaotic’  in this article.  \nIt is i nteresting to see a feature in pyroelectric current in the glassy magnetic phase. This does not arise \nfrom ferroelectricity , as mentioned earlier.  Therefore, an alternate explanation should be offered for the \nobservation of the peak in Ipyro. At this juncture, it may be recalled that such a dependence of the peak on \ndT/dt  has been explained in terms of ‘thermally stimulated d epolarization current (TSDC)’12,14 in the past \nliterature. This phenomenon can be explained as follows: The mobile charge carriers , presumably  \nintroduced by crystallographic defects due to intrinsic disorder described above, tend to organise \nthemselves to screen the applied electric field, and, with    a lowering of temperature, these charge carriers \nget trapped randomly forming electric dipoles and persist for a very long time after removal of the electric \nfield.  These carriers can be released thermally, which appears as a peak in Ipyro. The type of charge carriers \ntrapped determine the sign of Ipyro with respect to that of the electric field. For instance, positive Ipyro for a \nnegative electric field implies trapping of negative charges, whereas the same sign for both implies holes -\ntrapping. Thus, in DyMnO 3, such a p eak was attributed to holes14, whereas, in the case of yttrium iron \ngarnet13, electrons are responsible. Therefore, in the present case, considering the opposite sign of the \nelectric field and Ipyro, one can confidently state that the negative charges get trapped.    6 \n It is intriguing to note that the T-range over which this phenomenon occurs is essentially the same as \nthat of the spin -glass anomalies in ac χ. The observation that Ipyro feature is suppressed by a dc magnetic \nfield of 10 kOe as in the case of ac χ may signal some connection between these magnetic and elect ric \ndipole phenomena. However, it  is an open question whether this is truly the case or whether it is just \naccidental.  Intrinsic crystallographic disorder must be the root-cause of all these electric and magnetic \ndipole anomalies.  \nIn short, the present results reveal that the new Li -based compound,  Li3Ni2RuO 6, is not a simple \nferrimagnet, but is characterized by growing i nfluence of spin -glass dynamics with decreasing temperature \nin the magnetically ordered state. Due to strong chemical disorder, intrinsic to this compound, it appears \nthat spin -glass clusters form with apparently different freezing temperature s, rather tha n a single freezing \ntemperature. Thus, such multiple cluster -glass freezing temperature  for a crystalline compound is not \ncommonly reported  and we hope this compound would serve as an ideal system to model such a behavior . \nWe find pyroelectric  anomalies attributable to thermally stimulated depolarization mechanism, presumably \ndue to  trapping of negative char ge by crystallographic disorder, though the physical origin for possible \nconnection with cluster -glass dynamics is not obvious at present.   Thus this compound exhibits interesting \nmagnetic and pyroelectric anomalies.  \n \nMethods  \nPolycrystalline sample was prepar ed as described by Laha et al11 by a solid state reaction route. \nRequired amounts of high purity (>99.9%)  starting materials, Li 2CO 3, Ni oxalate (NiC 2O4.2H 2O), and \nRuO 2,  as per the stoichiometry of the compound, were mixed thoroughly, heated at 673 K for 4 h, at 1073 \nK for 12 h, and subsequently at 1198 K for 12 h with intermediate grindings between these three stages of \nheating. X -ray diffraction (XRD) pattern (Cu K α) confirms single phase nature of the compound. The \nbackscattered electron images of scanning electron microscope (SEM) have been obtained to check the \nhomogeneity of the sample. We have also performed energy dispersive  sca nning electron microscopic \nstudies to determine the composition, particularly for Ni and Ru, though it is not easy to obtain precise \ncomposition for low atomic number elements, Li and O, with the sensitivity of the SEM employed.  \nT-dependent dc magnetizati on studies were carried out with the help of a commercial supeconducting \nquantum interference  magnetometer (Quantum Design, USA) and ac χ study with different frequencies (ν= \n1.3, 13, 133, and 1333 Hz) with a ac field of 1 Oe  was also carried out with the same magnetometer. Heat -\ncapacity studies were carried with a commercial Physical Properties Measurement s System (Quantum \nDesign, USA). T he same system was used to measure complex dielectric permittivity using an Agilent \nE4980 A LCR meter with a home -made sample holder with several frequencies (1 kHz to 100 kHz) and \nwith a bias voltage of 5 V; the same sample holder was used for  pyroelectric studies with Keithley 6517B \nelectrometer by poling at 100 K with different electric fie lds. Unlike otherwise stated, all the measurements \nwere performed for zero -field-cooled condition ( ZFC from 300 K) of the specimen.  \nNeutron diffraction measurements were carried out on polycrystalline samples on a focusing crystal \nbased powder diffractome ter (PD -3) at Dhruva reactor, Trombay24. Sample was filled in a vanadium can \nsubjected to temperature variation using a closed cycle refrigerator. Neutrons at a wavelength of 1.48Å \nwere used for the diffraction experiments.  ND patterns were analyzed for n uclear (crystalline) and magnetic \nstructures by the Rietveld refinement method u sing the Fullprof program16,17,24  \n.  \n \n  7 \n Table 1:  The values of magnetic moments of Ni and Ru atoms at 4d and 4e sites at three \ntemperatures are tabulated.  Ni1 and Ni2 are at 4e: (0 y ¼) with different y -positions and \nNi3 is at 4d: (¼ ¼ ½). Ru is also found at site 4e, with different y position. The variation \nin the y -position with respect to temperature is also shown in the table along with the \nmagnetic moments of Ni and Ru at  these different sites.  \nTemperature \n(K) Ni1@4e  Ni2@4e  Ni3@4d  Ru2@4e  \n (y = 0.0913)  (y = 0.4230)  (¼ ¼ ½)  (y = 0.7423)  \n3 0.925  0.849  -0.485  0.601  \n (y = 0.0906)  (y = 0.4243)  (¼ ¼ ½)  (y = 0.7422)  \n10 1.22 1.439  -0.131  0.821  \n (y = 0.0910)  (y = 0.4223)  (¼ ¼ ½) (y = 0.7412)  \n30 1.11 1.22 -0.283  0.653  \n (y = 0.0962)  (y = 0.4199)  (¼ ¼ ½)  (y = 0.7442)  \n65 1.024  1.047  -0.137  1.199  \n \n \n  8 \n   \nReferences  \n1. Binder, K. & Young, A.P. Spin glasses: Experimental facts, theoretical concepts, and open \nquestions. Rev. Mod. Phys.  58, 801-976 (1986).  \n2. Mydosh, J. A. Spin glasses: An experimental introduction (Taylor & Francis, 1993).  \n3. Zvyagin, A. A. New physics in frustrated magnets: Spin ices, m onopoles, etc.  Low Temp. \nPhys.  39, 901-922 (2013).  \n4. Hardy, V. et al.  Temperature and time dependence of the field -driven magnetization steps \nin Ca3Co2O6 single crystals.  Phys. Rev. B  70, 064424 (2004).   \n5. Sampathkumaran, E. V. & Niazi, A. Superparamagnetic -like ac susceptibility behavior in the \npartially disordered antiferromagnetic compound  Ca3CoRhO 6. Phys. Rev. B 65, 180401(R) \n(2002).  \n6. Verbeek, B. H., Nieuwenhuys, G. J., Stocker, H., & Mydosh, J. A. Evidence for a \nFerromagnet —Spin-Glass Transition in  PdFeMn. Phys. Rev. Lett. 40, 586-589 (1978).  \n7. Jonason, K., Mattsson, J. & Nordblad, P. Chaos in t he Ferromagnetic Phase of a Reentrant \nFerromagnet. Phys. Rev. Lett.  77, 2562 (1996).  \n8. Wang, Y. T., Bai, H. Y., Pan, M. X., Zhao, D. Q. & Wang, W. H. Multiple spin -glass -like \nbehaviors in a Pr -based bulk metallic glass. Phys. Rev. B  74, 064422 (2006).  \n9. Mao, J. et al.  Evidence of two distinct dynamical freezing processes in single -layered \nperovskite La 0.7Sr1.3CoO 4. J. Phys.: Condens. Matter 23, 336001 (2011).  \n10. Phan, M.H. et al.  D. Magnetism and cluster -glass dynamics in geometrically frustrated \nLuFe 2O4. J. Appl. Phys.  105, 07E308 (2009).  \n11. Laha, S. et al.  New rock salt -related oxides Li 3M2RuO 6 (M=Co, Ni): Synthesis, structure, \nmagnetism and electrochemistry. J. Solid State Chem . 203, 160-165 (2013).  \n12. Bucci, C., Fieschi, R. & Guide, G. Ionic thermocurrents  in dielectrics . Phys. Rev. B. 148, 816-\n823 (1966).  \n13. Kohara, Y., Yamasaki, Y., Onose, Y. & Tokura, Y. Excess -electron induced polarization and \nmagnetoelectric effect in yttrium iron garnet. Phys. Rev. B  82, 104419 (2010).  \n14. Zou, T. et al.  Excess -hole induced high temperature polarized state and its correlation with \nthe multiferroicity in single crystalline DyMnO 3. App. Phys. Lett.  105, 052906 (2014).  \n15. Yu, C. L. et al. The structure of H 2TiO 3 — a short discussion on “Lithium recovery from salt \nlake brine by H 2TiO 3”. Dalton Trans.  44, 15721 -15724 (2015).   \n16. Carvajal, J. R. Recent advances in magnetic structure determination neutron powder \ndiffraction. Physica B  192, 55-69 (1993).  \n17. Rietveld, H. M. A Profile Refinement Method for Nuclear and Magnetic Structures. J.Appl. \nCryst.  2, 65-71 (1969).  \n18. Xu, Q.  et al.  Magnetic interactions in BiFe 0.5Mn 0.5O3 films and BiFeO 3/BiMnO 3 superlattices.  \nSci. Rep.  5, 9093  (2014).  \n19. Shvartsman, V.V., Bedanta, S. et al.  (Sr,Mn)TiO 3: A Magnetoelectric Multiglass.  Phys. Rev. \nLett. 101, 165704 (2008).  \n20. Bisht, V.  & Rajeev , K. P.  Memory and aging effects in NiO nanoparticles. J. Phys.: Condens. \nMatter  22, 016003 (2010).   \n21. Vasilakaki, M. et al. Memory effects on the magnetic behavior of assemblies of nanoparticles \nwith ferromagnetic core/antiferromagnetic shell morphology. Phys. Rev. B  88, 140402 (R) \n(2013).  \n22. Chakrabarty, T., Mahajan A. V. & Kundu, S., Cluster spin glass behavior in geometrically  \nfrustrated Zn 3V3O8. J. Phys.: Condens. Matter  26, 405601 (2014).  \n23. De, K., Thakur, M., Manna A. & Giri, S. Unusual glassy states in  LaMn 0.5Fe0.5O3: Evidence \nof two distinct dynamical freezing processes.  J. Appl. Phys.  99, 013908 (2006).  9 \n 24. Siruguri, V., Babu,  P. D. , Gupta, M., Pimpale, A. V. & Goyal, P. S. A high resolution powder \ndiffractometer using focusing optics. Pramana  71, 1197 -1202  (2008).  \n \nAuthor contributions  \nS.K.U prepared the sample and characterized the same. He performed ac and dc magnetization \nmeasurements  in association with P.L.P,  carried out heat -capacity and dielectric and electric \npolarization studies along with K.K.I and  analyzed the results. S.R. performed neutron diffraction \nstudies and analyzed this data.   E.V.S. proposed the proble m, formulated the manuscript and finalized \nin consultation with other authors.  \n \nCompeting financial interests  \nThe authors declare no competing financial interests.  \n \n \nFigure 1  | Magnetization data  for Li 3Ni2RuO 6.   Dc magnetic susceptibility as a function of temperature \nmeasured in a magnetic field of (a) 5 kOe, and (b) 100 Oe are plotted in (a) and (b) respectively.  In (a), \ninverse susceptibility is also plotted with a line through the Curie -Weiss region. In (b) the  curves obtained \nfor ZFC and FC conditions are shown  Low-field h ysteresis loops at 30 and 60 K and  isothermal \nmagnetization extended to high fields at 1.8, 30 and 60 K  are also shown in (c) and (d) respectively.  \n \nFigure 2  | Neutron diffraction patterns of Li 3Ni2RuO 6 measured at 150, 65 and 3K.  The data for 3 and \n65K include a fitting for the magnetic structure, as described in the text.  \n \n \nFigure 3  | Temperature dependence of different unit cell parameters obtained from the Rietveld   \nrefinement of neutron  diffraction patterns is plotted.  Lines are drawn through the data points as a guide \nto the eyes.  \n \nFigure 4  | The magnetic structure of Li 3Ni2RuO 6 at 3K.  The arrow in red colour represents magnetic \nmoment of Ni3 (at 4d site) ions, whereas arrows in blue a nd cyan represent magnetic moments of Ni1 and \nNi2 (both at 4e site) respectively. The Ru2 (at 4e site) moment is  shown in green colour.  \n \nFigure 5  | The values magnetic moments at different sites in Li 3Ni2RuO 6 structure are  plotted as a function \nof temperature.  \n \nFigure 6  | (a) Real and imaginary parts of ac susceptibility  measured with various frequencies (1.3, \n13, 133 and 1333 Hz) ,  (b) isothermal remnant magnetization at 1.8, 30 and 65 K, and (c)   the \ndifference in magne tization curves, ΔM, obtained  with and without waiting at 25 and 60 K for \nLi3Ni2RuO 6.  In (a), t he arrows show the direction in which the peaks shift with increasing frequency and \nwith the omission of data points through the lines,  In the inset of (c), he at-capacity as a function of \ntemperature is shown.  The χ  curves in (a) for 133 and 1333 Hz are shifted along y -axis  (by 0.01 and 0.02 \nemu/mol respectively), for the sake of clarity.  \n \nFigure 7  | Dc magnetization as a function of time  (waiting time depen dence or aging experiments) for \nzero-field-cooled (ZFC) and field -cooled (FC) protocols as described in the text for Li 3Ni2RuO 6 for 30 \nand 65 K.  \n \nFigure 8  | Temperature dependence of (a) dielectric constant ( ɛ) and loss factor (tan  ) shown for two \nfrequencies (1 and 100 kHz), (b) pyroelectric current, Ipyro, for two poling fields, obtained by increasing the \ntemperature at the rate of 2K/min, and (c) Ipyro as a function of T for different rates of change of T, after \npoling with 2.08 V/cm, for Li 3Ni2RuO 6. In (b), the curve obtained in a field of 10 kOe is also included.  10 \n  \n \n \n \n11 \n  \n  \n \n \n \n \n \n \n \n12 \n  \n \n \n13 \n   \n \n \n \n \n \n \n14 \n   \n \n \n \n15 \n  \n \n \n16 \n  \n  \n  \n17 \n Supplementary Information  \n \nA rock -salt-type Li-based oxide, Li 3Ni2RuO 6, exhibiting a chaotic ferrimagnetism with \ncluster spin -glass  dynamics  and  thermally frozen charge carriers  \n \nSanjay Kumar Upadhyay,1 Kartik K Iyer,1 S. Rayaprol2,  P.L. Paulose,1  and E.V. \nSampathkumaran1,* \n1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India  \n2UGC -DAE Consortium for Scientific Research, Mumbai Centre, R -5 Shed, BARC Campus, Trombay, \nMumbai – 400085, India  \n \n*Corresponding author: sampath@mailhost.tifr.res.in  \n \n \nHere, we show crystal structure of Li3Ni2RuO 6 along the [100] direction (Fig. S1) and also compare raw \nneutron diffraction data for three different ranges in an expanded sca le (Fig. S2) at three temperatures. In \nFig. S1, we ignored crystallographic disorder discussed in the article, for the sake of simplicity.  \nFig. S2 clearly brings out subtle variation s in the int ensity .  \n \n \nSupplementar y Figure S1: Crystal structure of Li 3Ni2RuO 6 viewed along the [100] dir ection.  \n \n18 \n  \nSupplementary Figure S2:  The raw data of the neutron diffraction patterns recorded at T = 3, 65 and 150K \nare plotted  on the same scale for three different regions.  \n \n"    },    {        "title": "1712.09973v1.Charge_ordering_and_ferrimagnetism_in_the_strongly_correlated__β__V__2_PO__5__single_crystal.pdf",        "content": "Charge ordering and ferrimagnetism in the strongly correlated \f-V2PO 5single crystal\nJie Xing,1Huibo Cao,2Arpita Paul,3Chaowei Hu,4Hsin-Hua Wang,4Yongkang Luo,4\nRaj Chaklashiya,4Jared M. Allred,5Stuart Brown,4Turan Birol,3and Ni Ni1,\u0003\n1Department of Physics and Astronomy and California NanoSystems Institute,\nUniversity of California, Los Angeles, CA 90095, USA\n2Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n3Department of chemical engineering and materials science,University of Minnesota, MN 55455, USA\n4Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA\n5Department of Chemistry and Biochemistry, University of Alabama, Tuscaloosa, AL 35487, USA\nA combined study of transport, thermodynamic, neutron di\u000braction, nuclear magnetic resonance\nmeasurements and \frst principles calculation were performed for \f-V2PO 5single crystal. It was\nshown to be a semiconductor with a band gap of 0.48 eV, undergoing a charge ordering (unusual\nV2+and V3+) phase transition accompanied by a tetragonal to monoclinic structural distortion at\n610 K and a paramagnetic to ferrimagnetic phase transition at 128 K with a propagation vector\nofk= 0. The easy axis is in the monoclinic acplane pointing 47(9)\u000eaway from the monoclinic a\naxis. This collinear ferrimagnetic structure and anisotropic isothermal magnetization measurements\nsuggest weak magnetic anisotropy in this compound. The \frst principles calculations indicate that\nthe intra-chain interactions in the face-sharing VO 6chains dominate the magnetic hamiltonian and\nidentify the \u0000+\n5normal mode of the lattice vibration to be responsible for the charge ordering and\nthus the structural phase transition.\nI. INTRODUCTION\nCharge ordering(CO), the long-range ordering of tran-\nsition metal ions with di\u000berent oxidization states, is a\nprominent feature in mixed valent 3 dtransition metal\noxides [1]. Due to the strong Coulomb interaction in the\ncharge ordering state, a high symmetry to low symmetry\nstructural distortion can occur, accompanied with the\nsudden enhancement in the electrical resistivity arising\nfrom the charge localization. The competition between\nthis charge disproportionation and the exchange inter-\nactions among magnetic transition metal ions has led to\nemergent phenomena, such as colossal magnetoresistance\nin RE 1\u0000xAxMnO 3(RE = rare earth, A = alkaline earth)\n[2, 3], superconductivity in \f-Ag 0:33V2O5[4], etc.\nThe vanadium phosphorus oxide system (V-P-O) has\ndistinct structural stacking and variable valences of\nvanadium, providing a great avenue to investigate the\nstructure-property relationship, enriching our under-\nstanding on the competition of CO and various exchange\ninteraction. The fundamental building blocks of the V-\nP-O system consist of VO 6octahedra or VO 4tetrahedra\nlinked by PO 4tetrahedra with valence P5+. Rich 3d\nvanadium magnetism and valences have been observed.\nFor example, VPO 4with V3+ions, containing one di-\nmensional chains of edge-sharing VO 6octahedra, under-\ngoes an incommensurate antiferromagnetic (AFM) phase\ntransition at 26 K and then a commensurate AFM phase\ntransition at 10.3 K [5]. \u000b-VO(PO 3)2with one dimen-\nsional chains of corner-sharing VO 6octahedra, is AFM\nat 1.9 K with valence V4+[6]. Mixed valence V3+and\nV4+antiferromagnetically couple together below 5 K in\n\u0003Corresponding author: nini@physics.ucla.eduV2(VO)(P 2O7)2, where segments of edge-sharing VO 4\ntetrahedra and VO 6octahedra exist [7]. Alternating V4+\nspin-chain model can be used to describe the magnetism\nin (VO) 2P2O7with corner and edge-sharing VO 6octa-\nhedra ladders [8{12].\nIn this article, we investigated \f-V2PO 5. In the tetrag-\nonal phase in Ref. [13] (Fig. 3(b)), it contains chains of\nface-sharing VO 6octahedra linked by PO 4tetrahedra.\nThese chains are stacked in layers along the caxis, run-\nning alternately along the aorbaxis in the adjacent\nlayers (Fig. 3b). This material is intriguing in three as-\npects. Firstly, the valence analysis with P5+and O2\u0000in-\ndicates remarkably low valence V2:5+in this compound,\nwhich may suggest possible CO of V3+and very uncom-\nmon valence V2+[14]. Secondly, the face-sharing VO 6\noctahedra in the building block is very rare for vana-\ndium oxides, implying unusually strong intra-chain in-\nteraction between V ions. What's more, although the\nparallel chains in each layer do not share any oxygen,\nthe perpendicularly-running chains in the neighboring\nlayers are corner-sharing. As a result, the other im-\nportant magnetic interaction is the inter-chain interac-\ntion between the corner-sharing V ions in neighboring\nlayers. Thirdly, a recent \frst-principles calculation sug-\ngested that the \f-V2PO 5is a ferromagnetic (FM) topo-\nlogical Weyl and node-line semimetal without any trivial\nband at the Fermi level [15], being a great material plat-\nform to study the emergent phenomena in FM topological\nsemimetals.\nDespite of these remarkable aspects we discussed\nabove, neither physical properties nor possible structural\ndistortion has been investigated for \f-V2PO 5, therefore,\nwe performed a combined study of the single crystalline\n\f-V2PO 5by x-ray and neutron di\u000braction as well as\nNMR, transport and thermodynamic measurements. We\ndiscovered that \f-V2PO 5is a semiconductor with a bandarXiv:1712.09973v1  [cond-mat.str-el]  28 Dec 20172\ngap of 0.48 eV. Upon cooling, a charge ordering phase\ntransition accompanied with a tetragonal to monoclinic\nstructural phase transition occurs at 610 K, followed by\na long range ferrimagnetic phase transition below 128 K.\nII. EXPERIMENTAL METHODS\nPrecursor\f-V2PO 5powder was made by solid state\nreaction. V 2O5powder and phosphorus chunks were\nweighed according to the stoichiometric ratio 1 : 1 and\nsealed in a quartz tube under vacuum. The ampule was\nslowly heated up to 600\u000eC and dwelled for 2 hours, and\nthen was increased to 1000\u000eC and stayed for 2 days be-\nfore it was quenched in water. The resultant \f-V2PO 5\n(\u00182 g) powder and iodine \rakes (10 mg / cm3) were\nloaded into a 15-cm long quartz tube and sealed under\nvacuum. Single crystals of \f-V2PO 5were then grown\nby chemical vapor transport method [13]. The hot end\nwas set at 1000\u000eC and the cold end was set at 900\u000eC.\nAfter two weeks, quite a few sizable three dimensional\nsingle crystals (\u00184 mm\u00024mm\u00022mm) were found at the\ncold end. The inset of Fig. 1(a) shows a \f-V2PO 5single\ncrystal against 1 mm scale.\nThroughout the paper, abplane is the plane where\nchains locate in. The (hkl) Tmeans the peak indexed\nin the tetragonal structure while (hkl) Mmeans the peak\nindexed in the monoclinic structure.\nMagnetic properties were measured in a Quantum\nDesign (QD) Magnetic Properties Measurement System\n(MPMS3). A single crystal around 20 mg with a pol-\nishedabsurface and a single crystal with as-grown (011)\nsurface were used. Temperature dependent heat capac-\nity was measured in a QD Dynocool Physical Properties\nMeasurement System (Dynoccol PPMS) using the relax-\nation technique at zero \feld. To enhance the thermal\ncontact and lower the measurement time, the \f-V2PO 5\nsingle crystal was ground into powder and then mixed\nwith silver powder according to the mass ratio of 1 : 1\n. The heat capacity of \f-V2PO 5was then obtained by\nsubtracting the heat capacity of silver [16]. Below 200\nK, the two wire ETO method was used for the electric\nresistivity measurement in PPMS. From 200 K to 400 K,\nthe electric resistivity was measured with standard four-\npoint method while above 400 K, it was measured in a\nhomemade high temperature resistivity probe.\nSingle crystal neutron di\u000braction was performed at\nthe HB-3A four-circle di\u000bractometer equipped with a\n2D detector at the High Flux Isotope Reactor(HFIR) at\nORNL. Neutron wavelength of 1.546 \u0017A was used from a\nbent perfect Si-220 monochromator [17]. The pyrolytic\ngraphite (PG) \flter was used before the sample to re-\nduce the half- \u0015neutrons. Representational analysis with\nSARAh [18] was run to search for the possible magnetic\nsymmetries. The nuclear and magnetic structure re\fne-\nments were carried out with the FullProf Suite[19]. Pow-\nder X-ray di\u000braction measurements were performed using\na PANalytical Empyrean di\u000bractometer (Cu K \u000bradia-tion). Using the Fullprof suit[19], Rietveld re\fnement\nwas carried out to re\fne the powder X-ray di\u000braction\ndata with the crystal structure determined by single crys-\ntal neutron di\u000braction.\nNuclear magnetic resonance (NMR) measurement was\ndone under a \fxed magnetic \feld of approximately 8.5 T,\napplied along the direction perpendicular to the abplane,\nwhere the chains locate in. The spectra were collected\nby performing an optimized \u0019=2-\u001c-\u0019spin-echo pulse se-\nquence. The spin-lattice relaxation time T1was mea-\nsured by integration of the phase corrected real part of\nthe spin echo using the saturation-recovery technique cite\nand spin echo decay time T2was measured by altering \u001c\nin the sequence. Spin-lattice relaxation time T1is ob-\ntained by the magnetization recovery \ftting to a single\nexponential form.\nFirst principles Density Functional Theory calcula-\ntions were performed to compare the energies of di\u000berent\nmagnetic con\fgurations. We used PAW as implemented\nin VASP with PBEsol exchange correlation functional\n[20{22]. A 8\u00028\u00028 k-point grid and energy cut o\u000b of\n500 eV ensures convergence in the primitive cell with 4\nformula units.\nIII. EXPERIMENTAL RESULTS\nA. Magnetic, transport and thermodynamic\nproperties\nFigure 1 (a)-(c) show the anisotropic magnetic prop-\nerties of\f-V2PO 5. Figure 1(a) presents the temperature\ndependent M=H taken atH= 1 kOe from 2 K to 250\nK in zero-\feld-cooled (ZFC) warming and \feld-cooled\nmode with Hparallel and perpendicular to the abplane.\nThe sharp upturns of the curves and the bifurcation in\nZFC and FC data for both directions below TIindicate\nthe existence of ferromagnetic component. The smooth\nZFC and FC curves suggest no other magnetic transi-\ntion below TI. Comparing with the other V-O-P mate-\nrials, the magnetic transition temperature is quite high\n[5{7], suggesting strong exchange interactions. Figure 1.\n(b) shows the temperature dependent M=H (blue) and\nH=M (black) measured from 300 K to 1000 K with H\n// (0 1 1) TatH= 10 kOe. Firstly, we see a subtle but\ndiscernable enhancement in M=H at the characteristic\ntemperature TII= 610 K, suggesting a possible phase\ntransition here. Secondly, upon cooling, linear Curie-\nWeiss behavior can be clearly seen from 1000 K to 500 K\ninH=M . By \ftting H=M from 1000 K to 500 K using the\nCurie-Weiss formula H=M =C=(T\u0000\u0012cw), whereCis the\nCurie constant and \u0012cwis the Weiss temperature, we ob-\ntained\u0016eff= 3.7(2)\u0016B/V and\u0012cw=\u0000900 K. The \u0016eff\nis larger than the one of V2:5+but comparable to the\none ofV2+. The large negative \u0012cwwithj\u0012cw=TIj\u00187.2,\nsuggests strong antiferromagnetic interaction. Thirdly,\ntheH=M fromTIto 500 K shows a crossover concave\nbehavior with temperature.3\nFIG. 1. (a) ZFC and FC M=H vs.TunderH= 1 kOe with H==ab andH?abfrom 2 K to 250 K. Inset: picture of \f-V2O5P\nsingle crystal against 1-mm scale. (b) H=M andM=H vs.TunderH=10 kOe with H==(011) T2 K to 1000 K. The red line\nis the Curie-Weiss \ft. (c) Isothermal M(H) curves at 2 K with H==ab andH?ab. Inset:M(H) curves at 300 K and 800\nK along with H==(011) T. (d) Speci\fc heat Cpvs.Tfrom 2 K to 200 K. The red line is the \ftting curve by Debye model.\nInset: Temperature dependence of magnetic entropy. (e) Resistivity \u001avs.Tfrom 130 K to 760 K. The red line emphasizes\nthe transition at TII. Inset:\u001avs. 1=T. Red line: the \ftting curve using the thermal excitation model. (f) The Arrott plot at\nvarious temperatures from 122 K to 137 K.\nFigure 1 (c) shows the anisotropic \feld dependent mag-\nnetizationM(H) taken at 2 K with H==ab andH?ab.\nThe crystal orientation was determined by x-ray di\u000brac-\ntion (Fig. S1) [23]. Clear hysteresis can be observed\nin both directions, con\frming the existence of ferromag-\nnetic component. Both curves show very similar shape\nand magnitude, suggesting weak magnetic anisotropy in\nthis system. The coercive \felds are around 1.3 kOe for\nboth. The remanent moments are 0.22 \u0016B/V forH?ab\nand 0.25\u0016B/V forH==ab and the saturation moments\nare 0.27\u0016B/V forH?aband 0.31\u0016B/V forH==ab ,\nwhich are so much smaller than the saturation moment\nof V2+(d3) and V3+(d2) ions, suggesting that instead\nof ferromagnetism, this material is likely ferrimagnetic or\ncanted antiferromagnetic below TI. The inset of Fig. 1(c)\nshows theM(H) curves taken at 300 K and 800 K with\nH// (0 1 1) T. Both curves are linear with the applied\nmagnetic \feld without hysteresis. The Arrott plot ( M2\nvs.H=M ) has been widely used to determine the fer-\nromagnetic phase transition temperature [24, 25], where\nthe curve of M2vs.H=M passes through the origin of\nthe plot at the transition temperature. To determine the\nvalue ofTI, isothermal M(H) curves are measured from\n122 K to 137 K. The Arrott plot are calculated and shown\nin Fig. 1(f), which suggests that TI\u0018128 K.\nFigure 1 (d) shows the temperature dependent Cp=T\ndata (blue) taken from 2 K to 200 K. A heat capacityanomaly featuring a second order phase transition ap-\npears around 128 K, accompanying with the magnetic\nphase transition observed in Fig. 1 (a). To estimate the\nmagnetic entropy, Debye model is used to \ft the heat\ncapacity to provide the non-magnetic background. The\n\ftted curve is shown in red in Fig. 1(d) and the \ftted\nDebye temperature is \u0002 D= 620 K. By subtracting the\nnon-magnetic background from \ftting, we obtained the\nmagnetic entropy as SM= 5.6 J/mol V-K2. This value is\nsigni\fcantly smaller than RLn4 of V2+andRLn3 of V3+,\nbut rather approximate RLn2. This may be caused by\nthe strong V-O covalency which lowers the moment size\nof V or a result of the entropy release above 128 K due\nto the chain structure and strong intrachain interaction\n[26, 27].\nFigure 1(e) shows the resistivity ( \u001a) of the\f-V2PO 5\nsingle crystals vs. temperature from 130 K up to 760 K.\nInstead of the semi-metal suggested by the theoretical\nprediction [15], it is a semiconductor. A semiconductor\nto semiconductor phase transition is discernable at 610\nK, which con\frms the possible phase transition at TII\nsuggested by the subtle susceptibility increase shown in\nFig. 1 (b). \u001avs. reciprocal temperature is plotted in\nthe inset of Fig. 1(d). By \ftting the data between 280\nK to 130 K with the thermal excitation model \u001a(T) =\n\u001a(0)exp(Eg=2kBT), the estimated gap size of \f-V2PO 5\nis 0.48 eV. The gap value is similar to 0.45-0.57 eV of the4\nVanadium phosphate glass [28].\nB. Structural phase transition and charge ordering\nBased on the transport, magnetic and heat capacity\nmeasurements, we have shown that \f-V2PO 5has one\nphase transition at 610 K and the other magnetic phase\ntransition at 128 K. To investigate the nature of these\ntwo phase transitions, single crystal neutron di\u000braction\nand NMR measurements are performed, which are sum-\nmarized in Fig. 2.\nFigure 2 (a) and (c) show the order parameter plot\nof the (1 1 4) Tneutron peak up to 650 K and (1 1 0) T\nneutron peak up to 450 K, respectively and Fig. 2 (b)\npresents the rocking curve scan for the (1 1 4) Tpeak. It is\ntwinned structure below 610 K, so we keep the tetragonal\nindex for convenience. The order parameter plot of the\nrelative stronger peak (1 1 4) T(Fig. 2(a)) indicates two\nphase transitions occurring at 610 K and 128 K, respec-\ntively, which is consistent with the order parameter plot\nof the (1 1 0) Tpeak (Fig. 2(c)) and (1 0 1) Tpeak (Fig.\nS2) [23]. The full data were collected at 4.5 K, 300 K,\nand 650 K to cover all three phase regions. At 650 K, the\ndata can be well \ftted in I41/amd symmetry (Table I).\nSince both (1 1 0) Tand (1 1 4) Tpeaks are symmetry dis-\nallowed re\rections in the tetragonal I41/amd structure,\nthe fact that we observed these peaks at room temper-\nature (Fig. 2(b)) suggests possible structural/magnetic\nphase transitions at 610 K.\nTo identify if the phase between 128 K and 610 K has\na magnetic component, phosphorus-31 NMR (31P-NMR)\nmeasurements were carried out. These are summarized\nin Fig. 2(e)-(f). In Fig. 2(d), we report the31P-NMR\nspectra at various temperatures from 170 K to 300 K.\n31P nuclear spin I= 1=2, and all sites are equivalent\nforB?ab. At 300 K, the spectra shift from the Lar-\nmor frequency by Ks= 0:493\u00060:002%, where the mean\nand uncertainty were calculated using the gaussian \ft-\nting. This value is on the same order of another VPO\nsample with V3+and about twice as much as that with\nV4+[29, 30], where a similar shift and broadening were\nobserved. The observations are interpreted as evidence\nfor no long range magnetic ordering in the intermediate,\ncharge-ordered phase. Below 190 K, a minor absorption\npeak at 146.4 MHz is resolved. Since it accounts for\nonly 3% of the total spin intensities, we expect the signal\nto be extrinsic due to the sites at twin boundaries and\nexclude it in our analysis. From the slope of K- \u001fplot\nshown in Fig.2 (e), we estimate the hyper\fne coupling\nconstant to be Acc= (10:23\u00060:87) kOe/\u0016B, which is\ntransferred from the unpaired electrons from the second\nnearest neighbors of P [31]. Figure 2(f) shows the tem-\nperature dependence of the spin-lattice relaxation time\nT1and spin-spin relaxation time T2. The relatively short\nand constant T1is consistent with a paramagnetic phase\nfrom 150 k to 300 K [32]. Meanwhile, T2starts drop-\nping rapidly below 210 K as the system approaches the\nFIG. 2. (a) The (1 1 4) Tneutron peak intensity vs. T. (b)\nThe (1 1 4) Tneutron peak intensity vs. !. (c) The (1 1 0) T\nneutron peak intensity vs. T. (d) P NMR frequency spectra.\nThe spectra are conserved after corrections for T 2. (e) Knight\nshift K and the peak width \u0001F obtained from (a) vs. mag-\nnetic susceptibility \u001f. Dashed line shows a linear \fttings of\nK. (f) Spin-lattice relaxation time T1and spin-spin relaxation\ntimeT2vs. T.\n128 K transition. This behavior is associated with slow\nlongitudinal \ructuations, which are likely related with\nthe onset of the 128 K magnetic phase transition. This\nis also consistent with the broadening observed in Fig.\n2(d) which appears as the magnetic correlation develops\nupon cooling. However, since 1/ T2is on the order of a\nfew to hundreds of KHz, we conclude that most of our\nbroadening, which is on the order of several MHz, is from\nthe inhomogeneous internal \feld.\nSince NMR shows that the phase transition at 610 K\nis not of magnetic origin, the phase transition at 610\nK should be a structural distortion. By including four\ntwinned structure domains, the room temperature neu-\ntron data can be well \ftted with the C2/cmonoclinic\nsymmetry, suggesting a tetragonal to monoclinic phase\ntransition at 610 K. Using this crystal structure, we re-\n\fned our powder X-ray di\u000braction taken at room tem-\nperature and obtained a very good \ft as shown in Fig.5\nFIG. 3. (a) The experimental and re\fned powder X-ray\ndi\u000braction patterns for \f-V2O5P at 300 K. Black: experi-\nmental pattern. Red: re\fned pattern. green: the di\u000berence\nbetween the experimental and re\fned patterns. Black ticks:\nthe Bragg peak positions in the monoclinic structure. Inset:\nenlarged view from 55.5\u000eto 58\u000e. (b)(c): The crystal structure\nof\f-V2O5P at 650 K (b) and 300 K (c).\n3(a). The detailed crystal structures at 4.5 K, 300 K\nand 650 K are summarized in Table I. The high temper-\nature tetragonal and low temperature monoclinic struc-\ntures are visualized in Fig. 3(b) and (c), respectively. In\nthe monoclinic phase, the monoclinic caxis is 121.45\u000e\nfrom theabplane which is the plane where the chains\nsit in. The unique V site in the tetragonal structure sep-\narates into V1 and V2 sites with the V1 atoms and V2\natoms alternately locating along each chain direction (V1\nand V2 sites are labeled in Fig. 3(c)), as a result, the av-\nerage bond length of VO 6octahedra on V1 site increases\nwhile that on V2 site decreases. Bond-valence analysis\nof the monoclinic crystal structure assigns charges of 2.0\nand 2.9 toV1andV2respectively, which is a smoking gun\nproof of the charge order [33, 34].\nC. Ferrimagnetic structure below 128 K\nThe magnetic order onsets at 128 K while the charge\norder continues to develop below the magnetic transi-TABLE I. The crystal structure of the \f-V2OPO 4phase at\n4.5 K, 300 K and 650 K, respectively.\n\f-V2PO 5at 4.5 K monoclinic C2/c\na= 7.563 \u0017A b=7.563 \u0017A c=7.235 \u0017A\f= 121.51\u000e\nRF2=0.0691 wRF2=0.0841 R F=0.044 \u001f2=19.2\nsite x/a y/b z/c\nV1 0 1/2 0\nV2 1/4 1/4 0\nO1 0.066(3) 0.749(2) 0.632(2)\nO2 0.317(2) 0.496(2) 0.598(1)\nO3 0 0.652(2) 1/4\nP 0 0.121(2) 1/4\n\f-V2PO 5at 300 K monoclinic C2/c\na= 7.570 \u0017A b=7.570 \u0017A c=7.232 \u0017A\f= 121.56\u000e\nRF2=0.0734 wRF2=0.0956 R F=0.0473 \u001f2=25.1\nsite x/a y/b z/c\nV1 0 1/2 0\nV2 1/4 1/4 0\nO1 0.065(3) 0.749(2) 0.630(2)\nO2 0.318(2) 0.496(2) 0.599(1)\nO3 0 0.652(2) 1/4\nP 0 0.119(3) 1/4\n\f-V2PO 5at 650K Tetragonal I41/amd\na= 5.357(2) \u0017A b= 5.357(2) \u0017A c=12.373(4) \u0017A\nRF2=0.0707 wRF2=0.0888 R F=0.0421 \u001f2=5.12\nsite x/a y/b z/c\nV1 1/4 1/4 1/4\nP1 0 3/4 1/8\nO1 0 -0.012(1) 0.193(4)\nO2 0 3/4 5/8\nFIG. 4. (a)-(c): Three magnetic structure models showing\nferrimagnetism with di\u000berent easy axis. Model (a) is collinear\nsuggesting weak magnetic anisotropy while Models (b) and\n(c) are noncollinear indicating strong magnetic anisotropy.\nModel (a) is the magnetic structure of \f-V2PO 5.6\ntion. Since no observed sharp change can be determined\nby the structure re\fnement at 4 K (see Table I), no fur-\nther structural phase transition below 128 K is discern-\nable. The magnetic propagation vector is k=0, which\nmeans that the magnetic scattering signal appears on\ntop of the nuclear Bragg peaks. To determine the mag-\nnetic structure more precisely, the magnetic signals were\nextracted by subtracting the data measured just above\n128 K from that at 4 K. During the procedure, to se-\nlect peaks which are insensitive to the thermal displace-\nments and charge ordering, we compared the data mea-\nsured at 300 K and 450 K and only selected a peak if\nthe change of its intensity is much smaller than the ex-\ntracted magnetic intensity. The selected re\rections are\nlisted in Table SI [23]. Since (1 1 0) T, (1 1 4) T, (1 0\n1)T, (0 0 2) Tand (0 0 4) Tpeaks were measured with a\nlong counting time and also tracked upon warming, they\nare highly reliable as indicated by their small error bars\nshown in the Table SI. We then performed the represen-\ntational analysis that determines the symmetry-allowed\nmagnetic structures for a second-order magnetic tran-\nsition. It yielded two magnetic symmetries, C2=cand\nC20=c0. Only the C20=c0can \ft our data (see Table SI).\nThe obtained magnetic structure is ferrimagnetic. Spins\non all V2+(V1 sites) atoms are parallel and so do the\nspins on all V3+(V2 sites) atoms while these two spin\nsublattices are antiparallel to each other. Since it is un-\nlikely for the V2+(d3) to be in a low spin state due to\nthe longer V-O bond length on V1 site and thus weaker\ncrystal electric \feld, the moment MV1> M V2. Figure\n4 shows the ferrimagnetic structure with three possible\neasy axis assignments where MV1> M V2. The calcu-\nlated peak intensity and goodness of \ft are summarized\nin Table SI. The model in the left panel of Fig. 4(a)\nis our pick for the \f-V2PO 5which gives the best \ft of\nthe data as shown in the right panel of Fig. 4(a). The\nMV1= 1:4(1)\u0016BandMV2= 1:2(1)\u0016B. The easy axis is\ninacMplane and 47(9) degrees away from aMtowards\ncM. The magnetic structure is collinear, suggesting weak\nmagnetic anisotropy. This is indeed consistent with the\nanisotropic M(H) measurements shown in Fig. 1(c). The\nother two models shown in Fig. 4(b) and (c) are with the\neasy axis along the chain direction (Fig. 4(b)) or perpen-\ndicular to the chain direction on the abplane (Fig. 4(c)).\nThese two magnetic models are non-collinear with strong\nmagnetic anisotropy. Since the goodness of \ft for these\ntwo latter models are poor, they are not the right mag-\nnetic structure for \f-V2PO 5.\nIV. DISCUSSION\nIt is of particular interest to ask if the charge order is\nsolely responsible for the reduction in the crystal symme-\ntry, or rather if it is a secondary order parameter to some\nother electronic phase transition. In order to preclude\nthis possibility and elucidate the nature of the charge or-\ndering transition at 610 K, we performed a group theo-TABLE II. Energies of di\u000berent magnetic con\fgurations from\n\frst principle.\nIntra-chain Inter-chain Energy (meV/f.u.)\nFerrimagnetic Ferromagnetic 0\nFerrimagnetic Antiferromagnetic 9\nFerromagnetic Ferromagnetic 60\nFerromagnetic Antiferromagnetic 71\nretical analysis of the lattice distortion using the Isotropy\nSoftware Suite.[35] The distortion from the high temper-\nature tetragonal structure ( I41=amd ) to the low temper-\nature monoclinic structure ( C2=c) can be caused by two\nnormal modes of lattice vibration, \u0000+\n5and \u0000+\n4. \u0000+\n5re-\nduces the symmetry from I41=amd toC2=c, and is the\nonly candidate for the primary structural order parame-\nter, whereas \u0000+\n4by itself reduces the symmetry to Fddd .\nC2cis a subgroup of Fddd , and as a result, \u0000+\n4is most\nlikely a secondary order parameter that is not important\nin the energetics of the phase transition. At the same\ntime, the charge order itself, which is a di\u000berentiation\nof the neighboring V ions in the same chain, transforms\nas the \u0000+\n5irreducible representation does for the high\nsymmetry structure. Figure 5(a) shows the displacement\nof the oxygen atoms in the VO 6face-sharing chain due\nto the \u0000+\n5normal mode. This mode breaks the symme-\ntry between the V ions that are symmetry equivalent at\nI41=amd , and decreases the V-O bond length for V2while\nincreasing it for V1(Fig. 5(b)) as expected in a charge\nordering transition. We therefore conclude that to re-\nduce the symmetry to the monoclinic phase, the charge\norder, by itself, is su\u000ecient and no other magnetic or\nelectronic mechanisms are necessary. This is consistent\nwith the fact that the TCOis almost 5 times of the Tmag.\nWe also note that \u0000+\n5is a Raman active mode, and as a\nresult, signature of the charge ordering transition should\nbe visible in the Raman spectrum of \f-V2PO 5.\nWe performed the \frst principles calculation using\nDFT+U with U = 4 eV to correct for the underestima-\ntion of the on-site coulomb interaction on the V ion [36].\nOur DFT calculations predict magnetic moments of 2.6\n\u0016Band 1.8\u0016Brespectively inside the V1andV2spheres,\nbut the band structure (Fig. 5(c)) shows no partially\n\flled bands. This signals strong hybridization between\ntheVand theOions. DFT gives a magnetic moment\n0.5\u0016BperVincluding the interstitials. This is a strong\noverestimation compared to the experimental value. The\nreason of this is likely the DFT+U's tendency to overes-\ntimate the ordered moments when there are dynamical\n\ructuations present, and it is possible that a more ad-\nvanced \frst principles method (such as the Dynamical\nMean Field Theory) can reproduce the experimentally\nobserved value of the local moments.\nTo understand the magnetic order, we calculated the\nenergies of phases with di\u000berent magnetic orders from\nDFT, as listed in Table II. The lowest energy phase is\npredicted to have ferrimagnetic intra-chain order, where7\nFIG. 5. (a) A sketch of the Oxygen anion displacements due\nto the \u0000+\n5mode which is responsible for charge ordering. (b)\nThe average V\u0000Obond length as a function of the \u0000+\n5normal\nmode amplitude. (c) DFT band structure in the ferrimagnetic\nstate. Majority and minority spin bands are shown in red and\nblue respectively.\nthe moments of the neighboring V1andV2ions on the\nsame chain are aligned anti-parallel, and ferromagnetic\ninter-chain order, so that, for example, the magnetic mo-\nments of all V1ions are parallel. This observation is in\nline with the experimental observation. The energy cost\nof having a magnetic phase where the di\u000berent chains\nhave antiparallel moments is bout 9-10 meV per formula\nunit, whereas the energy cost of having spins on the same\nchain parallel is 60 meV per formula unit. This suggests\nthat the intra-chain interactions between the V1andV2\nions in the face-sharing VO 6chains are the dominant\nterm in the magnetic hamiltonian.\nTo understand the crossover behavior in H/M shown\nin in Fig. 1(b), we calculated the energetics of di\u000ber-\nent magnetic phases in the high temperature tetrago-\nnal structure ( I41=amd ) with the same parameters (not\nshown), and found that similar couplings apply to mag-netic moments in that structure too. This explains the\ncross-over: Above the charge ordering temperature, the\nmagnetic moments on all V ions are equal, and its Curie-\nWeiss behaviour is that of an antiferromagnet, with a\nnegative Curie temperature. However, charge ordering\nmakes the moments unequal, and as a result, below 610\nK the Curie-Weiss behaviour is that of an ferrimagnet,\nwhich has a positive Curie temperature like a ferromag-\nnet.\nTo address if there is non-trivial topology in this com-\npound, the DFT band structure in Fig. 5(c) is calcu-\nlated in the ferrimagnetic ground state. Unlike the DFT\nband structure calculated in the ferromagnetic phase and\nthe tetragonal structure without U [15], there are no\nband crossing at the Fermi level. And more importantly,\nthe top of the valence and the bottom of the conduc-\ntion bands have opposite spin directions. This obser-\nvation precludes any possibility of topological phases in\n\f-V2PO 5.\nV. CONCLUSION\nIn conclusion, we have grown and characterized \f-\nV2PO 5single crystals. A tetragonal to monoclinic struc-\ntural phase transition at 610 K and a paramagnetic to\nferrimagnetic phase transition are revealed by transport,\nmagnetic, speci\fc heat, single crystal neutron di\u000braction\nand NMR measurements. Below 610 K, the single V site\nat the high temperature tetragonal phase distorts into\ntwo alternating V sites, leading to the increase of V-O\nbond length of one V site but the decrease of the other\nV site. Our \frst principles calculation shows that this\ndistortion was caused by the \u0000+\n5normal mode of lat-\ntice vibration. Accompanied with the distortion, charge\nordering of V2+and V3+is undoubtedly suggested by\nthe Bond-valence analysis. Below 128 K, the spins order\nparallel on each sublattice of V sites while the spins on\none sublattice order antiparallel to the other. With the\neasy axis being in the monoclinic acplane and 47(9)\u000e\naway from the monoclinic atowardscaxis., this gives a\ncollinear ferrimagnetic structure with the moment to be\n1.4(1)\u0016Bon V2+site and 1.2(1) \u0016Bon V3+site, suggest-\ning weak magnetic anisotropy and dominant role of the\nintra-chain V-V interaction in magnetism. No non-trivial\ntopology is suggested by our \frst principles calculation.\nACKNOWLEDGMENTS\nWork at UCLA (JX, CWH, RC, NN) was supported\nby NSF DMREF program under the award NSF DM-\nREF project DMREF-1629457. Work at ORNL HFIR\nwas sponsored by the Scienti\fc User Facilities Division,\nO\u000ece of Science, Basic Energy Sciences, U.S. Depart-\nment of Energy. Work at UMN was supported by NSF\nDMREF program under the award NSF DMREF project\nDMREF-1629260. Work at UCLA (HHW, YKL, SB) was8\nsupported by NSF DMR-1410343 and DMR-1709304.\nYKL would also like to thank the support from LANL\nLDRD program.\nNote : During the preparation of this manuscript, we\nnoticed a work on pollycrystalline \f-V2OPO 4was justaccepted but Journal of the american chemistry society\n(http://pubs.acs.org/doi/abs/10.1021/jacs.7b09441),\nwhich also revealed the charge ordering and simi-\nlar ferrimagnetism with di\u000berent easy axis in this\ncompound.\n[1] J. Att\feld. Charge ordering in transition metal oxides.\nSolid state sciences 8, 861 (2006).\n[2] E. Wollan, W. Koehler. Neutron Di\u000braction Study of\nthe Magnetic Properties of the Series of Perovskite-Type\nCompounds [(1-x)La,xCa]MnO 3.Phys. Rev. 100, 545\n(1955).\n[3] J. Goodenough. Theory of the Role of Covalence in\nthe Perovskite-Type Manganites [La,M(II)]MnO 3.Phys.\nRev.100, 564 (1955).\n[4] T. Yamauchi, M. Isobe, and Y. Ueda. Charge order\nand superconductivity in vanadium oxides. Solid State\nSciences 7, 874 (2005).\n[5] R. Glaum, M. Reehuis, N. St u\fer, U. Kaiser, and\nF. Reinauer. Neutron Di\u000braction Study of the Nuclear\nand Magnetic Structure of the CrVO 4Type Phosphates\nTiPO 4and VPO 4.J.Solid State Chem. 126, 15 (1996).\n[6] J. Kikuchi, N. Kurata, K. Motoya, T. Yamauchi,\nand Y. Ueda. Spin Di\u000busion in the S=1/2 Quasi\nOne-Dimensional Antiferromagnet \u000b-VO(PO 3)2via 31P\nNMR. J.Phys. Soc. Jpn.70, 2765 (2001).\n[7] J. Johnson, D. Johnston, H. King Jr., T. Halbert, J.\nBrody, and D. Goshorn. Structure and magnetic proper-\nties of V 2(VO)(P 2O7)2. A mixed-valence vanadium (III,\nIII, IV) pyrophosphate. Inorg. Chem. 27, 1646 (1988).\n[8] D. Johnston, J. Johnson, D. Goshorn, and A. Jacob-\nson. Magnetic susceptibility of (VO) 2P2O7: A one-\ndimensional spin-1/2 Heisenberg antiferromagnet with\na ladder spin con\fguration and a singlet ground state.\nPhys. Rev. B35, 219 (1987).\n[9] E. Dagotto, J. Riera, and D. Scalapino. Superconductiv-\nity in ladders and coupled planes. Phys. Rev. B45,5744\n(1992).\n[10] A. W. Garrett, S. E. Nagler, D. A. Tennant, B. C. Sales,\nand T. Barnes. Magnetic excitations in the S=1/2 alter-\nnating chain compound (VO) 2P2O7.Phys. Rev. Lett.79,\n745 (1997).\n[11] J. Kikuchi, K. Motoya, T. Yamauchi, and Y. Ueda. Coex-\nistence of double alternating antiferromagnetic chains in\n(VO) 2P2O7: NMR study. Phys. Rev. B60, 6731 (1999).\n[12] T. Yamauchi, Y. Narumi, J. Kikuchi, Y. Ueda, K. Tatani,\nT. C. Kobayashi, K. Kindo, and K. Motoya, Phys. Rev.\nLett. 83, 3729 (1999).\n[13] R. Glaum, and R. Gruehn. Synthese, Kristallstruktur\nund magnetisches Verhalten von V2PO5. Z.Kristallogr\n186, 91 (1989).\n[14] D. C. Johnston, Magnetic Susceptibility of Collinear and\nNoncollinear Heisenberg Antiferromagnets, Phy. Rev.\nLett., 109, 077201 (2012)\n[15] Y. Jin, R. Wang, Z. Chen, J. Zhao, Y. Zhao, and H.\nXu. Ferromagnetic Weyl semimetal phase in a tetragonal\nstructure. Phys. Rev. B96, 201102 (2017).\n[16] D. Smith, and F. Fickett. Low-temperature properties of\nsilver. J.Res. Natl. Inst. Stand. Technol. 100, 119 (1995).\n[17] B. Chakoumakos, H. Cao, F. Ye, A. Stoica, M. Popovici,M. Sundaram, W. Zhou, J. Hicks, G. Lynn and R. Riedel.\nFour-circle single-crystal neutron di\u000bractometer at the\nHigh Flux Isotope Reactor. J.Appl. Crystallogr. 44, 655\n(2011).\n[18] A. Wills. A new protocol for the determination of\nmagnetic structures using simulated annealing and rep-\nresentational analysis (SARA h).Phys. B:Condens.\nMatt. 276, 680 (2000), program available from\nwww.ccp14.ac.uk.\n[19] J. Rodriguez-Carvajal. Recent advances in magnetic\nstructure determination by neutron powder di\u000braction.\nPhys. B:Condens. Matt. 192, 55 (1993).\n[20] G. Kresse and J. Furthm uller. E\u000ecient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set. Phys. Rev. B54, 11169 (1996).\n[21] G. Kresse, and J. Hafner. 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A\ncomparative study of the e\u000bects of rare-earth oxides on\nthe physical, optical, electrical and structural properties\nof vanadium phosphate glasses. J.Materials Science 20,\n2207 (1985).\n[29] M. Sananes, and A. Tuel. Study by 31 P NMR spin echo\nmapping of vanadium phosphorus oxide catalysts. Solid\nstate nuclear magnetic resonance 6, 157 (1996).\n[30] M. Sananes, A. Tuel, and J. C. Volta. A study by 31P\nNMR spin-echo mapping of VPO catalysts: I. charac-\nterization of the reference phases. J.Catalys. 145, 251\n(1994).\n[31] J. Li, M. Lashier, G. Schrader, B. Gerstein. Oxida-\ntion states of vanadium in V-P-O oxidation catalysts\n31P NMR by spin-echo mapping. Appl. Catalys. 73, 83\n(1991).\n[32] T. Moriya. Nuclear magnetic relaxation in antiferromag-\nnetics. Prog. Theor. Phys. 16, 23 (1956).\n[33] N. Brese, and M. O'kee\u000be. Bond-valence parameters\nfor solids. Acta Crystallographica Section B:Structural\nScience 47, 192 (1991).9\n[34] I. David Brown, Bond valence parameters,\nhttps://www.iucr.org/resources/data/datasets/bond-\nvalence-parameters\n[35] H. Stokes, D. Hatch and B. Campbell. Isotropy. Retrievedfrom stokes. byu. edu/isotropy. html(2007).\n[36] A. Liechtenstein, V. Anisimov and J. Zaanen. Density-\nfunctional theory and strong interactions: Orbital order-\ning in Mott-Hubbard insulators. Phys. Rev. B,52, R5467\n(1995)."    },    {        "title": "2108.10881v1.The_domain_wall_motion_driven_by_a_rotating_field_in_a_ferrimagnet.pdf",        "content": "The domain-wall motion driven by a rotating \feld in a ferrimagnet\nMunsu Jin,1,\u0003Ik-Sun Hong,2,\u0003Duck-Ho Kim,3Kyung-Jin Lee,1and Se Kwon Kim1,y\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n2KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Republic of Korea\n3Center for Spintronics, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea\n(Dated: August 25, 2021)\nWe theoretically study a ferrimagnetic domain-wall motion driven by a rotating magnetic \feld.\nWe \fnd that, depending on the magnitude and the frequency of the rotating \feld, the dynamics\nof a ferrimagnetic domain wall can be classi\fed into two regimes. First, when the frequency is\nlower than a certain critical frequency set by the \feld magnitude, there is a stationary solution\nfor the domain-wall dynamics, where a domain-wall in-plane magnetization rotates in-phase with\nthe external \feld. The \feld-induced precession of the domain wall gives rise to the translational\nmotion of the domain wall via the gyrotropic coupling between the domain-wall angle and position.\nIn this so-called phase-locking regime, a domain-wall velocity increases as the frequency increases.\nSecond, when the frequency exceeds the critical frequency, a domain-wall angle precession is not\nsynchronous with the applied \feld. In this phase-unlocking regime, a domain wall velocity decreases\nas the frequency increases. Moreover, the direction of the domain-wall motion is found to be reversed\nacross the angular compensation point where the net spin density of the ferrimagnet changes its\nsign. Our work suggests that the dynamics of magnetic solitons under time-varying biases may serve\nas platform to study critical phenomena.\nI. INTRODUCTION\nSpintronics is the \feld which aims at advancing in-\nformation technology beyond what has been achievable\nwith charge-based electronics by exploiting spin degree of\nfreedom [1]. A natural venue to look for spin-based func-\ntionality is magnet materials, which are known to exhibit\nvarious excitations that can be used for information car-\nriers such as spin waves and topological solitons [2, 3].\nIn particular, a magnetic domain wall, which is a proto-\ntypical soliton in easy-axis magnets, has been a subject\nof intensive studies in spintronics due to its technological\nutilities as topologically robust information carriers as\nwell as intriguing physics [4, 5]. For example, a domain-\nwall racetrack memory, where a series of domain walls are\nmoved along the one-dimensional racetrack while carry-\ning information, has been shown to have potential for\nfast, nonvolatile, and three-dimensional solid-state mem-\nory architecture [5, 6]. In addition to practical utilities,\na domain wall is known to exhibit various fundamentally\ninteresting nonlinear phenomena. One example is given\nby the so-called Walker breakdown, which refers to a phe-\nnomenon of sudden drop of the domain-wall velocity at\ncertain critical strength of the driving force due to the\nonset of the precession motion of domain wall [7]. The\nWalker breakdown in the \feld-driven domain-wall motion\nhas been experimentally demonstrated in ferromagnetic\nwires, e.g., in Ref. [8].\nIn e\u000borts to expand material platforms for spintronics\nfrom ferromagnets that have been conventional material\n\u0003These authors contributed equally to this work.\nysekwonkim@kaist.ac.krplatform for spintronics, antiferromagnets have been re-\nceiving much attention in spintronics as alternative ma-\nterial choices due to their certain advantages over ferro-\nmagnets [9{11]. For example, the dynamics of antifer-\nromagnets are known to exhibit THz intrinsic frequency,\nwhich is generally faster than ferromagnetic dynamics\nwhich is on the order of GHz. Also, the absence of the\nequilibrium magnetization of antiferromagnets allows for\nthe development of denser spintronic devices compared to\nferromagnet-based devices which su\u000ber from strong cross-\ndevice interactions mediated by the stray \feld. In partic-\nular, antiferromangetic domain wall has been shown to be\nfundamentally di\u000berent from ferromagnetic counterparts\nand thus has been studied intensively in the last decade.\nFor example, antiferromagnetic domain wall is shown to\nnot exhibit the Walker breakdown unlike a ferromagnetic\ncase and thus can be driven with higher velocities [12].\nAlso, when the antiferromagnetic domain-wall velocity is\nclose to the maximum magnon group velocity, it has been\nexperimentally demonstrated to exhibit the Lorentz-like\ncontraction by shrinking its width according to the rel-\nativistic kinematics [13{18]. Despite the fundamental\ninterest and technological potentials, however, it is still\nexperimentally challenging to detect and control antifer-\nromagnetic dynamics due to its zero net magnetization,\nalthough there have been some progress enabled by x-ray\nabsorption spectroscopy [19{21], spin-polarized scanning\ntunneling microscopy [10, 22], and quantum sensing with\nsingle spins [23, 24].\nRecently, ferrimagnets, which consist of two or more\ninequivalent magnetic sublattices that are coupled anti-\nferromagnetically, have emerged in spintronics as mate-\nrial platforms that can o\u000ber advantages of both ferro-\nmagnets and antiferromagnets [25]. They generally have\na small, but \fnite magnetization and thus can be de-arXiv:2108.10881v1  [cond-mat.mes-hall]  24 Aug 20212\nH\n<latexit sha1_base64=\"JAu09ZBtP9D6nViMwzzD+tLiHj4=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaJUY9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1mu1C9K1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOiDjQM=</latexit>x<latexit sha1_base64=\"6GKBgagDkYhxv0Vb75HqYGOGHiM=\">AAAB7HicbVBNS8NAEJ3Ur1q/qh69BItQLyUpoh6LXjxWMLXQhrLZbtqlm03YnQgl9Dd48aCIV3+QN/+NmzYHbX0w8Hhvhpl5QSK4Rsf5tkpr6xubW+Xtys7u3v5B9fCoo+NUUebRWMSqGxDNBJfMQ46CdRPFSBQI9hhMbnP/8YkpzWP5gNOE+REZSR5yStBIXreO55VBteY0nDnsVeIWpAYF2oPqV38Y0zRiEqkgWvdcJ0E/Iwo5FWxW6aeaJYROyIj1DJUkYtrP5sfO7DOjDO0wVqYk2nP190RGIq2nUWA6I4Jjvezl4n9eL8Xw2s+4TFJkki4WhamwMbbzz+0hV4yimBpCqOLmVpuOiSIUTT55CO7yy6uk02y4l43m/UWtdVPEUYYTOIU6uHAFLbiDNnhAgcMzvMKbJa0X6936WLSWrGLmGP7A+vwBiGKN2g==</latexit>X(t)\n<latexit sha1_base64=\"xLUWPYX7rlo/qu0PuV8VJNMmK8s=\">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeiF49V7Ae0oWy2m3bpZjfsboQQ+g+8eFDEq//Im//GTZuDtj4YeLw3w8y8IOZMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WiCG0TyaXqBVhTzgRtG2Y47cWK4ijgtBtMb3O/+0SVZlI8mjSmfoTHgoWMYGOlh7QyrNbcujsHWiVeQWpQoDWsfg1GkiQRFYZwrHXfc2PjZ1gZRjidVQaJpjEmUzymfUsFjqj2s/mlM3RmlREKpbIlDJqrvycyHGmdRoHtjLCZ6GUvF//z+okJr/2MiTgxVJDFojDhyEiUv41GTFFieGoJJorZWxGZYIWJseHkIXjLL6+STqPuXdYb9xe15k0RRxlO4BTOwYMraMIdtKANBEJ4hld4c6bOi/PufCxaS04xcwx/4Hz+AB6bjRg=</latexit>y\n<latexit sha1_base64=\"sp78TYbCPXvrzZe6qPTFMBbL3Yo=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaJUY9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1mu1C9K1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOuLjQU=</latexit>z<latexit sha1_base64=\"1RDDYAJp1kSOoDJkoclpt3bOYuk=\">AAAB9HicbVDLSgNBEJyNrxhfUY9eBoMQL2E3iHoM6sFjBPOA7BJmJ73JkNmHM72BEPIdXjwo4tWP8ebfOEn2oIkFDUVVN91dfiKFRtv+tnJr6xubW/ntws7u3v5B8fCoqeNUcWjwWMaq7TMNUkTQQIES2okCFvoSWv7wdua3RqC0iKNHHCfghawfiUBwhkby3DuQyNz6QJTxvFss2RV7DrpKnIyUSIZ6t/jl9mKehhAhl0zrjmMn6E2YQsElTAtuqiFhfMj60DE0YiFobzI/ekrPjNKjQaxMRUjn6u+JCQu1Hoe+6QwZDvSyNxP/8zopBtfeRERJihDxxaIglRRjOkuA9oQCjnJsCONKmFspHzDFOJqcCiYEZ/nlVdKsVpzLSvXholS7yeLIkxNySsrEIVekRu5JnTQIJ0/kmbySN2tkvVjv1seiNWdlM8fkD6zPH+kHkYs=</latexit>\u0000\u0000(t)\n<latexit sha1_base64=\"E6KkbiyVf1wjS3ObY2CYJ6pTGLY=\">AAAB7nicbVBNS8NAEJ34WetX1aOXxSLUS0mKqMeiF48V7Ae0oWy2m3bpZhN2J0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2M72Z++4lrI2L1iJOE+xEdKhEKRtFK7V5jJCp40S+V3ao7B1klXk7KkKPRL331BjFLI66QSWpM13MT9DOqUTDJp8VeanhC2ZgOeddSRSNu/Gx+7pScW2VAwljbUkjm6u+JjEbGTKLAdkYUR2bZm4n/ed0Uwxs/EypJkSu2WBSmkmBMZr+TgdCcoZxYQpkW9lbCRlRThjahog3BW355lbRqVe+qWnu4LNdv8zgKcApnUAEPrqEO99CAJjAYwzO8wpuTOC/Ou/OxaF1z8pkT+APn8weDtI8J</latexit>\u0000(t)\nFIG. 1. Schematic of the magnetization con\fguration of a fer-\nrimagnetic domain wall, where blue and red arrows represent\nthe magnetic moments of two sublattices of the ferrimagnet.\nThe domain-wall position is denoted by X(t) and the in-plane\ndomain-wall magnetization angle is denoted by \b( t). The ex-\nternal \feld H(t) rotates within the xyplane and the domain-\nwall in-plane magnetization lags behind the external \feld by\n\u0001\b(t).\ntected and controlled by conventional methods used for\nferromagnets. Also, under suitable conditions, their dy-\nnamics resembles the dynamics of antiferromagnets since\ntheir magnetic sublattices are antiferromagnetically cou-\npled similarly to antiferromagnets [26]. In other words,\nantiferromagnet-like dynamics of ferrimagnets is control-\nlable and detectable due to its \fnite magnetization. This\nfeature of ferrimangets, typi\fed by rare-earth transition-\nmetal (RE-TM) ferrimagnets, allows for the fast domain-\nwall motion [27{30] and ultrafast magnetization switch-\ning [31{33]. In this work, we are interested in the dy-\nnamics of a domain wall motion in a ferrimagnet.\nMost of the studies on a domain-wall motion have fo-\ncused on the e\u000bects of DC biases such as an external \feld\nand a current. In searching for novel magnetic phenom-\nena, domain-wall motion by oscillating biases have been\nreceiving increasing attention in the \feld. For example,\nthe motion of an antiferromagnetic domain wall by a ro-\ntating \feld has been studied in Refs. [34, 35]. Also, it\nhas been shown that a ferromagnetic domain-wall mo-\ntion driven by time-periodic \feld or current can exhibit\na sudden drop of the domain-wall velocity akin to the\nWalker breakdown [36, 37]. In Ref. [37], this Walker-like\nbreakdown of an AC-bias-driven ferromagnetic domain-\nwall motion has been explained by phase-locking and\nphase-unlocking transition, which mimics an analogous\nphenomenon in an electric RLC circuit discovered by\nAdler [38]. In spintronics, the phase-locking of the spin-\ntorque oscillator to an AC current has also been explained\nby invoking its analogy to the Adler equation [39{41]. Al-\nthough the AC-bias-driven domain-wall motion has been\nstudied for ferromagnets and antiferromagnets, the cor-\nresponding problem for a ferrimagnetic domain wall has\nnot been studied.\nIn this paper, we theoretically investigate a ferrimag-\nnetic domain-wall motion driven by a rotating \feld,\nwhich is schematically illustrated in Fig. 1. We \fnd that,\nwhen the frequency is below a certain critical frequency,\nthe precession of the in-plane magnetization inside the\ndomain wall is synchronized with the applied rotating\n\feld. In this low-frequency regime, the domain-wall ve-\nlocity increases linearly as the frequency increases. Whenthe frequency exceeds the critical frequency, on the other\nhand, the domain-wall motion cannot keep pace with the\nrotating \feld, making its motion asynchronous with the\nrotating \feld. In this case, the domain-wall velocity de-\ncreases as the frequency increases. We refer the former\nand the latter regimes as the phase-locking regime and\nthe phase-unlocking regime, respectively. The analyti-\ncal solutions are checked by performing the numerical\nsimulations, which show good agreement in both phase-\nlocking and phase-unlocking regimes. The unique feature\nof the ferrimagnetic domain-wall motion occurs in the\nvicinity of angular momentum compensation point ( TA):\nthe direction of the domain-wall motion is reversed as the\nferrimagnet passes across TAdue to the sign \rip of the\nnet spin density at TA. For experimental feasibility of us-\ning a rotating \feld to drive a domain wall, we would like\nto mention that there has already been an experimental\ndemonstration of the chirality reversal of a vortex domain\nwall induced by a rotating magnetic \feld [42].\nThis paper is organized as follows. In Sec. II, we de-\nvelop a theory for the dynamics of a ferrimagnetic do-\nmain wall in the presence of a rotating \feld within the\nLandau-Lifshitz-Gilbert-like equations of motion for fer-\nrimagnets. The main analytical results are the critical\nfrequency [Eq. (8)] that separates between the phase-\nlocking and the phase-unlocking regimes, the domain-\nwall velocity as a function of the frequency in the phase-\nlocking regime [Eq. (13)] and the velocity in the phase-\nunlocking regime [Eq. (15)]. In Sec. III, we present our\nnumerical simulation results and compare them with the\nanalytical solutions. In Sec. IV, we summarize our work.\nII. THEORY FOR DOMAIN-WALL DYNAMICS\nDRIVEN BY A ROTATING FIELD\nIn this section, we develop a theory for the dynam-\nics of a ferrimagnetic domain wall driven by a rotating\n\feld in close connection to the existing theory for the dy-\nnamics of a ferromagnetic domain wall driven by an AC\n\feld [37]. For concreteness, we consider a RE-TM ferri-\nmagnets, where RE magnetic moments and TM magnetic\nmoments are exchange-coupled antiferromagnetically.\nA. Analytical model\nThe dynamics of ferrimagnets is described by the\nLandau-Lifshitz-Gilbert (LLG)-like equation, which is\ngiven by [43{49]\n\u000es_n\u0000\u000bsn\u0002_n\u0000\u001an\u0002n=\u0000n\u0002he\u000b; (1)\nwhere nis the unit vector in the direction of the mag-\nnetization of the TM sublattice, \u000es=sTM\u0000sREis the\nnet spin density of a ferrimagnet along n,s=sTM+sRE\nis the sum of spin densities of two sublattices, \u000b > 0\nis the Gilbert damping constant, he\u000b\u0011\u0000\u000eU=\u000enis the3\n\feld conjugate to the order parameter n, and\u001ais the mo-\nment of inertia of the antiferromagnetic dynamics for the\nstaggered magnetization n[11], which is inversely pro-\nportional to the microscopic exchange energy between\nthe two magnetic sublattices.\nWe consider a quasi-one-dimensional ferrimagnet in a\nrotating \feld, which can be modeled by the potential\nenergyU=R\ndx[fA(@xn)2\u0000K(nz)2+Ky(ny)2g=2\u0000\nMH\u0001n], whereAis the exchange coe\u000ecient, K > 0 is the\neasy-axis anisotropy (also called perpendicular magnetic\nanisotropy), Ky>0 is hard-axis anisotropy that captures\nthe shape anisotropy induced by the magnetostatic inter-\naction,M=MTM\u0000MREis the net magnetization of the\nferrimagnet, H=H(cos(!t);sin(!t);0) represents the\nrotating \feld about the zaxis at the frequency given by\n!. Without loss of generality, we consider the cases with\nH > 0 and! >0. In this work, we neglect the nonlocal\ndipolar interaction since, due to the antiferromagnetic\nalignment of the two magnetic sublattices, the net mag-\nnetization of ferrimagnets is orders of magnitude smaller\nthan that of ferromagnets.\nDue to the easy-axis anisotropy, the ferrimagnet sup-\nports a stable nonlinear soliton solution with boundary\ncondition n(x!\u00061 ) =\u0006^z, which is a called a domain\nwall. An equilibrium domain-wall solution is given by\nthe following Walker ansatz [7]:\nn=\u0012\ncos \b sechx\u0000X\n\u0015;sin \b sechx\u0000X\n\u0015;tanhx\u0000X\n\u0015\u0013\n;\n(2)\nwhere\u0015=p\nA=K is the parameter for the domain-wall\nwidth,Xrepresents the domain-wall position, and \b\nis the in-plane angle of the domain-wall magnetization.\nSee Fig. 1 for the schematic illustration of the domain\nwall. The domain-wall position Xrepresents a zero-\nenergy mode associated with the spontaneous breaking\nof the translational symmetry of the system. By plug-\nging the domain-wall solution to the potential energy U,\nwe obtain the following energy of the domain wall:\nU(\b) =\u0000\u0019\u0015MH cos(\b\u0000!t) +\u0015Kysin2\b:(3)\nThis result indicates that when the external \feld is suf-\n\fciently strong, H\u001dKy=M, the domain-wall angle \b\nwill follow!t, i.e., the phase of the external \feld, closely\nto minimize the Zeeman energy. When the anisotropy\ndominates the external \feld Ky\u001dMH, the domain-wall\nangle will be kept closely to 0 or \u0019and there would be no\nappreciable e\u000bect of the external \feld on the domain-wall\ndynamics.\nThe low-energy dynamics of the domain wall can be\ndescribed the dynamics of the two collective coordinates,\nX(t) and \b(t). Within the collective-coordinate ap-\nproach, we can derive the following coupled equations\nfrom the LLG-like equation [50{53]\n\u00002\u000bs_X+ 2\u000es\u0015_\b\u00002\u001aX= 0; (4)and\n\u00002\u000bs\u0015_\b\u00002\u000es_X\u00002\u001a\u0015\b\n= 2\u0015Kysin \b cos \b + \u0019\u0015MsHsin (\b\u0000!t):(5)\nIn this work, we are interested in the time-averaged dy-\nnamics of the domain wall over su\u000eciently long time. By\ntaking time-average of Eq. (4), we obtain the following\naverage domain-wall velocity:\nh_Xi=\u0015\u000es\n\u000bsh_\bi; (6)\nwherehXiandh\biare set to be zero by assuming that the\ndomain-wall dynamics is periodic such that the velocity\n_X(t) and the angular velocity _\b(t) are periodic functions\nof timet. The domain-wall velocity h_Xiis linearly pro-\nportional to the angular precession of the magnetization\nh_\bi, which is rooted in the gyrotropic coupling between\nXand \b [52]. Note that the net spin density \u000esappears in\nthe proportionality constant. In ferrimagnets, the value\n\u000esvaries when the temperature changes. In particular,\nit changes sign across the angular momentum compensa-\ntion pointTA, which will be invoked below to argue that,\nfor the given rotating \feld, the sign of the domain-wall\nvelocity \rips as the temperature varies across TA.\nThe coupling [Eq. (6)] between h_Xiandh_\bienables us\nto drive the domain wall by a rotating \feld. For exam-\nple, when the \feld magnitude His su\u000eciently strong and\nthe \feld rotation is su\u000eciently slow, the in-plane mag-\nnetization inside the domain wall will be mostly parallel\nto the \feld direction H(t) =H(cos(!t);sin(!t);0). This\nmeans that the domain-wall angle \b follows !tclosely,\nleading toh_\bi\u0019!. Then, when the net spin density is\n\fnite\u000es6= 0, the domain wall should move at average\nvelocity given by h_Xi\u0019\u0015\u000es!=(\u000bs). Understanding the\ndomain-wall dynamics for general situations, e.g., with\nhigher frequencies, requires more sophisticated analysis,\nwhich we present below.\nThe time-averaged Eq. (5) can be solely written in\nterms of the domain-wall angle \b by replacing h_Xiby\n\u0015\u000es\n\u000bsh_\bi, which results in\nh_\bi=\u0000!Hhsin (\b\u0000!t)i\u0000!Khsin 2\bi; (7)\nwhere\n!H\u0011\u000bs\u0019MsH\n2f(\u000bs)2+\u000e2sg; (8)\nis the characteristic frequency determined by the external\n\feld and\n!K\u0011\u000bsKy\n2f(\u000bs)2+\u000e2sg; (9)\nis the characteristic frequency determined by the hard-\naxis anisotropy. This equation describes the dynamics\nof the domain-wall angle \b driven by a rotating \feld\nin thexyplane. In this work, we are interested in4\nthe e\u000bect of the rotating \feld on the domain-wall dy-\nnamics. Therefore, we will restrict our attention to the\nsituations where the external \feld dominates the hard-\naxis anisotropy term so that we can set Ky= 0 and\n!K= 0. With this approximation, Eq. (7) is reduced\ntoh_\bi=\u0000!Hhsin (\b\u0000!t)i. Instead of solving this av-\neraged version, we will present an exact solution of the\nfollowing equation\n_\b =!Hsin (!t\u0000\b); (10)\nand will use the solution to obtain the domain-wall ve-\nlocityh_Xias a function of the \feld magnitude Hand\nthe \feld frequency !. Our analysis results, which will\nbe obtained below, will be compared with the simulation\nresults in Sec. III.\nB. Phase-locking and phase-unlocking regimes\nTo solve Eq. (10), let us introduce a new parameter\n\u0001\b\u0011!t\u0000\b. The physical meaning of \u0001\b is the phase\ndi\u000berence between the domain-wall angle and the rotat-\ning \feld. The equation of motion for \u0001\b is given by\nd\u0001\b\ndt=!\u0000!Hsin \u0001\b: (11)\nThis equation has been studied in the \feld of nonlin-\near dynamics [54]. Note that two frequencies appear\nin the equation: the \feld rotation frequency !and the\nmagnitude-related frequency !H. Depending on the rel-\native magnitude of these two frequencies, the dynamics\nis divided into two regimes.\nFirst, let us consider the cases where !H>!, i.e., the\ncases where the \feld is su\u000eciently strong or the frequency\nis su\u000eciently slow. In this regime, the equation permits\na steady-state solution given by\nsin\u00001!\n!H= \u0001\b;for!<!H: (12)\nIn this regime, \u0001\b is constant, which means that the\ndomain-wall angle evolves in time while keeping its\nphase di\u000berence with the rotating \feld constant. Since\nthe phase di\u000berence between the external \feld and the\ndomain-wall angle is locked due to the strong external\n\feld, this regime is called a phase-locking regime. In this\nregime, according to Eq. (6), the average domain-wall\nvelocity is given by\nh_Xi=\u0015\u000es\n\u000bs!; for!<!H: (13)\nThis our \frst main result: The domain-wall velocity is\nlinearly proportional to the rotating-\feld frequency for\n! < !H. Note that the sign of the velocity depends\non the sign of the net spin density \u000es. Therefore, for\nthe given rotating \feld, the domain-wall velocity should\nchange its sign when \u000eschanges the sign, i.e., when thetemperature crosses the angular momentum compensa-\ntion pointTA.\nSecond, let us consider the cases with ! >!H, where\nthe rotation frequency is large compared to !H. In this\ncase, Eq. (11) does not possess a steady-state solution.\nIt still permits an exact solution given implicitly by\ntan\u0001\b\n2=!H\n!+r\n1\u0000!2\nH\n!2tanp\n!2\u0000!2\nH(t\u0000t0)\n2;\n(14)\nwheret0is an arbitrary constant. Note that the period\nof the solution is given by T= 2\u0019=p\n!2\u0000!2\nH, which\nis longer than the period of the applied \feld 2 \u0019=!, im-\nplying that the evolution of the domain-wall angle \b is\nnot in-phase with the rotating \feld. For this reason, the\ndomain-wall dynamics with ! > !His referred to be in\nthe phase-unlocking regime. The averaged angular ve-\nlocity is given by h\u0001_\bi= 2\u0019=T =p\n!2\u0000!2\nH, and thus,\nfrom Eq. (6), the averaged domain-wall velocity is given\nby\nh_Xi=\u0015\u000es\n\u000bs\u0012\n!\u0000q\n!2\u0000!2\nH\u0013\n;for!>!H;(15)\nwhich is a decreasing function of !. This is our second\nmain result: In the phase-unlocking regime ( ! > !H)\nwhere the external \feld rotates too fast for the domain\nwall to keep pace with it, the average domain-wall veloc-\nity decreases as the frequency increases.\nIII. NUMERICAL ANALYSIS\nTo con\frm the analytical results, particularly the\ndomain-wall velocity in the phase-locking regime\n[Eq. (13)], and one in the phase-unlocking regime\n[Eq. (15)], we performed the atomistic spin simula-\ntions by solving the two coupled Landau-Lifshitz-Gilbert\nequations for two antiferromagntically-coupled sublat-\ntices representing TM and RE magnetizations. The\nmaterial parameters that used in the simulations are\nA= 2:5\u000210\u00007erg/cm,K= 9:5\u0002107erg/cm3, and\nKy= 3\u0002103erg/cm3, and\u000b= 0:002. The used gyro-\nmagnetic ratios of the TM and the RE sublattices are\n\rTM= 1:936\u0002107s\u00001Oe\u00001and\rRE= 1:76\u0002107\ns\u00001Oe\u00001, respectively. The cell size was 0 :4\u000250\u00021\nnm3, corresponding to x,y, andzaxis, respectively.\nThe system size is 400 \u000250\u00021 nm3. Table I shows\nthe magnetization and the spin density parameters that\nwe used to model the e\u000bect of the temperature. The\ntemperature T4corresponds to the magnetization com-\npensation point TMwhere the magnetizations of the two\nsublattice are equal and thus the net magnetization van-\nishes. The temperature T7corresponds to the angular\nmomentum compensation point TAwhere the spin den-\nsities of the two sublattices are equal and thus the net\nspin density vanishes. The applied \feld strengths are\n3000 Oe for T1;T2;T3, 1000 Oe for T4;T5, and 200 Oe\nforT6;T7;T8;T9. The \feld magnitude is chosen for each5\nTABLE I. Material parameters used for simulations. MTM,MRE,sTM,sRE,\u000es, andsare the magnetization of TM elements,\nthe magnetization of RE elements, the spin density of TM elements, the spin density of RE elements, the net spin density, and\nthe total spin density, respectively. T4andT7representTMandTA, respectively.\nIndex T1 T2 T3T4(TM)T5 T6T7(TA)T8 T9\nMTM(emu/cm3) 1170 1140 1110 1080 1050 1020 990 960 930\nMRE(emu/cm3) 1260 1200 1140 1080 1020 960 900 840 780\nsTM(erg\u0001s/cm3) 6:04\u000210\u000055:89\u000210\u000055:73\u000210\u000055:58\u000210\u000055:42\u000210\u000055:27\u000210\u000055:11\u000210\u000054:96\u000210\u000054:8\u000210\u00005\nsRE(erg\u0001s/cm3) 7:16\u000210\u000056:82\u000210\u000056:48\u000210\u000056:14\u000210\u000055:8\u000210\u000055:45\u000210\u000055:11\u000210\u000054:77\u000210\u000054:43\u000210\u00005\n\u000es(=sA\u0000sB)\u00001:1\u000210\u00005\u00009:3\u000210\u00006\u00007:4\u000210\u00006\u00005:6\u000210\u00006\u00003:7\u000210\u00006\u00001:9\u000210\u000060 1:86\u000210\u000063:72\u000210\u00006\ns(=sA+sB) 1:3202\u000210\u000041:2707\u000210\u000041:2211\u000210\u000041:1715\u000210\u000041:1219\u000210\u000041:0723\u000210\u000041:0557\u000210\u000049:7314\u000210\u000059:2355\u000210\u00005\ntemperature such that the resultant domain-wall veloci-\nties are comparable.\nFigure 2(a) shows the domain-wall velocity h_Xias a\nfunction of the frequency of the rotating \feld for various\ncon\fgurations. The lines represent the analytical results,\nwhich are given by Eq. (13) for !<!Hand Eq. (15) for\n!>!Hwith the critical frequency !Hgiven by Eq. (8).\nThe dots represent the simulation results. The analytical\nresults and the simulation results agree with each other\nreasonably well. Several features are noteworthy. First,\natTMwhere the magnetization is zero, the domain-wall\nvelocity vanishes, which is due to the absence of the cou-\npling of the external \feld and the domain wall. Second,\natTAwhere the net spin density is zero, the velocity\nvanishes. In our analytical results, the domain-wall ve-\nlocity is proportional to \u000es, and thus it is expected to\nvanish atTA. Physically, this is due to the absence of\nthe gyrotropic coupling between the domain-wall posi-\ntionXand the domain-wall angle \b at TA. Thirdly, the\nsign of the domain-wall velocity, i.e., the direction of the\ndomain-wall motion depends on the sign of the net spin\ndensity\u000es. ForT1;T2;\u0001\u0001\u0001;T6where\u000es<0, the sign of\nthe velocity is negative, and for T8andT9where\u000es>0,\nthe sign of the velocity is positive. This means that for\nthe given rotating \feld, if we vary the temperature of the\nferrimagnet across TA, the direction of the domain-wall\nmotion should reverse exactly at TA, which may be ex-\nploited to detect TAexperimentally. Figure 2(b) shows\nthe domain-wall velocity as a function of the net spin den-\nsity\u000eswithin the phase-locking regime for the frequency\nf=!=2\u0019= 90 MHz. The analytical solutions [Eq. (13)]\nand the simulation results are depicted by square and\ntriangle symbols, respectively. Note that the sign of the\nvelocity changes, i.e., the direction of the domain-wall\nmotion reverses, as the net spin density changes its sign.\nThe results with all the temperatures except the magne-\ntization compensation point T4(TM) are used [55].\nFigure 3(a) shows the domain-wall velocity as a func-\ntion of the frequency for the con\fguration T5(see Ta-\nble I for the de\fnition). The critical frequency !H\nobtained from Eq. (8) is shown as a vertical dashed\nline. Figure 3(b) and (c) show the evolution of my(t),\nthey-component of the magnetization evaluated at the\ndomain-wall center and Hy(t), they-component of the\n<latexit sha1_base64=\"hZ5JwVhShyuy5pjFmZ/UjK2iWL4=\">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyWpoB6LXjxWsB/QhLLZbtqlm03YnYgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5QSK4Rsf5tgpr6xubW8Xt0s7u3v5B+fCoreNUUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmsfyAScJ8yMylDzklKCRPA/ZEwZhVg3Op/1yxak5c9irxM1JBXI0++UvbxDTNGISqSBa91wnQT8jCjkVbFryUs0SQsdkyHqGShIx7Wfzm6f2mVEGdhgrUxLtufp7IiOR1pMoMJ0RwZFe9mbif14vxfDaz7hMUmSSLhaFqbAxtmcB2AOuGEUxMYRQxc2tNh0RRSiamEomBHf55VXSrtfcy9rFfb3SuMnjKMIJnEIVXLiCBtxBE1pAIYFneIU3K7VerHfrY9FasPKZY/gD6/MH0+eRjA==</latexit>(b)<latexit sha1_base64=\"eOAdPlFCe9DfNsZuabxSV0AkUm0=\">AAAB83icbVBNS8NAEJ3Ur1q/qh69BItQLyWpoB6LXjxWsB/QhLLZbtqlm03YnYgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5QSK4Rsf5tgpr6xubW8Xt0s7u3v5B+fCoreNUUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmsfyAScJ8yMylDzklKCRPA/ZEwZhViXn03654tScOexV4uakAjma/fKXN4hpGjGJVBCte66ToJ8RhZwKNi15qWYJoWMyZD1DJYmY9rP5zVP7zCgDO4yVKYn2XP09kZFI60kUmM6I4EgvezPxP6+XYnjtZ1wmKTJJF4vCVNgY27MA7AFXjKKYGEKo4uZWm46IIhRNTCUTgrv88ipp12vuZe3ivl5p3ORxFOEETqEKLlxBA+6gCS2gkMAzvMKblVov1rv1sWgtWPnMMfyB9fkD0mGRiw==</latexit>(a)\n50100150200250300-600-400-2000200 T9 T8 T7 (TA) T6 T5 T4 (TM)  T3 T2 T1Domain wall velocity (m/s)\nFrequency of the field (MHz)\n-1.0x10-5-5.0x10-60.05.0x10-6-400-2000200 analytical solution simulationDomain wall velocity (m/s)\nNet spin density (erg⋅s/cm3)FIG. 2. (a) Domain-wall velocity h_Xias a function of the\nrotating-\feld frequency f=!=2\u0019for various con\fgurations\nwith the parameters shown in Table I. The lines are analyt-\nical solutions, Eq. (13) for ! < ! H(phase-locking regime\nwhere a domain-wall magnetization precesses at the same fre-\nquency as the external \feld) and Eq. (15) for !>! H(phase-\nunlocking regime where a domain-wall magnetization rotates\nslower than the external \feld). The dots represent simulation\nresults. Note that the domain-wall velocity changes its sign\nas the temperature varies across the angular momentum com-\npensation point TA. (b) Domain-wall velocity as a function\nof the net spin density \u000eswithin the phase-locked regime for\nthe frequency f= 90 MHz. The data from all the temper-\naturesT1;T2;\u0001\u0001\u0001;T9except the magnetization compensation\npointT4(TM), where the frequency f= 90 MHz belongs to\nthe phase-unlocking regime, is used. The squares and the tri-\nangles represent the analytical solutions [Eq. (13)] and the\nsimulation results, respectively.6\n(a)(b)(c)0501001502002503000-50-100-150-200Domain wall velocity (m/s)\nRotating field frequency (MHz) T5Phase-locking regimePhase-unlocking regime\n!!/2$70 MHz260 MHz051015202530-1.0-0.50.00.51.0Normalized amplitude\nTime (ns) my Hy my (analytical solution)\n051015202530-1.0-0.50.00.51.0Normalized amplitude\nTime (ns) my Hy my (analytical solution)\nFIG. 3. (a) Domain-wall velocity h_Xias a function of the rotating-\feld frequency f=!=2\u0019for the case of T5(de\fned in\nTable I). The lines and the dots represent the analytical solutions and the simulation results, respectively. The critical frequency\n!Hwhich separates the two distinct regimes of domain-wall dynamics is calculated from Eq. (8) and is shown as the vertical\ndashed line. When the frequency is below the critical frequency, the dynamics of the domain wall is in the phase-locking\nregime, where the domain wall precesses at the same frequency of the rotating \feld and thus the velocity increases linearly\nas a function of the frequency. When the frequency is above the critical frequency, the dynamics of the domain wall motion\nis in the phase-unlocking regime and the resultant velocity decreases as the frequency increases. (b, c) Time evolution of the\ny-component of the magnetization myat the domain-wall center and the external \feld Hyfor the rotating-\feld frequency of\n(b) 70 MHz and (c) 260 MHz. The black solid line and the red dashed line are obtained from the simulations. The blue dotted\nlines are the analytical solutions obtained from (b) Eq. (12) and (c) Eq. (14).\nrotating \feld at the frequencies of 70 MHz and 260 MHz,\nrespectively. The black solid lines and the red dashed\nlines represent the simulation results for myandHy, re-\nspectively. The dashed blue lines show the analytical so-\nlutions given by (b) Eq. (12) and (c) Eq. (14). In the\nphase-locking regime shown in Fig. 3(b), the domain-\nwall magnetization precesses at the same frequency of\nthe external \feld and thereby the domain-wall velocity\nincreases linearly as the frequency increases as expected\nfrom the analytical solution [Eq. (13)]. In the phase-\nunlocking regime shown in Fig. 3(c), the time duration\nformyto change between 1 and \u00001 is much longer than\nthe period of Hy, meaning that the domain-wall preces-\nsion is much slower than the applied \feld. In this case,\nthe domain-wall velocity decreases as the frequency in-\ncreases [Eq. (15)]. There are some deviations between the\nanalytical solution and the simulation result in Fig. 3(a,\nb), which are presumably due to the approximations that\nwe take to obtain the analytical results such as neglecting\nXand\b in Eq. (4) and Eq. (5).\nIV. SUMMARY\nWe have studied the dynamics of a ferrimagnetic do-\nmain wall driven by a rotating \feld analytically by using\nthe Landau-Lifshitz-Gilbert-like equations for ferrimag-\nnets and also by numerically solving the coupled LLG\nequations. We have found that, depending on the fre-\nquency of the \feld rotation, there are two distinct regimes\nof the dynamics of a domain wall. In the phase-lockingregime, where the frequency is below the critical fre-\nquency, the domain-wall velocity is proportional to the\nfrequency of the rotating \feld. In the phase-unlocking\nregime where the frequency is above the critical fre-\nquency, the domain-wall velocity decreases as the fre-\nquency increases. In addition, we have found that the\ndirection of the domain-wall motion depends on the sign\nof the net spin density of the ferrimagnet. This results\nin the reversal of the domain-wall velocity sign as the\ntemperature varies across the angular momentum com-\npensation point TA.\nACKNOWLEDGMENTS\nThis work was supported by Brain Pool Plus Pro-\ngram through the National Research Foundation of Ko-\nrea funded by the Ministry of Science and ICT (Grant\nNo. NRF-2020H1D3A2A03099291), by the National Re-\nsearch Foundation of Korea funded by the Korea Govern-\nment via the SRC Center for Quantum Coherence in Con-\ndensed Matter (Grant No. NRF-2016R1A5A1008184).\nK.J.L. was supported by the National Research Foun-\ndation of Korea (Grant No. NRF-2015M3D1A1070465).\nD.H.K was supported by the POSCO Science Fellowship\nof POSCO TJ Park Foundation, by the Korea Institute\nof Science and Technology (KIST) institutional program\n(No. 2E31032), and by the National Research Coun-\ncil of Science & Technology (NST) grant (Project No.\n2N45290) funded by the Korea government (Ministry of\nScience and ICT).\n[1] I. \u0014Zuti\u0013 c, J. 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Strogatz, Nonlinear Dynamics and Chaos\n(Addison-Wesley, Reading, MA, 1994).\n[55] The frequency f= 90 MHz belongs to the phase-\nunlocking regime for the magnetization compensation\npoint, and thus it is excluded from Fig. 2(b)."    },    {        "title": "1207.5084v1.Quantum_phase_transitions_in_alternating_spin__1_2__5_2__Heisenberg_chains.pdf",        "content": "arXiv:1207.5084v1  [cond-mat.str-el]  21 Jul 2012Quantum phase transitions in alternating spin-(1\n2,5\n2) Heisenberg chains\nAntˆ onio S. F. Ten´ orio, R. R. Montenegro-Filho, and M. D. Coutinh o-Filho\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, CEP 50670-901, Recife , Pernambuco, Brazil\nThe ground state spin-wave excitations and thermodynamic p roperties of two types of ferrimag-\nnetic chains are investigated: the alternating spin-1/2 sp in-5/2 chain and a similar chain with a\nspin-1/2 pendant attached to the spin-5/2 site. Results for magnetic susceptibility, magnetization\nand specific heat are obtained through the finite-temperatur e Lanczos method with the aim in\ndescribing available experimental data, as well as compari son with theoretical results from the semi-\nclassical approximation and the low-temperature suscepti bility expansion derived from Takahashi’s\nmodified spin-wave theory. In particular, we study in detail the temperature vs. magnetic field\nphase diagram of the spin-1/2 spin-5/2 chain, in which sever al low-temperature quantum phases are\nidentified: the Luttinger Liquid phase, the ferrimagnetic p lateau and the fully polarized one, and\nthe respective quantum critical points and crossover lines .\nPACS numbers: 75.10.Pq, 75.10.Jm, 75.30.Kz, 75.40.Mg\nI. INTRODUCTION\nQuasi-one-dimensionalmagnetic materialsform aclass\nof compounds with magnetic properties that above a\ncharacteristictemperaturecan be described throughone-\ndimensional models [1]. These include systems with a ro-\ntationally invariant singlet ground state (GS), modeled,\nfor example, through spin-1 gapped and gapless critical\nspin-1/2 chains [1], as well as more complex structures,\nsuch us ladders [1] and spin tubes [2]. Typically, gap-\nless one-dimensional systems exhibit power-law decay of\nthe correlation functions and can be understood through\nthe Bethe ansatz [3] or the Luttinger Liquid theory [1].\nIn addition, in gapped one-dimensional (1D) systems the\napplication of an external magnetic field Bcan suppress\nthe gap and induce a quantum phase transition [4] to a\nLuttinger Liquid phase. In particular, an extensive study\nof theB- temperature ( T) phase diagram of a spin-1/2\ngapped ladder system was recently carried out [5, 6].\nContrary to the above mentioned systems, quasi-1D\nferrimagnetic compounds display GS spontaneous mag-\nnetization and have ferromagnetic and antiferromagnetic\n(AF) spin-wave excitations. Usually the AF spin-wave\nmode is gapped and the magnetization curve exhibits\na plateau, which can be explained by topological argu-\nments [7]. Ferrimagnetism can arise from the topology\nof the unit cell [8], as in the phosphates with chemical\nformula A 3Cu3(PO4)4, where A = Ca, Sr or Pb. These\nmaterials have three Cu2+spin-1/2 ions [9] and can be\nmodeled by a line ofspin-1/2 trimer clusters [10, 11] with\nAF exchange couplings. Another class of ferrimagnets\nare mixed-spin compounds of type (A-X-B-X-) n, where\nA and B are two different magnetic components (single\nions or more complex molecules) and X is a bridging lig-\nand. In particular, we are interested in compounds that\ncan be modeled by spin-1/2 spin-5/2 chains (sS chains);\nthis includes, for example, systems built from Mn2+and\nCu2+ions linked through a dithioxalato ligand [12, 13].\nFurther, in the composition of some ferrimagnets, mag-\nneticelementscanbe organicradicalslikethenitronylni-troxide free radicals (NITR), where R stands for an alkyl\n(methyl, ethyl) or aromatic group (phenyl). A family\ninto this category consists of the Mn-NITR compounds\n[14], for which there is an AF exchange coupling between\nthe spin-1/2 radicals and the spin-5/2 Mn2+ions.\nIn this work we present a numerical study of the\nthermodynamic properties of the ferrimagnetic chains\nillustrated in Fig. 1: spin-1/2 spin-5/2 alternating\nchain (sS chain) and the spin-1/2 spin-5/2 alternat-\ning chain with a spin-1/2 pendant attached to the\nspin-5/2 site (ssS chain). These chains can be re-\nspectively used to model the ferrimagnetic compounds\nCuMn(S 2C2O2)2·7.5H2O (denoted by CuMnDTO) [13]\nand [Mn(NITIm)(NITImH)]ClO4(denoted by MnNN)\n[15], whose crystal structures belong to the centrosym-\nmetric monoclinic space group P21/c(C2h). The 3D or-\ndered phase observed [13, 15, 16] in these compounds,\nand in similar ones [17], at very low- Thave been inten-\nsively investigated. In fact, magnetization measurements\n[15, 16] suggest that the canting of the ferrimagnetic mo-\nments of the chains give rise to a 3D weak ferromag-\nnetism below the critical temperature, although neutron-\ndiffraction experiments [17] in similar compounds in-\ndicate a canted AF structure. A common feature in\nthese compounds is that the Mn2+ion has a6S5/2GS,\nthereby leading to a single ion anisotropy with no zero-\nfield splitting in first order of perturbation theory. Very\nlow-Tmagnetization measurements in a noncentrosym-\nmetric orthorhombic compound [17], belonging to the\nspace group P212121, suggest a single-ion anisotropy\nD/kB≈40 mK, which is much smaller than the intra-\nchain AF exchange couplings of the referred compounds\n[13, 15–17]. Therefore, a proper description of the 1D-3D\nmagnetic transition may require, in general, anisotropic\ncouplings, including the dipolar interaction. However,\nsimilarly to previous analysis [13, 15], in order to de-\nscribe the 1D ferrimagnetic properties of the compounds,\nwe disregard anisotropy effects.\nThe GS and the low-energy magnetic excitations are\ncalculated through the Lanczos exact diagonalization2\n´(a): sS chain\n(b): ssS chainJ\nJ\nJ\nFIG. 1: (Color online) Schematic representation of the ex-\nchange couplings and the GS long-range ferrimagnetic order -\ning for the (a) sS and (b) ssS alternating chains.\n(ED) algorithm, while thermal properties are obtained\nby the finite-temperature Lanczos method (FTLM) [18].\nWe also explore the field-induced quantum phase transi-\ntions of these systems and discuss our results in light of\nexperimental data, as well as predictions from the semi-\nclassical approximation [19] and the modified spin-wave\n(MSW) theory [3].\nThis work will unfold as follows: in Sec. II, we de-\nscribe the theoretical models and methods employed. In\nSec. III, we estimate the model parameters suitable to\ndescribe the experimental data (susceptibility and mag-\nnetization) ofthe relatedcompounds. In Sec. IV, the one\nmagnon bands and the specific heats of the two systems\nare presented and the main features discussed. In Sec. V\nwe exhibit the T−Bphase diagram of the sS chain and\ndiscuss in detail its quantum critical points and crossover\nlines, the Luttinger liquid phase and the plateau regions.\nIn Sec. VI, we analyze the low-temperature behavior of\nthe zero-field magnetic susceptibility and, finally, in Sec.\nVII we present a discussion of our relevant findings.\nII. MODELS AND METHODS\nThe sS chain with a uniform exchange interaction\nJ(>0) andNc[s,S] cells is described by the following\nHamiltonian:\nHsS=JNc/summationdisplay\ni(Si+Si+1)·si−gµBBSz,(1)\nwheres= 1/2 andS= 5/2, theg-factor is assumed uni-\nform,µBis the Bohrmagneton, Bis an applied magnetic\nfield in the zdirection and Szis the operator for the z\ncomponent of the total spin. This chain is bipartite, with\nNcsites with spin-5/2 in one sublattice and Ncsites with\nspin-1/2 in the other; therefore, the Lieb and Mattis the-\norem [20] assures that the GS total spin, SGS, is given\nbyNc|S−s|= 2Nc, i. e., spin 2 per unit cell. The GS\nmagnetic ordering of this chain is sketched in Fig. 1(a).The ssS chain with Nc[s,S,s′] cells is described by the\nfollowing Hamiltonian:\nHssS=JNc/summationdisplay\nl=1sl·(Sl+Sl+1)+J′Nc/summationdisplay\nl=1Sl·s′\nl−gµBBSz,(2)\nwheres=s′= 1/2 andS= 5/2, whileJ >0 and\nJ′>0. TheLiebandMattistheoremassuresthat SGS=\nNc|S−2s|= 3Nc/2, i. e., spin3\n2per unit cell. The GS\nmagnetic order of this chain is sketched in Fig. 1(b).\nThe FTLM [18] is based on the Lanczos diagonaliza-\ntion technique and random sampling. The fundamental\nrelations used in FTLM for the calculation of an static\nquantity associated to an operator Aare\n/angbracketleftA/angbracketright ≈Nst\nZRR/summationdisplay\nr=1M/summationdisplay\nj=0e−βǫr\nj/angbracketleftr|ψr\nj/angbracketright/angbracketleftψr\nj|A|r/angbracketright,\nZ≈Nst\nRR/summationdisplay\nr=1M/summationdisplay\nj=0e−βǫr\nj|/angbracketleftr|ψr\nj/angbracketright|2, (3)\nwhere the sampling is carried over Rrandom states |r/angbracketright,\ntaken as initial states for a M-step Lanczos procedure\nwhich results in Mapproximate eigenvalues ǫr\njwith re-\nspective eigenvectors |ψr\nj/angbracketrightin theNst-dimensional Hilbert\nspace. The method allow us to calculate the tempera-\nture dependence of the magnetization per unit cell mc,\nmagnetic susceptibility per unit cell χ, and specific heat\nCthrough:mc=gµB/angbracketleftSz/angbracketright\nNc, χ=g2µ2\nB/angbracketleft(Sz)2/angbracketright−/angbracketleftSz/angbracketright2\nNckBT,and\nC=/angbracketleftH2/angbracketright−/angbracketleftH/angbracketright2\nkBT2,wherekBis the Boltzmann constant.\nThe total number of sites N= 2Ncfor the sS chain and\nN= 3Ncfor the ssS chain. In the computation we have\nused periodic boundary conditions, M= 50 for both\nchains and R= 40000 (50000) for the sS (ssS) chain. A\nfull diagonalization study of the specific heat and suscep-\ntibility for the sS chain with Nc= 3 can be found in Ref.\n[21].\nIII. MAGNETIC SUSCEPTIBILITY AND\nMODEL PARAMETERS\nThrough a semiclassicalapproach, in which the S spins\nare treated as classical variables, Seiden [19] derived a\nclosed formula for the magnetic susceptibility χ. In par-\nticular, the quantity Tχ(T) has a minimum at a temper-\natureTminwhich is generally situated in a region where\nβJS <1, a feature which has been known to be typi-\ncal of 1D ferrimagnets. Similarly, a closed expression for\nthe susceptibility of the ssS chain can also be established\n[15].\nIn Fig. 2(a) we present data for the magnetic suscepti-\nbility of the compound CuMnDTO (from Refs. [13, 19])\ntogether with FTLM, with J/kB= 44.8 K andg=\n1.90, and semiclassical-approximation [19] results, with\nJ/kB= 59.7 K andg= 1.9, for the sS chain. For3\n0 100 200\nT(K)024681012χmT(cm3K mol-1)\nExperimental\nFTLM \nSemiclassical \n(a)\n075 150 225 300375\nT(K)0481216χmT (cm3K mol-1)\nExperimental\nFTLM \nSemiclassical \n(b)\nFIG. 2: (Color online) Molar susceptibility χmtimes temper-\natureTof the (a) sS and (b) ssS chains as a function of T.\nExperimental: data of (a) the compound CuMnDTO from\nRef. [13] and of (b) the MnNN compound from Ref. [15].\nFTLM: (a) N= 16,g= 1.90 andJ= 44.8K, (b)N= 18,\ng= 2.0,J/kB= 150 K and J′/kB= 255 K. Semiclassical:\n(a) Ref. [19] (with J/kB= 59.7 K, S= 2.5 andg= 1.9) and\n(b) Ref. [15] (with J/kB= 141 K, J′/kB= 250 K, S= 2.5\nandg= 2.0).\nthe FTLM results, the estimation of Jis made by us-\ning the value of the minimum of the experimental curve:\nTmin= 130K.We seethatboth the FTLM andthe semi-\nclassical approach agree with the experimental data in\nthe mid- and high-temperature regimes. As the temper-\nature is lowered below Tmin,χTincreases and presents a\nmaximum at Tmax= 7.5 K, which marks the onset of the\ntridimensional ordering. For a strictly one-dimensional\nsystem it is expected, from the Mermin-Wagner theo-\nrem [22], that long-range order (LRO) may occur only at\nT= 0. We remark that for quantum ferromagnetic [23]\nchainsthe correlationlength divergesas 1 /Tand the sus-\nceptibility as 1 /T2; further, the low-lying magnetic ex-\ncitations of ferrimagnetic chains present a ferromagnetic\ncharacter (see below) and the same referred critical be-\nhavior is shown [24] to hold, which explains the increase\nin the curve of χTjust belowT=Tmin.\nInFig. 2(b)wepresentFTLMandsemiclassicalresults\n[15] for the ssS chain, and experimental data [15] of the\nMnNN compound. A profile similar to that of Fig. 2(a)\nis observed with Tmin= 255 K. Taking g= 2.0, our\nestimative for the model parameters, JandJ′, areJ′=\n1.7J, withJ/kB= 150 K and J′/kB= 255 K. We also0 1 2 3 4 5 6\nB ( T )04812162024Mm (103 G cm3 mol-1)\nExperimental ( T = 4.2 K)\nFTLM \nBP(a)\n0 10 20 30 40 50\nB ( T )05101520Mm (103 G cm3 mol-1)\nFTLM\nBP(b)\nFIG. 3: (Color online) (a) Molar magnetization Mmof the sS\nchain as a function of BatT= 4.2 K. Experimental data of\nthe compound CuMnDTO from Ref. [13]. FTLM results for\na chain with N= 16,J/kB= 44.8 K and g= 1.93. Brillouin\nparamagnet ( BP) with total spin S= 15.0. (b) FTLM results\nforMmof the ssS chain with N= 18 as a function of Bat\nT= 15 K. BPwithS= 8.5.\nnote that the FTLM curve for this chain depart from the\nexperimental one at a higher temperature than the curve\nfor the sS chain. This behavior is in fact a finite size\neffect since the number of unit cells used in the FTLM\ncalculation for the ssS chain (6 unit cells, with 18 sites)\nis effectively less than the number used for the sS chain\n(8 unit cells, with 16 sites).\nIn Fig. 3 we compare our results at T= 4.2 K with ex-\nperimentaldatafromRef. [13]. Thetemperatureis lower\nthan the one in which the maximum of the χTcurve is\nobserved,Tmax≈7.5 K. However, due to the low value\nof the coupling between chains Jinter-chain/kB∼0.1 K,\nwe expect that at T= 4.2 K and for fields higher than\n∼0.1 T, the ferrimagnetic correlations along the chain\nare the relevant ones to determine the behavior of the\nmagnetization as a function of B. Since the correlation\nlength along the chains diverges [24, 25] as 1 /T, a finite\nnumber of unit cells are correlated at 4.2 K. Thus, we\ncan treat the system as composed of independent lin-\near clusters, each cluster carrying a total spin S, and\na superparamagnetic behavior is expected for the mag-\nnetization curve. Within this context, we try to esti-\nmate the number of correlated cells in the chain from the\nexperimental data shown in Fig. 3(b) by comparing it\nwith the FTLM data and the molar magnetization of a\nBrillouin paramagnet (BP) with total spin S, given by\nMm(B,T) =NAgµB(S−s)BS(x), whereNAis the Avo-\ngadro constant and BS(x) is the Brillouin function. As\nshown in Fig. 3(b), the experimental data is well de-\nscribed by the BP curve with S= 15, indicating that\napproximately 8 unit cells (size used in the FTLM calcu-\nlation) are ferrimagnetically correlated at 4.2 K. This en-\nforces the one-dimensional description of the experimen-\ntalmagnetizationforthistemperatureandfieldvalues,as\nwell as the superparamagneticbehavior. We remark that\nthe authors of Ref. [13] estimate that approximately 10\ncells of the compound CuMnDTO are ferrimagnetically\ncorrelated at T= 7.9 K (just above Tmax) forB= 0.\nFor the ssS system, we find no published experimental\ndata for the magnetization. However, considering the4\n0 1 2 3q050100150200(Energy/gµB) ( T )\nN=20\nN=24\nFree Spin Wavesω+\nω−(a)\n0 1 2 3q0100200300400500600(Energy/gµB) ( T )\nN=18\nN=21ω+\nω−(b)\nFIG. 4: (Color online) ED results for the lower energy one-\nmagnon bands, in units of magnetic field (using g= 1.93), of\nthe (a) sS and (b) ssS chains for the indicated values of N.\nFull lines are non-interacting spin-wave results from Ref. [26],\nwhile dashed lines are guide to the eyes.\nFTLM results, Fig. 3(b), we estimate that at T= 15\nK the number of ferrimagnetically correlated unit cells is\n∼6 (size used in the FTLM calculation), due to the good\nagreement between the FTLM results and the BP curve\nwithS= 8.75.\nIV. ONE-MAGNON BANDS AND SPECIFIC\nHEAT\nDue to the ferrimagnetic order of the GS, there are\ntwo kinds of elementary excitations in the systems: fer-\nromagnetic magnons, which lowers the total spin by one\nunit and AF magnons, which increases the total spin by\none unit. The dispersion relations of the lower energy\nmagnons are calculated, respectively, through\nω−(q) =Emin(SGS−1,q)−EGS (4)\nω+(q) =Emin(SGS+1,q)−EGS,(5)\nwhereEmin(St,q) indicates the lowest energy in the\ntotal-spin sector Stand lattice wavenumber q=\n2πl/Nc,withl= 0,1,2,...,Nc−1.\nIn Figs. 4(a) and 4(b) we display the lower energyone-\nmagnon bands, in units of magnetic field, of the sS and\nssS chains. For the sS chain, we also plot in Fig. 4(a)\nnon-interacting spin-wave (SW) results [26]:\nω−\nSW(q) =−J(S−s)+ωq+gµBB, (6)\nω+\nSW(q) =J(S−s)+ωq−gµBB, (7)\nwhereωq=J/radicalBig\n(S−s)2+4sSsin2(q/2),s= 1/2 andS= 5/2. We notice that in zero field the ferromag-\nnetic excitation is gapless (which is expected from the\nspontaneously broken symmetry of the GS) and display\na quadratic dispersion relation in the long wavelength\nlimit, as predicted by conformal invariance [27], while a\ngap ∆ exists for the AF excitation. The ferromagnetic\nbranch obtained through non-interacting SW theory for\nthe sS chain is in good agreement with the ED data,\nwhile for the AF branch the value of the zero field gap\n∆ = 2J(S−s) = 4Jdepartures from the ED value, as is\noften the case in other ferrimagnetic systems [28], due to\nquantum fluctuations effects. In fact, we estimate that\nin the thermodynamic limit ∆ = 4 .9046J(3.88J) for the\nsS (ssS) chain.\nIn Fig. 5(a) we show the specific heat of the sS and ssS\nchains in zero field. Due to the LRO ferrimagnetic state\natT= 0 =B, with low-energy gapless ferromagnetic ex-\ncitations, it is expected that C∼√\nT. Another feature is\nthe occurrence of double peaks [29–31]; it turns out that\nthe main peak is well described by the Schottky formula\n[29, 30]:C\nNckB=A(δ/2kBT)2\ncosh2(δ/2kBT), whereδis the Schottky\ngap andAis the amplitude parameter. The Schottky\ngap for the ssS chain δssS≈4.1Jis in accord with the\nAF spin-wave gap ∆ ssS≈3.9J. However, for the sS\nchain the value of δsS(≈3.4J) significantly departures\nfromtheAFspin-wavegapvalue: ∆ sS≈4.9J,indicating\nstrong influence of the lower-energy ferromagnetic exci-\ntations. Since these states have a total spin ( St=Sg−1)\nlowerthan the one ofthe AF branch( St=Sg+1), we ex-\npect that a field Bcan wash it out. In fact, the center of\nthe AF branch [see Fig. 4(a)] is found at ¯∆sS(0) = 5.4J\nand is loweredin the presence of a magnetic field through\n¯∆sS(B) =¯∆sS(0)−gµBB. In Fig. 5(b) we present the\nspecific heat for fields up to 103.8 T, and in Fig. 5(c) we\ncompare ¯∆sS(B) with the Schottky gap δsS(B). As we\ncan see in the figure, the values of the two quantities are\nnearly equal for moderate values of B.\nV. T - B PHASE DIAGRAM\nThe GS magnetization per unit cell, mc, of one-\ndimensional systems under an applied magnetic field can\nexhibit plateaus at values such that Sc−mc= integer,\nwhereScis the maximum total spin of a unit cell [7].\nThis condition implies that a plateau can be observed\nat values of mcdiffering from its saturated value by an\ninteger number of spin flips. In particular, a magneti-\nzation plateau at 1/3 of the saturation magnetization\nwas observed in the magnetization curve of the min-\neral azurite [32], which is generally modeled through\nthe distorted diamond chain [33]. Other compounds ex-\nhibiting the 1/3 magnetization plateau are the trimer\nchain systems Cu 2(P2O6OH)2[34] and the phosphates\n[9] A3Cu3(PO4)4, where A = Ca, Sr or Pb. Further,\nthe thermal properties of a variety of models [35–37] pre-\nsenting plateaus in their magnetization curves were ana-\nlyzed in recent years and it was evidenced that the 1/35\n0 2 4 6\nkBT / J00.51C / Nc kB\nssS\nsS0 200.51 (a)\n0 100 200 300\nT(K)05101520Cm (JK-1mol-1) 103.8\n  86.1\n  69.2\n  51.9\n  34.6\n  17.3\n    0.0(b)B ( 104 G )\n050100150\nB ( 104 G )050100150200(Energy/gµB) (10 4G)δsS(B)\n∆sS(B)\n(c)\nFIG. 5: (Color online) FTLM results. (a) Specific heat of the\nsS and ssS chains for B= 0 as a function of temperature T;\nfull lines are the respective Schottky specific heats. (b) Mo lar\nspecific heat Cmof the sS chain for the indicated values of B\n(usingg= 1.93); (c) B-dependent Schottky gap δsS(B) and\nspin-wave gap ¯∆sS(B). Dashed lines are guide to the eyes.\nmagnetization plateau is also a characteristic feature of\nfrustrated spin-S chains [38].\nFor the sS chain studied here, possible plateaus should\nbe observed at mc=mLM=gµB(S−s) [Lieb-Mattis\n(LM) magnetization] and mc=mFP=gµB(S+s)\n(fully polarized magnetization), as confirmed by the nu-\nmerical results shown in Fig. 6 (a). In zero field the\nGS is ferrimagnetic with gapless ferromagnetic excita-\ntions [Eq. (6)], while for B/negationslash= 0 this mode acquires the\ngap ∆ F(B) =gµBB. Also, as Bincreases, the gap\nfor the AF mode [Eq. (7)] decreases linearly with B:\n∆AF(B) = ∆ sS−gµBB, and forgµBB≥gµBBm=\n(∆sS/2)≈2.45Jits gap is equal to the ferromagnetic\none. AtgµBB= ∆sS≡gµBBc,AFthe AF gap van-\nishes and the system undergoes a quantum phase transi-\ntion (condensation of AF magnons , each carrying a spin\n+1) to a gapless Luttinger Liquid (LL) phase [40], with\npower-law decay of the transverse correlation functions.\nIn fact, the quantum critical point Bc,AFseparates an\nincompressible phase (plateau) from a compressible one\n(LL phase). For B/greaterorsimilarBc,AF, a low-density of magnons\nis found in the system and the asymptotic singular form\nof the magnetization can be obtained [41] by considering\nthe system as a free Fermi gas or hard-core bosons. In\nthis limit, the magnons will occupy single particle states\nwithq→0 and the dispersion relation, Eq. (7), can be\nused by replacing the linear spin-wave gap, ∆ SW, by the\ncomputed gap in Fig. 4(a), ∆ sS= 4.9046J:\nω+\nAF=−µ+v2\n2∆sSq2, q→0, (8)\nwherev=J√\n2sS=J/radicalbig\n5/2,µ=gµBB−∆sS=0 2 4 68 10\ngµBB/J23mc/gµB\nEq. (11)\nEq. (13)(a)\nLM PlateauFP Plateau\n  LL\nPhaseB = Bc,AFB = Bc,FP\n0 2 4 68\ngµBB/J0123\n0.10\n0.20\n0.30\n0.40\n1.00\n2.00kBT / J mc/gµB\nJχ/g2µ2\nB(b)\nFIG. 6: (Color online) FTLM results for the sS chain. (a)\nGS magnetization per cell mcas a function of BforN= 16:\nfull circles indicate the midpoints in the steps of the magne ti-\nzation of the finite-size system and edges of thermodynamic-\nlimit plateaus [39], while colored full lines indicate the r esults\nfrom the free-fermion model (see text). (b) mc(curves with\nsymbols) and χ(full curves) as a function of Bfor the listed\nvalues of T;Bmis indicated by the red diamond.\ngµB(B−Bc,AF) and ∆ SW= 2J(S−s). The energy\ndensity can thus be written (fermionic map) as\nε=/integraldisplaykF\n−kFdk\n2π(ǫk−µ), (9)\nwhereǫk=v2k2/2∆sS,kF=πn, andnis the density\nof particles. The value of nfor a prescribed µcan be\nobtained from the condition ∂nε= 0:\nn=√gµB/radicalbigg\n2Bc,AF\nπ2v2√µ, (10)\nwhich implies\nmc\ngµB= 2+gµB\nJ/radicalbigg\n4Bc,AF\n5π2/radicalbig\nB−Bc,AF.(11)\nIn Fig. 6(a) we show the very good agreement between\nthe numerical data and mcgiven by Eq. (11) in the LL\nPhase.6\nFIG. 7: (Color online) FTLM results for the low- Tphase\ndiagram of the sS chain: contour plot indicates the magneti-\nzation per cell mc. The critical point at B= 0 (black dia-\nmond), the inflection point of mcatgµBB=gµBBm= ∆/2\n(red diamond), the quantum-critical points (black triangl es)\natB=Bc,AFandB=Bc,FP, crossover lines (white circles)\nand their asymptotic behavior (full lines) are also indicat ed.\nThe gapless LL phase ends at the quantum critical\npointB=Bc,FP: the system becomes fully polarized\n(FP) and presents gapped low energy excitations. The\ntwo one-magnon excitations from the FP state, both car-\nryingaspin-1, canbe exactlyobtained[42]andthe lower\none has a dispersion relation given by\nωFP=−J(s+S)−J/radicalbig\n(S−s)2+4sScos2(q/2)+gµBB,\n(12)\nwhich implies gµBBc,FP= 2J(s+S) = 6J, in accord\nwith the numerical results [Fig. 6(a)]. For B/lessorsimilarBc,FP, a\nlow density of magnons is observed in the system and\nthe same arguments used to obtain Eq. (10) can be\nused in this case. For q→0, Eq. (12) can be writ-\nten as Eq. (8) with v=J√\n2sS=J/radicalbig\n5/2, ∆F=\n2J(s+S) = 6Jandµ=gµB(Bc,FP−B), which implies,\nfrom Eq. (10), that the density of magnons is given by\nn=gµB\nJ/radicalBig\n4Bc,FP\n5π2/radicalbigBc,FP−B, andmcnow reads:\nmc\ngµB= 3−gµB\nJ/radicalbigg\n4Bc,FP\n5π2/radicalbig\nBc,FP−B,(13)\nwhich is plotted in Fig. 6(a) and is also in very good\nagreement with the numerical data.\nInFig. 6(b) wepresentFTLM datafor mcandχvs.B\nforT/negationslash= 0. We first notice that the magnetization in zero\nfieldisnullandthesystemisinthethermalparamagnetic\nstate, as expected from the Mermin-Wagner theorem\n[22]. Increasing Binthelow-temperatureregime,theLM\n(or ferrimagnetic) plateau is exponentially reached and\nthe magnetizationexhibitsan inflectionpoint at B=Bm\n[red diamond in Fig. 6(b)] that marks the changing ofthe gapped low-energy excitations from ferromagnetic\n(B≤Bm) to AF magnons ( B≥Bm): the ferromag-\nnetic (antiferromagnetic) magnons are exponentially ac-\ntivatedand mcislower(higher)than mLM=gµB(S−s).\nAlso, by the same token, the FP plateau is exponentially\nreached from below for fields higher than Bc,FP. Fur-\nthermore, the singular form of the magnetization near\nthe quantum critical points ( B=Bc,AFandB=Bc,FP,\nwithT= 0), which implies χ→ ∞, are thermally\nsmoothed out and the singularities in the susceptibility\nevolve into local maxima, thus providing the determina-\ntion of the crossover lines. The LL phase, with linear\ndispersion relation ∼q, is expected [40] between the two\nlocal maxima for a given T[see, e. g., the susceptibility\ncurves forkBT= 0.10Jand 0.20Jin Fig. 6(b)] with\nthe two local maxima indicating a crossover to a region\nin which the excitations follow a non-relativistic disper-\nsion relation ∼q2, as previously discussed. On the other\nhand, asTincreases, the LL phase ends and a single\nmaximum is observed in the susceptibility curves (see,\ne. g., the susceptibility for kBT= 0.40J). This single\nmaximum defines a crossover from the regime in which\nthe physics is determined by the excitations from the LM\nplateau to a regime in which the FP plateau is the rele-\nvant one. For sufficiently high temperatures, the system\nlooses all information about the T= 0 LM magnetiza-\ntion plateau and the effect of Bis to bring the system\nfrom the thermal paramagnetic state to the FP state at\nhigher magnetic fields (see the case kBT= 2.00J).\nIn Fig. 7 we present the contour plot of mcin the\nT−Bplane and a schematic phase diagram. The T−B\ncrossover lines enclosing the region of the LL phase, lim-\nitedatT= 0byB=Bc,AFandB=Bc,FP, areobtained\n[43] from the local extrema of mc(T) vs.Tfor a given\nB, as shown in Fig. 8(a). Further, as B→Bcthese\ncrossover lines follow a universal function [43]: a|B−Bc|\nwitha= 0.76238; as shown in Fig. 7, our numerical data\nconfirm this asymptotic behavior for the two quantum\ncritical points at B=Bc,AFandB=Bc,FP. More-\nover, asTincreases beyond the crossover lines of the\ntwo plateaus, gapless phases are reached [40, 44]. In\naddition, by increasing Bunder a fixed T, local max-\nima are observed in the specific heat C(B) per spin, as\ndisplayed in Fig. 8(b). These features are used to es-\ntimate [5] the crossover lines related to the LM plateau\nand FP plateau shown in Fig. 7; in particular, we notice\nthatT∼ |B−Bc|as the lines reach the corresponding\nquantum critical points [40, 44]. Last, we stress that the\ncrossover lines and the LL instability lines meet at the\nquantum critical points, thus delimiting the respective\nquantum critical region [40, 43–45]; in each region the\nsystem is thus governed by the quantum critical point\nwith dynamical exponent z= 2 associated with the ex-\ncited magnons, as discussed above. On the other hand,\nthe magnon densities [40], n, given by ( mc/gµB)−2\nand 3−(mc/gµB) for the quantum critical point at\nB=Bc,AFandB=Bc,FP, respectively, follow a univer-7\n00.5 11.52\nkBT/J22.5mc/gµB(a)\n0 2 4 6 8 10\ngµBB/J00.51C/NckBkBT/J(b)\n0.30\n0.40\n0.50\n0.60\n0.70\n0.80\n0 1 2\ngµB|B-Bc,AF| / kBT00.51n=[(mc/gµB)-2]/(kBT/J)1/2\n0.15\n0.25\n0.35\n0.45\n0.55\n0.65\n0.75\n0.85\n0.95kBT / J(c)\nB < Bc, AFB > Bc, AF\n0 1 2\ngµB|B-Bc,FP| / kBT00.51n=[3-(mc/gµB)]/(kBT/J)1/2(d)\nB < Bc,FP\nB > Bc,FP\nFIG. 8: (Color online) FTLM results for the sS chain. (a)\nMagnetization percell mcas afunctionof Tfor fields gµBB/J\nfrom 4.85 to 5.95 in steps of 0 .05 (from below to top); trian-\ngles indicate local maxima associated with the LL crossover\nlines (see text). (b) Specific heat per cell Cas a function\nofBfor the indicated values of temperature. Scaling of the\nmagnon density naround the quantum critical points at (c)\nB=Bc,AFand (d)B=Bc,FP.\nsal function of Tand|B−Bc|/T:\nn=/radicalbigg\nkBT\nJf/parenleftbigg|B−Bc|\nT/parenrightbigg\n, (14)\nas shown in Figs. 8 (c) and 8 (d). A better scaling\nbehavior is observed for B <B c,AF(B >B c,FP) in Fig.\n8 (c) [8 (d)] since for B > B c,AF(B < B c,FP) the zone\nof influence of the quantum critical point at B=Bc,FP\n(B=Bc,AF) merges with the zone of influence of the\npoint atB=Bc,AF(B=Bc,FP). The guideline kBT=\ngµBBin Fig. 7 is discussed below.\nIn Fig. 9 we present the contour plot of C/Tin the\nT−Bphase diagram [5], including the above-discussed\ncrossover lines. At the plateaus, C/T→0 asT→0\ndue to the gaps, as evidenced in the plot. As we canFIG. 9: (Color online) FTLM results for the low- Tphase\ndiagram of the sS chain: contour plot indicates C/T. The\ncritical points and crossover lines are indicated as in Fig. 7.\nsee, the guideline kBT=gµBBdo not coincide with\nthe local maxima of C(B) in the low- Bregion [see Fig.\n8(b)] due to the LRO ferrimagnetic state at T= 0 =\nB: sinceC∼√\nT,C/T→ ∞asT→0 atB= 0\nand an enhancement in the intensity of C/Tis observed\nnearT= 0 =B. In spite of this fact, the plot shows\na depression in the values of C/Tnear theT= 0 LM\nplateau which, by increasing T, varies in a symmetrical\nfashion with respect to B=Bm(dome-shaped) and is\nlimited by the kBT=gµBBandkBT=gµB|Bc,AF−B|\nasymptotic crossover lines. Further, the LL dome is also\nclearly seen and the crossover lines of the FP and LM\ngapped phases can be visualized.\nNext, we exhibit in Fig. 10 the magnetization of the\nssS chain at T= 0. For this chain, the first plateau is\nfound atmc=gµB(S−2s), i. e., the LM plateau, and\nthesecondisthe FPplateauat mc=gµB(S+2s); theLL\nphaseis expected to occurbetween these twoplateaus. A\nthird plateau could be found [7] at mc=gµBS; however\nour numerical shows no evidence of this plateau. We\nremark that we did not perform a detailed analysis of\ntheT−Bphase diagram of this chain, but we expect\nthat it should display similar features already reported\nfor the sS chain.\nThehugevaluesofthe quantumcriticalmagneticfields\nofthe CuMnDTO (sS chain)and MnNN(ssS chain)com-\npounds, make the experimental investigation of the full\nT−Bphasediagramofthesesystemsverydifficult. How-\never, magnetic phase transitions induced by very large\nmagnetic fields (up to 400 T) in the low-temperature\nregime have been reported [46]. Further, materials phys-\nically described by similar models may have lower values\nfor the exchange coupling and thus a more experimen-\ntally accessible phase diagram.\nWe also mention that ferrimagnetism can be destabi-\nlized by competing (or frustrating) interactions [47, 48],8\n0 2 4 68\ngµBB/J1234mc/gµB\nLM PlateauFP Plateau\n  LL\nPhaseB = Bc,AFB = Bc,FP\nFIG. 10: (Color online) FTLM results for the magnetization\nper cellmcof the ssS chain as a function of field BatT= 0\nforN= 18. Full circles indicate the midpoints in the steps\nof the magnetization of the finite-size system and edges of\nthermodynamic-limit plateaus [39].\n00.25 0.5 0.75 1\nkBT / J01/240.10.20.30.40.5χT2kB2 / Jg2µB2\nFTLM, N=8\nFTLM, N=24 \nFitting\nMSW00.050.1\nkBT / J01/240.1\nFIG. 11: (Color online) χT2vs.Tfor the spin-1/2 ferromag-\nnetic chain in the low and very low ( kBT << J) temperature\nregions. FTLM results for N= 8 and N= 24. Fitting of the\nFTLM results for N= 8 using [1\n24+a0(kBT\nJ)1\n2+a1(kBT\nJ)]\n(see text). MSW results up to second order in kBT/J. The\ninset shows the results for kBT≤0.1J.\nwhich can give rise to other critical points. Unconven-\ntional ferrimagnetism (non-bipartite lattices) was indeed\nfound in one-dimensional frustrated structures [49, 50]\nand in the Kagom´ e lattice [51]. Further, the mag-\nnetocaloric effect in the kinetically frustrated diamond\nchain was recently investigated [52].\nVI. LOW-TEMPERATURE MAGNETIC\nSUSCEPTIBILITY\nWe now consider the temperature regime where ferro-\nmagnetic excitations tend to be a predominant feature.\nIn order to test and illustrate the accuracy of the FTLM\nin describing the susceptibility behavior at very low tem-\nperatures, we have calculated the susceptibility of the\nspin-1/2 linear ferromagnetic chain; the results for χT200.511.52\nkBT / J012345χT2 kB2/ Jg2µB2\nExperimental ( J/kB= 59.7 K, g = 1.9)\nExperimental ( J/kB = 44.8 K, g = 1.88)\nFTLM, N = 16\nFitting\nMSW\nSemiclassical\nFIG. 12: (Color online) χT2vs.Tfor the sS chain. Exper-\nimental data of the compound CuMnDTO from Ref. [13].\nFTLM results for N= 16. Fitting of the FTLM results for\nN= 16 using [5\n3+a0(kBT\nJ)1\n2+a1(kBT\nJ)] (see text). MSW\nresults up to second order in kBT/Jfrom Ref. [29]. The\nsemiclassical results are from Ref. [19].\nas a function of Tare shown in Fig. 11 for systems with\n8 and 24 sites. The crossoverto zeroof the FTLM results\nasT→0 is due to finite size effects. We note that for\n(kBT/J)/greaterorsimilar0.3 the curves for the two chain sizes super-\nimpose, thus suggesting that the thermodynamic-limit\nbehavior has already been within numerical accuracy.\nAlso, in the temperature range 0 .06/lessorsimilar(kBT/J)≈0.1,\nthe results for the larger system is in good agreement\nwith the expansion formula from Takahashi’s MSW the-\nory [23], which up to second order in t≡kBT/Jreads:\nχJ\n(gµB)2=t−2/bracketleftBig\n2\n3s4−21\n2s5\n2At1\n2+sA2t+O/parenleftBig\nt3\n2/parenrightBig/bracketrightBig\n, where\nA=ζ(1\n2)/√\n2π≈ −0.582597 and g= 2. Fors= 1/2 we\nobtain\nχJ\n(gµB)2=t−2/bracketleftbigg1\n24+0.145649t1\n2+0.16971t+O/parenleftBig\nt3\n2/parenrightBig/bracketrightbigg\n(15)\nWe stress that in the range 0 <(kBT/J)<0.1, Eq.\n(15) is in very good agreement with predictions from\nthe Bethe-ansatz approach [53], while the fitting of the\nFTLMresultsfor N= 8and0.5<(kBT/J)<0.9,yields\na0= 0.140 anda1= 0.186, in good agreement with the\nMSW coefficients.\nWe now turn our attention to the low-temperature\nregime of the sS-chain susceptibility displayed in Fig. 12.\nFirstly,wenotethatfor( kBT/J)/greaterorsimilar0.5theFTLMresults\nforN= 14(not shown)and N= 16(8 cells) coincide, in-\ndicating that the thermodynamic limit has been attained\nin this temperature range. The experimental data nor-\nmalized by J/kB= 44.8K (g= 1.88)andJ/kB= 59.7 K\n(g= 1.9) show the expected agreement with the FTLM\nresults and the semiclassical formula, respectively, as al-\nready displayed in Fig. 2(a). The MSW results comes\nfrom the expansion formula derived by Yamamoto et al.9\n[29], which up to second order in treads:\nχJ\n(gµB)2=t−2/bracketleftbiggSs(S−s)2\n3−(Ss)1\n2(S−s)3\n2At1\n2\n+(S−s)A2t+O/parenleftBig\nt3\n2/parenrightBig/bracketrightBig\n. (16)\nA relevant aspect of this expansion is that for S= 2s\nwerecovertheTakahashiexpansionforthe ferromagnetic\nlinear chain of spin s, which reinforces that the ferromag-\nnetic excitation is the relevant one at low temperatures\n[29]. Setting s= 1/2 andS= 5/2 in Eq. 16, we obtain\nχJ\n(gµB)2=t−2/bracketleftbigg5\n3+1.842334t1\n2+0.678840t+O/parenleftBig\nt3\n2/parenrightBig/bracketrightbigg\n,\n(17)\nThe FTLM results can be fitted by a function of the\nform [5\n3+a0(kBT\nJ)1\n2+a1(kBT\nJ)]. Guided by our studies\non the spin-1/2 ferromagnetic chain, we have chosen the\ninterval 0.5≤(kBT/J)≤0.9 to fix the values of a0and\na1:a0= 1.28 anda1= 0.69, which can be compared\nwith those in Eq. (17), and implies a good agreement for\nthe integer-power coefficient and an order-of-magnitude\nagreement for the half-integer power coefficient.\nOne should notice that the FTLM results and the ex-\nperimental data crossover to zero as ( kBT/J)→0, in-\nstead of approaching the constant value sS(S−s)2/3 =\n5/3. Here one must distinguish two effects: with re-\nspect toFTLM, thisis evidentlyamanifestationoffinite-\nsize effects, while for the experimental data one can at-\ntribute this to the 1D/3D crossover that takes place be-\nlowT= 7.5 K (see Sec. III). In fact, in the 3D region\nthe susceptibility behaves as χ∼T−γ, with the critical\nexponentγ <2, implying that χT2→0 asT→0.\nVII. SUMMARY AND DISCUSSION\nWe have presented a thorough numerical study of\nthe GS and thermodynamic properties of two one-dimensional models related to quasi-one-dimensional fer-\nrimagnetic compounds: CuMnDTO and MnNN. In fact,\nthe models are associated to two types of ferrimagnetic\nchains: the alternating spin-1/2 spin-5/2 chain and the\nspin-1/2 spin-5/2 alternating chain with a spin-1/2 pen-\ndant attached to the spin-5/2 site. The finite temper-\nature Lanczos method proved quite reliable, except at\nvery low temperatures where finite-size effects hinder its\naccuracy. A particular feature of these systems is the\npresence of gapless ferromagnetic and gapped AF spin-\nwave (magnon) branches in zero field. As the magnetic\nfield is increased, the low-energy excitation changes from\nferromagnetic to AF and the magnetic field vs. tem-\nperature phase diagram displays characteristic crossover\nlines which distinguish these systems from spin-1 Hal-\ndane chains and two-leg ladder models. In particular,\nfor the sS chain we have identified the quantum criti-\ncal points and the crossover lines, the Luttinger liquid\nphase, the ferrimagnetic (LM) and the fully polarized\nplateaus. The values of the exchange coupling param-\neters of the compounds discussed in the text are in-\ndeed very high. However, magnetic phase transitions\ninduced by very large magnetic fields (up to 400 T) in\nthe low-temperature regime have been experimentally\ninvestigated[46]. Also, other compounds described by\nsimilar models can have lower values of the exchange\nparameters and a more experimentally accessible phase\ndiagram. We expect that this work stimulates experi-\nmental and theoretical research with focus on the phase\ntransitions induced by an applied magnetic field in the\nlow-temperature regime of the large class of quasi-one-\ndimensional ferrimagnetic compounds.\nVIII. ACKNOWLEDGMENTS\nThisworkwassupportedbyCNPq,FACEPE,CAPES,\nand Finep (Brazilian agencies).\n[1] T. Giamarchi, Quantum physics in one dimension (Ox-\nford University Press, USA, 2004).\n[2] N. B. Ivanov, J. Schnack, R. Schnalle, J. Richter,\nP. K¨ ogerler, G. N. Newton, L. Cronin, Y. Oshima, and\nH. 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Nagar, Kalapet, Pondicherry-605014, India\nThe present report is dedicated to show that ferromagnetic L a0.67Ca0.33MnO3\n(LCMN) particles can be better described in the framework of ferrimagnetic model.\nTo confirm the ferrimagnetic signature in ferromagnetic LCM N particles, the tem-\nperature dependence of the inverse of magnetic susceptibil ity in the paramagnetic\nstate of the samples was taken as a tool of data analysis. The o bserved ferrimag-\nnetismisunderstoodasaneffectofofthecore-shellspinstru ctureinLCMNparticles.\nKey Words: A. Ferromagnetic nanoparticle; B. Mechanical Mi lling; C. Ferrimag-\nnetism; D. Core-shell spin structure\nPACS:75.47.Lx,75.30.Cr,75.50.-y,75.50.Gg,75.50.Tt\nI. INTRODUCTION\nMagnetic nanomaterials are continued to be at the center of curre nt research interests\ndue to their huge technological applications and incomplete undesta nding of many discov-\nered phenomena. For example, superparamagnetic blocking of mag netic moments below\nthe conventional paramagnetic to ferromagnetic transition temp erature (T C), appearance of\nunconventional spin glass behaviour at lower temperatures, decr ease of effective magnetic\nmoment of the material, exchange bias effect, quantum tunnelling of magnetization, and low\nfield magnetoresistance have been observed when the particle size of ferromagnetic materi-\nals decreases into nanosize dimension (below 100 nm)[1, 2]. Various me chanisms have been\nintroduced in literature to describe the magnetic features of nano particles, e.g., core-shell\n∗E-mail address for correspondence:\nrnbhowmik.phy@pondiuni.edu.in2\nstructure, dipole interactions, inter-particle interactions, exch ange anisotropy [3]. Among\nthe proposed mechanisms, the core-shell concept is world wide acc epted to explain the fea-\ntures of nanoparticle magnetism. In a magnetic nanoparticle the ce ntral part, known as\ncore, is assumed to be identical to the structure and property of bulk material with mi-\ncron sized particles. The structure and property of the outer pa rt of the particle, known\nas shell, are drastically different in comparison with core [3, 4]. If the b ulk material is a\ntypical long ranged ferromagnet (antiferromagnet), then core is assumed to be long ranged\nferromagnet(antiferromagnet) and disorder is introduced in the shell part of the particle.\nThis means the property of a magnetic nanoparticle is basically heter ogeneous in charac-\nter (i.e., consisting of two different magnetic components or equivale nt to two magnetic\nsublattices) over a length of particle dimension and also in the whole dim ension of the ma-\nterial when the particles are in contact. The common phenomena du e to the heterogeneous\nmagnetic structure in ferromagnetic nanoparticles are the reduc tion of particle moment and\nmagnetic blocking/freezing at lower temperatures. On the other h and, antiferromagnetic\nnanoparticles have shown many enhanced properties mainly due to d ifferent magnetic struc-\nture of shell part in comparison with bulk counter material [4]. This s hows that core-shell\nstructure plays an important role in the properties of magnetic mat erials, immaterial of\nferromagnetic or antiferromagnetic particles. Hence, proper un derstanding of the effects\nof core-shell structure is not only the long standing problem, but a lso useful in designing\nthe application oriented materials. To understand the effects of co re-shell structure in dif-\nferent types heterogeneous magnetic structures, e.g., ferrom agnetic core is surrounded by\nantiferromagnetic/paramagnetic/ferrimagnetic shell or antifer romagnetic core is surrounded\nby ferromagnetic/ferrimagnetic shell have been synthesized and reported in literature [5–8].\nThe effect of shell disorder and spin frustration has also been discu ssed in many spin-bilayer\nmagnetic systems [3]. G. Bouzerar et al. [9] discussed the effect of competition between\nintroduced superexchange (antiferromagnetic) interactions in lo ng ranged double exchange\nferromagnetic matrix. They argued that in the lower limit of anitiferr omagnetic superex-\nchange interactions the long ranged ferromagnetic state is not alt ered significantly; rather a\ncanted ferromagnetic phase or induced new magnetic phase is appe ared in the spin system.\nThe induced magnetic phases may be either stable or unstable depen ding on the quantum\nof magnetic disorder and frustration. Some report also studied co re-shell structure in a\ncomposite material consisting of ferrimagnetic core and ferroelec tric shell [10].3\nRecently, La 0.67Ca0.33MnO3nanoparticles in crystalline and amorphous structural phases\nhave shown many interesting magnetic properties, related to magn etic disorder at core-\nshell structure of the particles [11]. A proper knowledge of magnet ic interactions between\ncore-shell spins would be useful not only to realize the colossal mag netoresistance and inter-\ngrain tunneling of polarized spins, but also relevant to realize the effe ct of disorder on\ndouble exchange ferromagnetism in manganites. In the present wo rk, we demonstrate that\nthe modified magnetism in ferromagnetic La 0.67Ca0.33MnO3nanoparticles is identical to\nthe typical features of ferrimagnetic materials. The evidence of f errimagnetic signature in\nLa0.67Ca0.33MnO3nanoparticles is also discussed by comparing the features already o bserved\nin ferrimagnetic (Mn 0.5Ru0.5Co2O4and MnCr 2O4) particles.\nII. EXPERIMENTAL\nDetails of the sample preparation of La 0.67Ca0.33MnO3(perovskite) particles and their\ncharacterization have been reported elsewhere [11]. In brief, the polycrystalline bulk\nLa0.67Ca0.33MnO3sample was prepared by solid state sintering (maximum temperature\n13800C) method. The bulk sample was subjected to mechanical milling in Argo n atmo-\nsphere upto 200 hours using Fritsch Planetary Mono Mill ”Pulveriset te 6” to synthesize\nthe material in nanocrystalline and amorphous phase. The structu ral phase of the samples\nwas confirmed from room temperature XRD spectrum. The XRD spe ctrum indicated that\ncrystalline nature of the material decreases significantly for the m illing time more than 61\nhours and amorphous phase dominates in the spectrum for milling time more than 98 hours.\nBoth bulk and milled samples (upto mh98) are in similar crystallographic p hase and found\nto be matching with orthorhombic structure with Pnma space group . The temperature\ndependence of magnetization under zero field cooled condition was m easured using SQUID\nmagnetometer (MPMS-Quantum Design, USA). The temperature d ependence of dc mag-\nnetization at 100 Oe in the temperature range 100 K to 400 K was also reproduced using\nvibrating sample magnetometer (Lakeshore 7404 model).4\nIII. EXPERIMENTAL RESULTS\nDetails of the temperature and field dependence of dc magnetizatio n have been reported\nelsewhere [11]. In summary, the paramagnetic to ferromagnetic Cu rie temperature (T C)\nfor bulk (LCMN) sample is nearly 281 K and T Cdecreases to 262 K, 250 K, 238 K, 225\nK and 212 K for mechanical milled mh25(nanocrystalline, particle size ∼65 nm), mh61\n(nanocrystalline, particle size ∼12 nm), mh98 (nanocrystalline+amorphous, particle size ∼\n16 nm), mh146 (amorphous, particle size ∼60 nm) and mh200 (amorphous, particle size\n∼90 nm) samples. At the same time, the long ranged ferromagnetic or der (spontaneous\nmagnetization ∼3.6µB) of bulk LCMN sample decreases to 2.17, 0.87, 0.35, 0.17, 0.10\n(inµB) unit) for mh25, mh61, mh98, mh146 and mh200 samples, respectiv ely. These are\nsome typical features of the magnetic disorder effect in ferromag netic materials. In the\npresent paper, we would like to show some specific magnetic feature s of thesamples based on\nmagnetizationdataat(highertemperature)paramagneticregime . InFig. 1, thedcmagnetic\nsusceptibility ( χdc= M/H) of bulk LCMN sample sharply increases above the magnetizatio n\npeak temperature T p∼260 K. On the other hand, magnitude of susceptibility, as well as\nsharp increase of magnetization below the respective T Csystematically decreases for mh25,\nmh98 and mh200 samples. The decrease of the χ(T) variation in milled samples reflects\nthe increasing magnetic disorder in the ferromagnetic material and realized in the previous\nwork [11]. Interestingly, a typical ferrimagnetic sample, e.g., Mn 0.5Ru0.5Co2O4(RuMn)\nspinel oxide in the inset of Fig. 1, also exhibits the similar χdc(T) behaviour above its\nmagnetization peak temperature T p∼100 K. This means only the shape of χdc(T) curve\nin the paramagnetic state can not determine the nature of magnet ic order in the samples,\nwhether ferromagnet or ferrimagnet. The nature of magnetic or der can be confirmed in\nconvincingmannerfromthetemperaturedependenceoftheinver seofmagneticsusceptibility\nintheparamagneticstate. Forthispurpose, we extended thedcm agnetizationmeasurement\nup to 400 K. First, we confirm the difference of the temperature de pendence of the inverse of\nsusceptibility curve inparamagnetic regime between bulk LCMN (ferr omagnetic) andRuMn\n(ferrimagnetic) samples. Fig. 2 shows that the inverse of dc susce ptibility ( χ−1\ndc= H/M)\ndata for bulk LCMN sample at high temperatures (T ≥300 K) are fitted with a simple\nCurie-Weiss law:\nχ=C/(T−θw) (1)5\nApplication of this equation confirms the ferromagnetic order in bulk LCMN sample. The\nobtained parameters are Curie constant (C ∼0.0196) and paramagnetic Curie temperature\n(θw∼+ 270 K). In contrast, the inverse of dc susceptibility ( χ−1) for Mn 0.5Ru0.5Co2O4\nspinel oxide at high temperatures is fitted with a typical equation:\n1/χ= (T−θ1)/Ceff−ξ/(T−θ2) (2)\nIn general, this equation is applicable for ferrimagnet [12]. The obta ined parameters ( θ1\n∼-1320 K, C eff∼0.076,ξ∼200850,θ2∼+ 112 K), in particular the positive value of\nθ2(slightly larger than T C∼100 K) and a high negative value of θ1, clearly indicate the\nferrimagnetic order in Mn 0.5Ru0.5Co2O4spinel oxide. Similar ( χ−1\ndc(T))character was also\nnoted in many other ferrimagnetic materials [12, 13].\nNow, we analyze the temperature dependence of the inverse of dc susceptibility data for\nmechanical millled nanoparticle samples. As shown in Fig. 3, the data ar e well fitted with\na simple Curie-Weiss law (equation (1)) above 330 K. The fit paramete rs of Curie-Weiss\nlaw (C and θw) are shown in Table I. On the other hand, the hyperbolic shape of χ−1(T)\ncurves (with down curvature) above the Curie temperature of th e samples suggests that\nmilled samples belong to the class of either ferrimagnet or double exch ange ferromagnet\n[14]. We noted that the χ−1(T) curves of the present nanoparticle samples are identical to\nthe ferromagnetic MnCr 2O4nanoparticle samples [13]. To clarify the ferrimagnetic nature\nof the nanoparticle (NP) samples, we have fitted the χ−1(T) data in the temperature range\n330 K-400 K using equation (2). We followed a non-linear curve fitting method. Initially,\nthe parameters ( θ1, Ceff,ξandθ2) were allowed to take initial values and iterated 10 times.\nAs soon as the fitted curve comes close to the experimental curve , we start to restrict the\nparameters one by one. Finally, best fit curve was obtained by allowin g all parameters to\nvary, except θ2keeping constant. The experimental data in the paramagnetic sta te of the\nsamples fitted with equation (2) are shown in Fig. 3. The fit on suscep tibility data in the\nparamagnetic regime according to equation (2) is excellent. The fit p arameters are shown\nin Table I. A comparative fits applying equation (1) and (2) for mh98 a nd mh200 samples\nsuggests that equation (1) may be well valid at higher temperature , but equation (2) is more\nappropriate to describe the magnetic behaviour over a wide temper ature range above T C.\nWe noted using equation (1) that the paramagnetic Curie temperat ure (θw) systematically\ndecreases asthematerial transformsfrombulk polycrystalline ph ase tonanocrystalline (NC)6\nphase and then, to amorphous (AMP) phase. The θwvalues remained positive for bulk as\nwell as mh25, mh61 and mh98 samples, where as θwbecomes negative for mh146 and mh200\nsamples. The negative value of θwindicates the introduction of antiferromagnetic exchange\ninteractions in the material as the particle size and crystalline phase changed. As discussed\nin earlier report [11], the magnetic dynamics of the present material strongly depends on\nthe structural phase transformation, rather than the particle size effects. The θ1(obtained\nusing equation (2)) also follows the pattern of θw, showing positive values only for mh25 and\nmh61 samples. The spin glass like feature in amorphous (mh146) samp le clearly proves the\nthe reduction of ferromagnetic (FM) exchange interactions or de velopment of antiferromag-\nnetic (AFM) exchange interactions in nanocrystalline and amorphou s samples, because spin\nglasslike featureneeds sufficient amount of bothmagnetic disorder andcompetitionbetween\nFM/AFM interactions. On the other hand, θ2is always positive and change is not drastic\n(within 9 K considering all milled samples). The positive value of θ2suggested the retain-\ning of a strong double exchange ferromagnetic interactions [14] b oth in nanocrystalline and\namorphous phase of La 0.67Ca0.33MnO3nanoparticles [11]. At the same time, application of\nequation (2) suggests the ferrimagnetic character of mechanica l milled nanoparticle LCMN\nsamples. Similar magnetic behaviour was also observed in ferrimagnet ic MnCr 2O4nanopar-\nticles [13]. Some reports [15, 16] also attempted to explain the magne tization data in the\nparamagnetic regime of ferromagnetic nanomaterials by following a s imple Curie-Weiss law\n(equation (1)), but those data seem to be more appropriate to th e ferrimagnetic description\n(equation (2)).\nThe validity of ferrimagnetic equation (2) in our milled samples can be ex amined by con-\nsidering the core-shell spin structure of nanoparticles, already p roposed in earlier work [11].\nThe existence of strong ferromagnetic order, even in the nanocr ystalline and amorphous\nphase, is essentially due to ferromagnetic ordered core spins. On t he other hand, magnetic\ndisorder is confined mainly in the shell part for nanocrystalline partic les (NCR NP) and\nalso introduces in the core part for amorphous nanoparticles (AMP NP). The shell spins\nmay not be typical antiparallel with respect to core, but effective s pin moment of shell\nis obviously low in comparison with ferromagnetic core and schematica lly shown in Fig.4.\nSimilar magnetic modulation was previously proposed for antiferroma gnetic nanoparticle [4]\nand later applied for ferromagnetic manganite nanoparticles [17]. Th is allows us to consider\nthe magnetic contributions form shell and core of a nanoparticle eq uivalent to two unequal7\nmagnetic sublatticles (shown in lower diagram of Fig. 4), as usually see n in a typical long\nranged ferrimagnet. It must be differentiated that two different m agnetic sublattices as we\nsuggest here for the ferromagnetic particles is not due to differen t crystal environments,\ni.e., tetrahedral and octahedral lattice sites of a typical ferrite c onsisting of two magnetic\nsublattices of antiparallel directions [12, 13]. Based on the experime ntal observations, the\nconcept of two different magnetic structure could be a realistic app roach for describing the\nmagnetic properties of ferromagnetic nanomaterials. Recently, s imilar concept was modelled\nby C.R.H. Bahl et al. [5] and M. Vasilakaki et al. [6].\nIV. CONCLUSIONS\nLa0.67Ca0.33MnO3ferromagnet exhibited many interesting features in the nanocrys talline\nand amorphous phase, as an effect of increasing disorder in core-s hell spin morphology\nand lattice structure. The present work clearly provides the evide nce of ferrimagnetic\ncharacter in ferromagnetic La 0.67Ca0.33MnO3(LCMN) nanoparticles. The ferrimagnetic\nconcept, as propsed in this work, is interesting and could be applied f or the understanding\nof basic mechanism in many ferromagnetic nanoparticles. Especially, this approach could\nbe more effective for the proper demonstration of the effect of co re-shell spin structure in\nferromagnetic nanoparticles.\nAcknowledgement: We thank A. Poddar of Saha Institute of Nuclear Physics and CIF,\nPondicherry University for magnetic measurements.\n[1] A.P. Ramirez, S. W. Cheong, and P. Schiffer, J. Appl. Phys. 81, 5337 (1997).\n[2] S.P. Gubin, Y.A. Koksharov, G.B. Khomutov and G. Y. Yurko v, Russ. Chem. Rev. 74, 489\n(2005).\n[3] O. Iglesias, A.Labarta, and X. Batlle, J. Nanoscience an d Nanotech. 8, 2761 (2008).\n[4] R.N. Bhowmik,R. Nagarajan, and R. Ranganathan,Phys. Re v.B69, 054430 (2004).\n[5] C.R.H. Bahl, J. Garde1,, K. Lefmann, T.B.S. Jensen, P.-A . Lindg, D.E. Madsen, and S. Mrup,\nEur. Phys. J. B 62, 53 (2008).\n[6] M. Vasilakaki and K. N. Trohidou, Phys. Rev. B 79, 144402 (2009).8\n[7] M. Muroi, P.G. McCormick and R. Street, Rev. Adv. Matter. Sci.5, 76 (2003).\n[8] O. Masala and R. Seshadri, J. Am. Chem. Soc. 127, 9354 (2005).\n[9] G. Bouzerar, R. Bouzerar, and O. Cpas, Phys. Rev. B 76,144419 (2007).\n[10] V. Corral-Flores, D. Bueno-Baques, D. Carrillo-Flore s, and J.A. Matutes-Aquino, J. Appl.\nPhys.99, 08J503 (2006).\n[11] R. N. Bhowmik, A.Poddar, R. Ranganathan, and Chandan Ma zumdar, J. Appl. Phys. 105,\n113909 (2009).\n[12] J. S. Smart, Am. J. Phys. 23, 356 (1955).\n[13] R.N. Bhowmik, R. Ranganathan, and R. Nagarajan, Phys. R ev. B73, 144413(2006).\n[14] P.W. Anderson and H. Hasegawa Phys. Rev. 100, 675 (1955).\n[15] Y. H. Huang, J. Linden, H. Yamauchi, and M.Karppinen,Ch em. Mater. 16, 4337 (2004).\n[16] X.H. Li, Y.P. Sun, W.J. Lu, R. Ang, S.B. Zhang, X.B. Zhu, W .H. Song, and J.M. Dai Solid\nState Comm. 145, 98 (2008).\n[17] T. Zhang, T.F. Zhou, T. Quian, and X.G. Li, Phys. Rev. B 76, 174415 (2007).9\nTABLE I: The particle size (d) of the milled samples are deter mined from the TEM data. The fit\nparameters (C and θw) were obtained using simple Curie-Weiss law (equation 1). T he parameters\n(Ceff,θ1,θ2andξ) were obtained using equation (2) for different milled sample s. It may be\nmentioned that the present values of C and θw(K) using temperature range 330 K to 400 K are\nslightly different from the values C ∼0.0176 and θw∼275 K using temperature range 300 K to\n340 K and previously repored [11].\nSample d C (K g Oe/emu) θw(K) C eff(K g Oe/emu) θ1(K)θ2(K)ξ(arb. unit)\nBulk few µm 0.196 270 – – – –\nmh25 65 nm 0.0275 200 0.035(3) 120 ±4 251 106450 ±2200\nmh61 12 nm 0.0365 100 0.043(2) 33 ±3 250 53400 ±520\nmh98 16 nm 0.0244 20 0.043(2) -46 ±4 247 91800 ±1000\nmh146 60 nm 0.0251 -80 0.024(6) -87 ±2 243 80000 ±1200\nmh200 90 nm 0.0245 -97 0.024(3) -106 ±3 242 62000 ±800200 250 300 350 400 0.00 0.01 0.02 0.03 0.04 \n100 150 200 250 300 0.000 0.001 0.002 0.003 0.004 0.005 \nTp (260 K) \nmh200 mh98 mh25 bulk  \n χdc  (emu g/Oe) \nT (K) Tp (100 K) \nFig.1 Temperature dependence of dc susceptibility ( χdc  (T)) at 100 Oe for \nferromagnetic (bulk) LCMN sample and selected mille d samples. \nInset shows the ( χdc  (T)) data at 10 oe for ferrimagnetic Mn 0.5 Ru 0.5 Co 2O4.\n   χdc  (emu g/Oe) \nT (K) Mn 0.5 Ru 0.5 Co 2O4\n (ferrimagnet) 250 300 350 400 02000 4000 6000 8000 \nFig. 2 (Colour online) Inverse of dc susceptibility  vs. temperature data of bulk LCMN sample (left-bot tom axis) \nand RuMn sample (right-top axis) are fitted with eq uation 1 (red line) and equation 2 (blue line), res pectively. ferrimagnetic Mn 0.5 Ru 0.5 Co 2O4 sample \nχ-1 \ndc  (g Oe/emu) χ-1 \ndc  (g Oe/emu) \nT (K) 0 100 200 300 \n05000 10000 15000 20000 \nbulk LCMN ferromagnetic sample T (K)  \n 200 250 300 350 400 05000 10000 15000 20000 \nFig. 3 (Colour online) inverse of dc susceptibility  at high temperature regime for nanoparticle sample s. \nThe fit data show that Curie-Weiss law is applicabl e only at higher temperature, where as equation (2)  is \napplicable over a wide temperature range above T C of the samples. dotted lines represent the fit with Curie Weiss law  (equation 1) \nsolid lines represent the fit with equation (2) mh200 \n(90 nm) \nmh146 (60 nm) \nmh98 (16 nm) \nmh61 (12 nm) \nmh25 (65 nm) \n  χdc -1  (Oe g/emu) \nT (K) Fig. 4 (Colour online) shows a schematic diagram for long  ranged ferromagnetic \n(LRFM) ordered spins (a), core-shell magnetic ordered spi ns (b), and core-shell \nMagnetic ordered Spins (c) in the top diagram and corre sponding normalized \nsaturation magnetization (NM S) of different FM samples (nanocrystalline \nnanoparticle –NCNP, amorphous nanoparticle-AMPNP) compa red to the bulk \nLRFM sample.     NM S\n01\nLong ranged \nferromagnetic \nOrder among spins Core shell Core \n(a) LRFM (b) FM NC NP (c) FM AMP NP \nshell \nshell \nshell Core Core \nshell \nshell "    },    {        "title": "1209.0004v1.Magnetic_symmetry_of_the_plain_domain_walls_in_the_plates_of_cubic_ferro__and_ferrimagnets.pdf",        "content": "1 \n \n MAGNETIC SYMMETRY OF THE PLAIN DOMAIN WALLS IN THE PLATES OF \nCUBIC FERRO- AND FERRIMAGNETS \nB. M. Tanygin 1, O. V. Tychko 2 \nKyiv Taras Shevchenko National University, Radiophy sics Faculty, Volodumiurska 64,  \nKyiv, MSP 01601, Ukraine \n1E-mail: b.m.tanygin@gmail.com, 2E-mail: a.tychko@gmail.com  \nAbstract.  Magnetic symmetry of possible plane domain walls i n arbitrary oriented plates of \nthe crystal of hexoctahedral crystallographic class  is considered.  The symmetry classification \nis applied for ferro- and ferrimagnets. \nPACS: 61.50 Ah, 75.60 Ch \n1.  Introduction \nFor sequential examination of static and dynamic pr operties of domain walls (DWs) \nin magnetically ordered media it is necessary to ta ke into account their magnetic symmetry \n[1,2]. The complete symmetry classification of plan e 180°-DWs in magnetically ordered \ncrystals [1], similar classification of these DWs w ith Bloch lines in ferromagnets and ferrites \n[2] and magnetic symmetry classification of plane n on 180 0-DWs (all possible DW types \nincluding 0 0-DWs [3]) in ferro- and ferrimagnets [4] were carri ed out earlier. These DW \nsymmetry classifications allows arbitrary crystallo graphic point symmetry group of the \ncrystal. The influence of the spatially restricted sample surfaces on the DW magnetic \nsymmetry wasn’t considered in works [1-4]. The real  magnetic sample restricts a spatial (3D) \nmagnetization distribution. Therefore, it modifies the DW symmetry in general case. This \npaper presents the investigation of the influence o f the restricted sample surfaces on the \nsymmetry of the all possible (0 0-, 60 0-, 70.5 0-, 90 0-, 109.5 0-, 120 0- and 180 0-DW [4,5]) plane \n(i.e. DW with 0r>> δ, where 0r is the curvature radius of the DW [1]) DWs in an a rbitrary \noriented plate of the cubic (crystallographic point  symmetry group m3m) ferro- and \nferrimagnets. \n 2 \n \n 2.  Domain wall symmetry in the restricted sample \nThe DW symmetry can be described by the magnetic sy mmetry classes (MSCs) kG \nwhere k is a MSC number [1]. The MSC kG of DW is the magnetic symmetry group \nincluding all symmetry transformations (all transla tions are considered as unit operations) that \ndo not change the spatial distribution of magnetic moments in the crystal with DW. The \nabove-mentioned group is a subgroup of the magnetic  (Shubnikov’s) symmetry group ∞\nPG of \nthe crystal paramagnetic phase [6]. Total number of  MSCs of arbitrary type DWs (i.e. DWs \nwith arbitrary α2 angle (0 0≤α2≤180 0) between the unit time-odd axial vectors 1m and 2m \ndirected  along magnetization vectors 1M and 2M in neighboring domains) in ferro- and \nferrimagnets is equal to 64. General enumeration of  MSCs contains 42 MSCs ( 42 1≤≤k) of \n180°-DWs [1], 10 MSCs ( 13 7≤≤k and 18 16 ≤≤k) of non 0°/180°- DWs and 42 MSCs ( \nk=2, 13 6≤≤k, 19 16 ≤≤k, k=22, 24, 26, 30, 32, 37, 39 and 64 43 ≤≤k) of 0°-DWs [4]. \nThe MSCs ( k=25, 28, 37-41, 52, 54, 61-63) with six-fold symmet ry axes (including inversion \naxes) do not realized in the cubic crystals [4]. \nThe unified co-ordinate system z y x O~~~ is chosen as [][ ]W12 zyx naaeee ,, ,, ~~~ −=  where \nWn is the unit polar time-even vector along the DW pl ane normal [4]. For the 180°-DWs the \nvectors 1a and 2a are given early [1] as vectors 1τ and 2τ respectively. For the case of \n°≠180 2α the unit vector 1a coincides with the direction of the vector ()mnnm Δ−ΔWW  (at \n0≠Δb and 0=Σb) or []Wna×2 (at 0=Δb or 0≠Σb) where 12mmm−=Δ , \n[]mnΔ ×=Δ Wb , []Σ Σ×=mnWb . Here the unit vector 2a coincides with the direction of \nvector ()Σ Σ−mnnmWW  (at 0≠Σb) or []1Wan× (at 0≠Δb and 0=Σb) or else with an \narbitrary direction in the DW plane ( Wna⊥2 at 0 ==ΔΣbb ) where 2 1mmm +=Σ . The \nmutual orientation of the vectors 1m, 2m and Wn is determined by the parameters \n()Σ Σ=mnWa , ()mnΔ=Δ Wa , ()CW Camn= ,  Σb and Δb, where []21mmm ×=C . The mutual 3 \n \n orientation of the vectors 1m, 2m, Wn and Sn is determined by parameters ()Sana11=, \n()S 2 2ana=, ()SWnann= , []1S 1b an×= , []2S 2b an×=  and []WSnb nn×= , where Sn \nis sample plane normal. \nThe MSC PG of restricted sample of crystal in paramagnetic ph ase could be defined \nas S P P GGG ∩=∞ where sample shape MSC SG is 1/mmm ′ ∞  for volumetric plate. MSCs of \nDWs in volumetric plate should satisfy the conditio n PkGG⊂. The MSCs of the all \npossible plane DWs in the arbitrary oriented plate of cubic crystals of hexoctahedral class \n(crystallographic point symmetry group m 3 m  in the p aramagnetic phase [6]) are presented \nin table. Here symmetry axes are collinear with vec tors 1a and 2a and reflection planes are \nperpendicular to them. For MSCs with k =24, k =26, k =27, 29≤k≤36, 42 ≤k≤51, k =53, \n55≤k≤60 and k=64 only generative symmetry elements are represent ed.  \nGeneral enumeration of MSCs of 0°-DWs contains MSCs  with k=2, 13 6≤≤k, \n19 16 ≤≤k, k=22, 24, 26, 30, 32, 51 43 ≤≤k, k=53, 60 55 ≤≤k and k=64. The 60 0- and \n120°-DWs are represented by MSCs with k=10, 16, 18 and k=11, 13, 16 respectively. The \nMSCs of the 70.5 0-, 90 0- (both for <100> and <110> like easy magnetization  axis [5]) and \n109.5 0-DWs are the MSCs with 7< k<13, 16< k<18.  The general list of 180°-DWs includes \nMSCs with 42 1≤≤k except for k=25, 28, 37-41. \n \n \n \n \n \n \n \n 4 \n \n 3.  Conclusions \nThe complete collection of ( nml )-plates with all possible orientations includes th e full \nlist of MSCs of α2 -DWs in cubic m 3 m  crystal. For separate ( nml )-plates with fixed \ncombination of Miller indexes this list is limited.  Such limitation depends on plate orientation. \nIt is minimal and maximal for the samples with high -symmetry (such as (100)-, (110)- or \n(111)-plates) and low-symmetry (the ( nml )-plates, where indexes are non-zero and have \ndifferent absolute values) developed surface respec tively. Maximal quantity of MSCs of α2-\nDWs is for (100)-plates. The MSC with k=16 is the MSC of all above-mentioned α2-DWs \nin arbitrary oriented plate of cubic m 3 mcrystal.  \n \nReferences \n[1] V. Baryakhtar, V. Lvov, D. Yablonsky, JETP  87 , 1863 (1984) \n[2] V. Baryakhtar, E. Krotenko, D. Yablonsky, JETP  91 , 921(1986) \n[3]    R. Vakhitov, A Yumaguzin, J. Magn. Magn. Mater.  215-216, 52 (2000) \n[4]    B. M. Tanygin, O. V. Tychko, Physica B: Condensed Matter.  Article in Press. \n[5] A  Hubert and R. Shafer, Magnetic Domains. The Analysis of Magnetic \nMicrostructures , Springer, Berlin 1998 \n[6]     L. Shuvalov, Modern Crystallography IV : Phys. Prop. Cryst.,  Springer, Berlin 1988 \n \n \n \n \n \n \n \n 5 \n \n Table.  MSCs of the plane α2-DWs in plates of the cubic m3m crystal. \n \nk  \n{nml }-sample  1m, 2m, Wn and \nSn: mutual \norientation**. 1m, 2m and Wn: \nmutual orientation  Symmetry \n     elements*** MSC \nsymbol \n1 {100}, {110}  0=nb or 1b= 0  0===ΔΣΣaba  ()()1 , 12 , 2 , 2 , 112×n mmm  \n2 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  or \n0===ΔΣΣaba  n2 , 2 , 2 , 12 1′′  mm′2′ \n3 {100}, {110}  0=nb or 1b= 0  0===ΔΣΣaba  n2 , 2 , 2 , 112  mm 2 \n4 {nml }*  0=1a or 0=1b 0===ΔΣΣaba  1, '1,1' 2,12 /m 2′ \n5 {nml }*  0=na or 0=nb 0===ΔΣΣaba  1, '1,n' 2 ,n2 /m 2′ \n6 {nml }* 0a2=or 0b2= 0===ΔΣ Caaa \n22 , 1 m \n7 {100}, {110}  0=nb or 1b= 0  0==ΔΣaa  1, 12′,22,n2′ 22′2′ \n8 {nml }*  0=na or 0=nb 0==ΔΣaa  1, n2′ 2′ \n9 {100}, {110}  0=nb or 1b= 0  0===ΣΔbaaC  1, 12′,22′,n2  mm′2′ \n10  {nml }*  0=1aor 0=1b 0=Δa 1, 12′ 2′ \n11 {nml }* 0=na or 0=nb 0===ΣΔbaaC  1, n2 m \n12  {nml }* 0=1aor 0=1b 0=Ca 1, 12′ m′ \n13  {nml }*  0a2=or 0b2= 0=Σa 1, 22 2 \n14 {nml }* 0a2=or 0b2= 0==ΣΣba  1, 1,22′,22′ m / 2′′ \n15 Arbitrary Arbitrary 0==ΣΣba  1, '1 '1 \n16  Arbitrary  Arbitrary  Arbitrary  1 1 \n17  {100}, {110}  0=nb or 1b= 0  0===ΔΣbaaC  1, 12′,22,n2′ m′m′2 \n18  {nml }*  0=na or 0=nb 0===ΔΣbaaC  1, n2′ m′ \n19  {nml }*  0=na or 0=nb 0==ΣΔbb  1, n2 2 \n20 {nml }*  0=na or 0=nb 0 ===ΔΣΣbba  1, '1,n2,n2′ 2/m  \n21  {100}, {110}  0=nb or 1b= 0  0===ΔΣΣbba  1, 12 ,22 ,n2 222  \n22  {100}, {110}  0=nb or 1b= 0  0==ΣΔbb  1, 12′,22′,n2 m′m′2 \n23  {100}, {110}  0=nb or 1b= 0  0===ΔΣΣbba  ( )()' 1 , 12 , 2 , 2 , 121×n mmm'′′ \n24  {111} 0=nb 0==ΣΔbb  n3 3 \n26  {111} 0=nb 0==ΣΔbb  12 , 3′n m 3′ \n27 {111} 0=nb 0===ΔΣΣbba  12 , 3n 32 \n29 {111} 0=nb 0===ΔΣΣbba  12 , ' 3′n m ' 3′ \n30  {100}  0=nb 0==ΣΔbb  n4 4 \n31  {100}  0=nb 0 ===ΔΣΣbba  nn' 2 , 4  m / 4′ \n32  {100} 0=nb 0==ΣΔbb  12 , 4′n mm4′′ \n33 {100} 0=nb 0===ΔΣΣbba  12 , 4n 422 \n34 {100} 0=nb 0===ΔΣΣbba  nn' 2 , 2 , 41′ mm/m'4 ′′\n35  {100}  0=nb 0===ΔΣΣbba  n' 4  '4 \n36 {100} 0=nb 0===ΔΣΣbba  12 , ' 4n m2'4′ \n42  {111}  0=nb 0 ===ΔΣΣbba  n' 3  ' 3 6 \n \n Table.  MSCs of the plane α2-DWs in plates of the cubic m3m crystal (continue).  \n \nk  \n{nml }-sample  1m, 2m, Wn and \nSn: mutual \norientation. 1m, 2m and Wn: \nmutual orientation  Symmetry \nelements MSC \nsymbol \n43 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  ()()1 , 12 , 2 , 2 , 12 1×′′n mmm′′ \n44  {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  n2 , 2 , 2 , 12 1′′  mm′2′ \n45  {nml}* 0a2=or 02=b 0===ΣΔΔaba  1, 1,22,22 2/m \n46  {nml }*  0=na or 0=nb 0===ΣΔΔaba  1, 1,n2′,n2′ m / 2′′ \n47  {nml }*  0=1aor 0=1b 0==ΔΔba  1, 1,12′,12′ m / 2′′ \n48 Arbitrary Arbitrary 0==ΔΔba  1, 1 1 \n49 {nml }* 0=na or 0=nb 0===ΣΔΔbba  1, 1,n2,n2 2/m \n50 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔbba  1, 12′,22′,n2 22′2′ \n51 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔbba  ()()1 , 12 , 2 , 2 , 12 1×′ ′n mmm′′ \n53  {111} 0=nb 0===ΣΔΔbba  12 , 3′n 2 3′ \n55  {111} 0=nb 0===ΣΔΔbba  12 , 3′n m 3′ \n56  {100}  0=nb 0===ΣΔΔbba  nn2 , 4 4/m  \n57  {100}  0=nb 0===ΣΔΔbba  12 , 4′n 224′′ \n58  {100}  0=nb 0===ΣΔΔbba  nn2 , 2 , 41′ mm/m 4′′ \n59  {100}  0=nb 0===ΣΔΔbba  n4 4 \n60  {100}  0=nb 0===ΣΔΔbba  12 , 4′n m24′ ′ \n64  {111}  0=nb 0===ΣΔΔbba  n3 3 \n \n*   ( nml )-plates with arbitrary Miller indexes except non z ero values n≠m≠l≠n \n** At ()()02 1 = =ananW W   \n  *** The possible symmetry elements are rotations around two-fold symmetry axes n2 , n2′ or \n12, 12′ or else 22, 22′ that are collinear with the unit vectors Wn or 1a or else 2a, \nrespectively, reflections in planes n2, n2′ or 12 , 12′ or else 22 , 22′ that are normal to the \nabove mentioned vectors, respectively, rotations ar ound three-, four-fold symmetry axes n3 , \nn4  that are collinear with the vector Wn, rotations around three-, four-fold inversion \nsymmetry axes n3 ,n3′,n4 ,n4′ that are collinear with the vector Wn, inversion in the symmetry \ncenter 1, '1 and identity 1. Here an accent at symmetry element s means a simultaneous use \nof the time reversal operation [6]. \n "    },    {        "title": "1602.02239v1.Lieb_Mattis_ferrimagnetism_in_diluted_magnetic_semiconductors.pdf",        "content": "Lieb-Mattis ferrimagnetism in diluted magnetic semiconductors\nR.O. Kuzian,1, 2J. Richter,3M. D. Kuz'min,4and R. Hayn4\n1Institute for Problems of Materials Science NASU, Krzhizhanovskogo 3, 03180 Kiev, Ukraine\n2Donostia International Physics Center (DIPC), ES-20018 Donostia-SanSebastian, Spain\n3Institut f ur Theoretische Physik, Otto-von-Guericke-Universit at Magdeburg,\nPF 4120, D - 39016 Magdeburg, Germany\n4Aix-Marseille Universit\u0013 e, IM2NP-CNRS UMR 7334,\nCampus St. J\u0013 er^ ome, Case 142, 13397 Marseille, France\n(Dated: 11.10.15)\nWe show the possibility of long-range ferrimagnetic ordering with a saturation magnetisation of\n\u00181\u0016Bper spin for arbitrarily low concentration of magnetic impurities in semiconductors, provided\nthat the impurities form a superstructure satisfying the conditions of the Lieb-Mattis theorem.\nExplicit examples of such superstructures are given for the wurtzite lattice, and the temperature\nof ferrimagnetic transition is estimated from a high-temperature expansion. Exact diagonaliza-\ntion studies show that small fragments of the structure exhibit enhanced magnetic response and\nisotropic superparamagnetism at low temperatures. A quantum transition in a high magnetic \feld\nis considered and similar superstructures in cubic semiconductors are discussed as well.\nPACS numbers: 75.10.-b, 75.20.-g, 75.50.Gg, 75.50.Pp\nIn order to launch the engineering of a new generation\nof electronic devices, one needs new materials with spe-\ncial properties. For instance, spintronics has a need for\nroom-temperature ferromagnetic semiconductors1. Since\nthe discovery of high- TCferromagnetism in GaAs:Mn2\nand the prediction of room-temperature ferromagnetism\ninp-doped ZnO:Co,Mn systems3, a lot of attempts have\nbeen made to obtain ferromagnetism in transition metal\ndoped ZnO, GaN and in other oxides and nitrides. The\np-type carriers doping is necessary for the p-dZener ferro-\nmagnetic long-range interaction4. Up to now all attempts\nto obtain ZnO with p-type current carriers have failed.\nNevertheless, several reports of \\ferromagnetic\" room\ntemperature behavior have been published5{7. \\Perhaps\nthe most surprising development of the past decade in\nthe science of magnetic materials is the abundant ob-\nservations of spontaneous magnetization persisting to\nabove room temperature in semiconductors and oxides,\nin which no ferromagnetism was expected at any temper-\nature, particularly in the p-dZener model\"5.\nIn the absence of p-type current carriers, the inter-\naction between magnetic impurities is governed by the\nsuperexchange mechanism. Superexchange is often re-\ngarded as an obstacle in the way towards magnetic semi-\nconductors as it has antiferromagnetic (AFM) charac-\nter and tends to anti-align the interacting spins, leading\nto a cancellation of the net magnetization. In fact, the\nAFM interaction does notpreclude spontaneous mag-\nnetization. In a seminal paper8, E. Lieb and D. Mattis\nshowed that the ground state of an AFM system depends\non the topology of the interacting bonds and, under cer-\ntain conditions, it is ferrimagnetic rather than AFM. The\nLieb-Mattis theorem applies if there is no magnetic frus-\ntration in the spin system.\nIn this communication we study various structures\nformed by the interacting magnetic impurities in wurtzite\nsemiconductors. We take antiferromagnetic nearestneighbor interaction into account and consider diluted\nlattices without frustration, in order to remain within\nthe Lieb-Mattis scheme. First we construct several \fnite\nclusters that show an enhanced magnetic response at low\ntemperatures. Not alone do they possess a net magnetic\nmoment, they all share a further interesting peculiarity:\nbelow a certain temperature their magnetic susceptibility\nexceeds that of non-interacting spins. We call it isotropic\nsuperparamagnetic response9,10. Next we construct ex-\ntended lattices of these clusters, which undergo a fer-\nrimagnetic ordering transition at a \fnite temperature.\nThe average ground-state spin per magnetic ion of spin\nStends to a \fnite value (of about S=3) despite the low\nconcentration of magnetic ions. The extension of our idea\nto other lattices and the in\ruence of frustration will be\nbrie\ry discussed at the end of the communication.\nWe take the interaction in the form\n^H=1\n2X\nR;rJr^SR^SR+r; (1)\ni.e., we adopt the notation Jrfor the interaction be-\ntween one pair of spins11. We assume that only the\nnearest-neighbor (in the metal sublattice) interaction\nis nonzero. This assumption is relevant to magnetic\nsemiconductors, where the nearest-neighbor exchange\ndominates12{14. Two kinds of nearest neighborships are\npresent in wurtzites: those where both ions lie in the\nsame plane and those where they lie in two adjacent\nplanes. The corresponding exchange integrals, J1(in-\nplane) and J2(out-of-plane), are di\u000berent15{17.\nThe magnetic response of a system is characterized by\nits magnetic susceptibility. Talking about a compound\nA1\u0000xMxX (where X is a ligand of V or VI group, A is a\nmetal of IIId or IId group, and M is a transition metal),\nwe shall attribute all the magnetic moment to transition\nmetal ions (TMIs) only. We now introduce the magneticarXiv:1602.02239v1  [cond-mat.mtrl-sci]  6 Feb 20162\nsusceptibility per one spin,\n\u001f\u0011\u0016M\nH; (2)\nwhere\u0016Mis the average magnetic moment of one TMI.\nFor non-interacting spins, the susceptibility obeys the\nCurie law\u001fC= [(g\u0016B)2S(S+1)]=(3kBT);whereSis the\nspin of the TMI and gis its gyromagnetic ratio. Besides\nisolated spins, TMI impurities may form pairs, trimers,\ntetramers, and more complex structures (see Fig. 1). The\n1\n3\n4 4’3’a)2\nb)\nJ12J\nc)\nFIG. 1. (Color online) a): Complexes formed by transi-\ntion metal impurities (arrows): isolated ions (1), dimers (2),\ntrimers (3,30), tetramers (4,40). Black solid line segments de-\npict the nearest-neighbor interaction J1bonds. One wurtzite\nabplane is shown, blue circles denote non-magnetic host metal\nions, ligands are not shown. b), c) : More complex Lieb-\nMattis systems with ferrimagnetic ground state: linear chains\nof impurities in the abplane \"decorated\" by spins in adjacent\nplanes (gold arrows); pink line segments depict J2bonds.\nantiferromagneitc interaction depresses the magnetic re-\nsponse at high temperatures. For T\u001dJmaxS(S+ 1)\u0011\nTs, the susceptibility of an interacting system obeys the\nCurie-Weiss law \u001fCW= [(g\u0016B)2S(S+1)]=[3kB(T\u0000\u0012)]<\n\u001fC, with\u0000\u0012= [S(S+ 1)]=(3kBN)P\nR;r(R)Jr(R). Here\nNis the number of spins and Jmaxis the strongest ex-\nchange interaction in the system, Rruns over all spins of\nthe lattice, and rruns over all nearest neighbors of each\nspin.\nAt temperatures T.Ts, the response of the system\ndepends on its geometry. Analytic expressions for the\nsusceptibility can be obtained for small systems18. Fig.\n2a shows the results for the simplest S=1/2 case. We see\nthat atT\u0018Tsthe response of three spins arranged lin-\nearly30is larger than that of a triangular arrangement of\nthe same spins 3. For 4-spin systems we see the striking\ndi\u000berence between the response of a star arrangement 40\nand that of a rhombus 4.\nEven more interesting is the response of the complexes\nshown in Fig. 1b,c. Each one of these systems can be\ndecomposed into two sublattices A and B (denoted by\narrows \\up\" and \\down\"), the interaction being nonzero\nonly between sites that belong to di\u000berent sublattices.\nSuch a system satis\fes the requirements of the Lieb-\nMattis theorem8, and possesses a ferrimagnetic ground\nstate with total spin Sg=SjNA\u0000NBj. In this case, the\nterm \\ferrimagnetic\" refers to correlations of the spins\n 0 2 4 6 8 10\n 0 0.5  1 1.5  2 2.5  3g2µB2/χ\nkBT/|J|S(S+1)a)\n1 \n2 \n3’\n3 \n4’\n4 \n 0 1 2 3 4 5 6\n 0  0.5  1  1.5g2µB2/χ\nkBT/J1S(S+1)b)S=1/2\nS=1  \n 0 0.2 0.4 0.6 0.8 1 1.2\n 0  0.1  0.2  0.3  0.4g2µB2/χ\nkBT/J1S(S+1)c)J2/J1=1\n0.75\n0.33\n 0 0.2 0.4 0.6 0.8 1\n 0  0.1  0.2  0.3g2µB2/χ\nkBT/J1S(S+1)d)J2/J1=1\n0.75\n0.33FIG. 2. (Color online) Inverse susceptibility (per spin) \u001f\u00001\nfor the complexes shown in Fig. 1. Straight solid red line\nshows the Curie law \u001f\u00001\nC; straight dashed lines show the low-\ntemperature asymptotics: \\super\" -paramagnetic Curie laws\n(g\u0016B)2=\u001fg= 3NkBT=[Sg(Sg+ 1)] for Lieb-Mattis systems.\na): clusters shown in Fig. 1a with S= 1=2;b): the com-\nplex shown in Fig. 1b with two di\u000berent values of spin S;\nthe straight dash-dotted red line is the high- TCurie-Weiss\nasymptote. c)the same complex with S= 1 and various val-\nues ofJ2=J1;d)the complex shown in Fig. 1c with S= 1=2\nand various values of J2=J1.\nin the ground state, in the absence of a long-range mag-\nnetic order19. We have performed full exact diagonaliza-\ntion studies (ED) of thermodynamic properties of clus-\nters shown in Fig. 1b,c using J. Schulenburg's spinpack\nprogram20,21. The susceptibility \u001f(T) is calculated as the\nratio of the induced magnetization Mto the \"vanishing\"\nmagnetic \feld H= 10\u00005J1=g\u0016B. One observes in Figure\n2b,c,d that the response of the systems shown in Fig. 1b,c\nexceeds the response of non-interacting spins at low tem-\nperature. Thus, an antiferromagnetic interaction may\nresult in an enhancement of magnetic response if the ge-\nometry of spin arrangement favors the formation of a fer-\nrimagnetic ground state. Then for temperatures T\u001cTs\nthe susceptibility per spin shows superparamagnetic re-\nsponse\u001fg= [(g\u0016B)2Sg(Sg+ 1)]=[3kBT(NA+NB)]. Evi-\ndently, the enhancement of the low-temperature response\ntakes place, if\nK\u0011\u001fg\n\u001fC=jNA\u0000NBj(jNA\u0000NBjS+ 1)\n(NA+NB)(S+ 1)>1:(3)\nNot every system satisfying the requirements of the Lieb-\nMattis theorem and having a ferrimagnetic ground state\nhas an enhanced susceptibility. Thus, the clusters 30\n(NA= 1;NB= 2) and 40(NA= 1;NB= 3) both have\nK < 1, i.e. their response is weaker than that of the\nsame number of non-interacting spins.\nThe \"S\"-shape form of the T-dependence of the inverse3\nsusceptibility (Figure 2b) was previously reported for\nsmall fragments of ferrimagnetic superstructure in double\nperovskites10,22. It interpolates between the Curie-Weiss\nlaw\u001fCWatT\u001dTs, and the \"super\"-spin Curie law\n\u001fg=K\u001fCatT\u001cTs.\nIf impurity spins arrange themselves in a periodic su-\nperstructure having two (or more) non-equivalent spin\npositions, a ferrimagnetic ground state is possible for this\nsuperstructure. Let us denote the number of spins in the\nsuperstructure unit cell nA+nB, where A and B refer to\nthe non-equivalent positions. If the spins of the sublattice\nA interact (antiferromagnetically) only with the spins of\nthe sublattice B (absence of frustration), and nA6=nB,\nthe ground-state spin of the unit cell is Sc=SjnA\u0000nBj8.\nFor a fragment of such a ferrimagnetic superstructure\na) b)\nc)\nd)\nFIG. 3. (Color online) Examples of ferrimagnetic superstruc-\ntures a), b) : \rat and three-dimensional two-leg honey-\ncombs,L= 1; c): four-leg honeycomb, L= 2; d): a unit\ncell of a square network, it may be also regarded as a face of\ncubic unit cell. The notations is the same as in Fig. 1. The\ncyan rhombi show the unit cells.\ncontainingNccells, the ground-state spin is Sg=NcSc=\nNcjnA\u0000nBjS, and the enhancement ratio equals K=\njnA\u0000nBj(NcjnA\u0000nBjS+ 1)=[(nA+nB)(S+ 1)]. It is\nclear that for a su\u000eciently large number of cells Ncthe\nratioKwill be not only greater than 1, but can reach\nvery large values. Fig. 3a shows a honeycomb superstruc-\nture that may be formed by TMIs in the abplane of the\nwurtzite structure. The hexagon edge length is ah= 2a,\nabeing the lattice parameter of the wurtzite. It is easy\nto imagine superstructures with ah= 2La,L= 1;2:::, all\nof them being ferrimagnetic.\nFlat superstructures like those shown in Fig. 3a can be\nlinked together by some bridging spins to form a three-\ndimensional ferrimagnetic superstructure, which will un-dergo a ferrimagnetic phase transition, provided that the\nnumber of cells is macroscopically large. Figure 3b,c\n 0 2 4 6 8 10 12 14 16 18\n 0  1  2  3  4  5(gµB)2/χ\nkBT/J1S(S+1)a)\n1/2\n1  \n3/2\n2  \n5/2\n 0 2 4 6 8 10\n 0  0.5  1  1.5  2(gµB)2/χ\nkBT/J1S(S+1)b)\n1  \n0.75\n0.33\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.4  0.5  0.6  0.7(gµB)2/χ\nkBT/J1S(S+1)c) [4,4]\n[5,5]\n[5,6]\n[6,5]\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.4  0.5  0.6  0.7(gµB)2/χ\nkBT/J1S(S+1)d) 1\n 2\n 3\n 4\n5\n6\nFIG. 4. (Color online) Temperature dependence of inverse\nsusceptibility given by [5,5] Pad\u0013 e approximants for tenth-\norder high-temperature expansion (HTE) for ferrimagnetic\nsuperstructures: a) two-leg honeycomb ( L= 1), vari-\nous spin values are shown, solid (dash-dotted) straight red\nline shows Curie (Curie-Weiss) law; b)four-leg honeycomb\n(L= 2),S= 5=2 variousJ2=J1values are shown; c)four-leg\nsystem,L= 2,S= 2 various Pad\u0013 e approximants for eighth-\norder ([4,4])23,24, tenth-order([4,6], [5,5], [6,4])25, and eleven-\norder ([5,6], [6,5])26HTE; d)the vicinity of TCfor various\nhoneycomb superstructures with size parameter L= 1;2:::6,\nS= 5=2,J2=J1.\nshows examples of the structures. It is clear that this\nmotif may be repeated in an in\fnite number of varia-\ntions. Like the host wurtzite lattice, the unit cell of the\nsuperstructure contains metal ions in two planes. The\nmagnetic ions in one plane (green \\down\" and brown\n\\up\" arrows) form a honeycomb lattice with the hexagon\nedge 2La. In the second plane, the magnetic ions (gold\n\\up\" arrows) occupy the positions nearest to the green\n\\down\" arrows. The interaction between the ions in\nthe \frst plane is J1, whereas the interaction between\nthe ions in two adjacent planes is J2. We note that\nthe complexes shown in Fig. 1b,c are building blocks\nof the honeycombs. It will be demonstrated below that\nmany other Lieb-Mattis networks can be built of such\nblocks. The number of magnetic ions in the unit cell is\nnA+nB= 9L\u00001 , the ground state spin of the cell being\nSc=SjnA\u0000nBj=S(3L\u00001). Now the total number\nof ions in the cell is nc= 24L2. Thus, the concentra-\ntion of magnetic ions equals x= (9L\u00001)=(24L2), and\ncan be made very small for su\u000eciently large L. At the\nsame time, the average ground-state spin per magnetic\nion,hSRi=Sc=(nA+nB) =S(3L\u00001)=(9L\u00001), tends\nto a \fnite value, S=3, asL!1 .\nThe inverse magnetic susceptibility \u001f\u00001of such super-4\nstructures is presented in Fig. 4 as a function of nor-\nmalized temperature T=Ts. It was calculated using a\nprogram25based on the tenth-order high-temperature ex-\npansion (HTE)27. The program computes the exact coef-\n\fcients of the HTE as well as its Pad\u0013 e approximants (ra-\ntios of two polynomials), \u001f(T)\u0019[m;n] =Pm(T)=Pn(T).\nThe Pad\u0013 e approximants allow to extend the region of va-\nlidity of the HTE down to T\u00180:5Ts25(Fig. 4c). This\nextension sometimes fails if an approximant has a pole\nin the temperature region of interest. Our experience\nshows that the [5,5] approximant works well in almost\nall cases. Sometimes di\u000eculties arise for S= 1=2, and\nfor smallJ2=J1ratios, i.e., for the extreme quantum case.\nNevertheless, due to the weak dependence of the shape\nof the curve \u001f\u00001(T=Ts) on the spin value S(Fig. 4a),\nit can still be analyzed. At T&3Ts, the inverse sus-\nceptibility follows the Curie-Weiss asymptotic law with\n\u0012=\u0000[S(S+ 1)=3kB]12L(J1+J2)=(9L\u00001). ForT.Ts\nit sharply deviates from the asymptotic behavior and\nchanges sign at T=TC. This is the temperature of\nferrimagnetic ordering | the Curie temperature.\nThe precision of the determination of critical temper-\natures from the zero of \u001f\u00001(Fig. 4c) was estimated to\nbe about 10%25. Figure 4b shows that TCdecreases as\nthe ratio of out-of-plane to in-plane couplings, J2=J1, is\nreduced. At J2= 0 the system becomes a stack of non-\ninteracting two-dimensional planes, and TCshould van-\nish. This limit lies outside the range of applicability of\nthe HTE, and we postpone its study to future works.\nHere we mention only that magnetic anisotropy, which is\nneglected in our study, should act in the opposite direc-\ntion, i.e., it should enhance the TCas it depresses spin\n\ructuations.\nFigure 4d shows that the ordering temperature de-\ncreases very slowly as Lis increased. Note that the\nsuperstructure parameter values L= 1;2;3;4;5;6 corre-\nspond to the following concentrations of magnetic ions:\nx= 0:33;0:18;0:12;0:09;0:07;0:06. To get a closer rela-\ntion to experiments, we may consider, e.g., ZnO:Mn,Co,\nwhere the in-plain superexchange values are J1=kB\u0018\n50 K11,13,14andTs=J1S(S+ 1)=kB\u0018438(188) K for\nS= 5=2(3=2). For other Co-doped semiconductors 66 K\n.J1=kB.100 K12,17,28(and references therein), i.e., Ts\nlies within the interval 248 K .Ts.375 K. The Mn-\ndoped semiconductors have 12 K .J1=kB.32 K12,29,\nand 105 K .Ts.280 K.\nThus, a very diluted system may have an appreciable\nordering temperature ( TC&100 K) provided that the\nmagnetic ions are arranged in a Lieb-Mattis ferrimagnetic\nsuperstructure.\nIn many aspects, the behavior of a ferrimagnet in its\nordered state is similar to that of a ferromagnet with the\nsame value of spontaneous magnetization Ms. But in\na high magnetic \feld the ferrimagnet exhibits a transi-\ntion accompanied by reorientation of its sublattices30{32.\nAtT= 0 the magnetization per spin has a constant\nvalue,\u0016M;s=g\u0016BSjnA\u0000nBj=(nA+nB), up to a cer-\ntain critical \feld, Hc;1; then it grows up linearly to thesaturation value, \u0016M;max =g\u0016BS, which is reached at a\nsecond critical \feld, Hc;2. For a two-sublattice ferrimag-\nnet having the structure shown in Fig. 3a ( L= 1) and\nJ1=J2=Jwe \fndg\u0016BHc;1=JS, andHc;2= 5Hc;1.\nForJ=kB\u001820 K this gives Hc;1\u001837 T,Hc;2\u0018185 T.\nThe complexes shown in Fig. 1b,c may be arranged\nin many kinds of networks, to form Lieb-Mattis ferri-\nmagnetic superstructures in various host semiconductors.\nFigure 3d shows an example of a 2D square superstruc-\nture unit cell with L= 2, which is possible in a cubic\nhost. It has nA= 1 + 2(L\u00001) andnB= 4(L\u00001) + 2L.\nOne can also imagine a 3D cubic network; then Fig. 3d\ncorresponds to a face of the cubic unit cell having nA=\n1+3(L\u00001),nB= 3L+12(L\u00001), and the concentration of\nmagnetic ions x= (nA+nB)=nc= (9L\u00007)=(4L3). For-\nmation of such superstructures is possible in perovskite\nsolid solutions, like KMn xMg1\u0000xF333,34, or in solutions of\nmultiferroics PbFe 1=2Nb1=2O3or PbFe 1=2Ta1=2O3with\nferroelectric perovskites35{39.\nWe conclude that Lieb-Mattis ferrimagnetism is a\npossible route to obtaining long-range magnetic order\nin semiconductors containing transition metal ions as\nsubstitutional impurities, which requires no additional\ncharge carriers. A precursor of the ordering transition\nis the enhanced magnetic response of \fnite cluster show-\ning isotropic superparamagnetism. Our results for the\ninverse susceptibility show a characteristic \"S\"-like form\nof the curves, which could be used to identify the present\nmechanism. Adding the magnetic anisotropy to our the-\nory, we expect also other ingredients of superparamag-\nnetism, namely a \fnite blocking temperature and hys-\nteresis.\nThese superparamagnetic clusters serve as building\nblocks to create in\fnite sublattices of the wurtzite struc-\nture that obey the Lieb-Mattis rules. As we have al-\nready noted, there is an enormous wealth of such Lieb-\nMattis sublattices, our proposals (Fig. 3) may only serve\nas examples. We expect a \fnite transition tempera-\nture for all these lattices and we have shown it explic-\nitly for the subclass that we considered. Of course,\na question arises, whether frustration in a realistic di-\nluted semiconductor can in\ruence the above discussed\nscenario. First we argue that there are several numer-\nical studies showing that the Lieb-Mattis theorem, al-\nthough not rigorously valid, applies to many frustrated\nspin systems, see, e.g., Ref. 40. Furthermore, we know\nthat there are various frustrated 2D lattices with antifer-\nromagnetic nearest-neighbor exchange, such as the tri-\nangular or the Shastry-Sutherland lattices, which show\nground-state magnetic LRO41,42. Last but not least, the\nstability of the ferrimagnetic ground state against frus-\ntration has been demonstrated for several speci\fc ferri-\nmagnetic models, see, e.g., Refs. 43{45. Consequently,\nthere is ample evidence that the above sketched mech-\nanism should be robust against frustration. The \fnal\nproof that the here proposed mechanism can, indeed, be\nrealized in a real material demands further studies, in\nclose collaboration between experiment and theory.5\nIn this communication, we have considered only semi-\nconductors doped by one kind of magnetic ions, where\nferrimagnetism can appear due to the topology of inter-\nacting bonds. Another option is the co-doping with two\nkinds of ions having di\u000berent spin values. In both cases\na ferrimagnetic semiconductor may be a good alternativeto a ferromagnetic one.\nACKNOWLEDGMENTS\nThe projects NASc of Ukraine 07-02-15, and NATO\nproject SfP 984735 are acknowledged. The exact diago-\nnalization calculations were performed using J. Schulen-\nburg's spinpack .\n1I.\u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod.\nPhys. 76, 323 (2004), URL http://link.aps.org/doi/\n10.1103/RevModPhys.76.323 .\n2F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys.\nRev. B 57, R2037 (1998).\n3T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Fer-\nrand, Science 287, 1019 (2000).\n4C. Zener, Phys. Rev. 82, 403 (1951).\n5T. Dietl, Nat. Mater. 9, 965 (2010), URL http://dx.doi.\norg/10.1038/nmat2898 .\n6R. Janisch, P. Gopal, and N. A. Spaldin, Journal of\nPhysics: Condensed Matter 17, R657 (2005), URL http:\n//stacks.iop.org/0953-8984/17/i=27/a=R01 .\n7S. B. 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Mat. 140-144 , 1611 (1995).\n41J. Richter, J. Schulenburg, and A. Honecker, Lect. Notes\nPhys. 645, 85 (2004).\n42D.J.J. Farnell, O. G otze, J. Richter, R.F. Bishop, and\nP.H.Y. Li, Phys. Rev. B 89, 184407 (2014).\n43N.B. Ivanov, J. Richter, and U. Schollw ock, Phys. Rev. B\n58, 14456 (1998).\n44C. Waldtmann, H. Kreutzmann, U. Schollw ock,\nK. Maisinger, and H.-U. Everts, Phys. Rev. B 62,\n9472 (2000).\n45N.B. Ivanov, J. Richter, and D.J.J. Farnell, Phys. Rev. B\n66, 014421 (2002)."    },    {        "title": "1301.4754v3.Ferrimagnetism_of_dilute_Ising_antiferromagnets.pdf",        "content": "arXiv:1301.4754v3  [cond-mat.dis-nn]  18 Apr 2013Ferrimagnetism of dilute Ising antiferromagnets\nP. N. Timonin∗\nSouthern Federal University, 344090, Rostov-on-Don, Russ ia\n(Dated: September 3, 2018)\nIt is shown that nearest-neighbor antiferromagnetic inter actions of identical Ising spins on im-\nbalanced bipartite lattice and imbalanced bipartite hiera rchical fractal result in ferrimagnetic order\ninstead of antiferromagnetic one. On some crystal lattices dilute Ising antiferromagnets may also be-\ncome ferrimagnets due to the imbalanced nature of the magnet ic percolation cluster when it coexists\nwith the percolation cluster of vacancies. As evidenced by t he existing experiments on FepZn1−pF2,\nsuch ferrimagnetism is inherent property of bcc lattice so t hermodynamics of these compounds at\nlowpcan be similar to that of antiferromagnet on imbalanced hier archical fractal.\nThe system of the identical Ising spins on the sites\nof some crystalline lattices with the nearest-neighbor\nantiferromagnetic (AF) exchange may have magnetized\nground states. In such states there would be antipar-\nallel neighboring spins, as interaction dictates, but the\nwhole numbers of up-spins and down-spins would dif-\nfer. One such 2 dlattice is shown in Fig. 1(a). Here\ntwo sublattices with parallel up and down spins in the\nground state are shown by filled and empty circles cor-\nrespondingly. We see that in the unit cell there are\none filled circle and two empty ones so we get ±1/3\nmagnetizations in two globally-reversed ground states\nfor nearest-neighbor AF on this lattice. Thus this AF\nmodel has a couple of ferrimagnetic ground states with\nboth staggered L= (/angbracketleftSA/angbracketright−/angbracketleftSB/angbracketright)/2 and homogeneous\nM= (2/angbracketleftSA/angbracketright+/angbracketleftSB/angbracketright)/3 magnetizations.\nOne can easily show that this ordering persists up to\nfiniteTc. Summing the Gibbs function over spins on sub-\nlatticeA(empty circles)wegetthe Gibbs distributionfor\nthe spins on the sublattice B having effective ferromag-\nnetic Hamiltonian. Indeed, for each link with SAspin we\nhave (Jis AF exchange)\n/summationdisplay\nSA=±1exp[−SA(SB+S′\nBJ/T)] = 2cosh[( SB+S′\nBJ/T)]\n= 2exp[KB(SBS′B+1)],2KB≡lncosh(2 J/T)\nHence, the ordering of SBspins is described by the ferro-\nmagnetic Ising model on the square lattice, so /angbracketleftSB/angbracketright /negationslash= 0\nforKB>1\n2ln/parenleftbig√\n2+1/parenrightbig\n[1] or\nT < T c= 2J/ln/parenleftBigg\n√\n2+1+/radicalbigg\n2/parenleftBig√\n2+1/parenrightBig/parenrightBigg\n.\nAs/angbracketleftSB/angbracketrightis a linear combination of LandM, the ordered\nphase is a ferrimagnetic one. This implies that homoge-\nneous magnetic field Hhas a part conjugated with the\norder parameter so magnetic susceptibility diverges at Tc\nwhenH= 0and the transition becomessmearedat finite\nH.\nThus we have a simple example showing that nearest-\nneighbor AF interaction of identical Ising spins may re-\nsult in the macroscopic ferrimagnetic order. This is inab\ncd\nA\nAB\nFIG. 1. Examples of imbalanced bipartite graphs with dif-\nferent numbers of sites in sublattices A (open circles) and B\n(filled circles), NA> NB. (a) - fragment of regular 2 dlattice,\ndotted line shows the unit cell; (b, c, d) - clusters of dilute\nsquare lattice. In the ground state short-range Ising AF on\nthem would have parallel spins on A and B sublattices and,\nhence, a nonzero magnetization.\napparent distinction with conventional ferrimagnets hav-\ning several different magnetic moments in a cell. It may\nlook rather exotic in the realm of real crystals yet such\nsituation can be frequent in disordered Ising AF, first of\nall, in dilute Ising AF on bipartite lattices. These lattices\ncan be divided in two subsets of sites, A and B, such that\nall bonds are of the A-B type, i. e. there are no bonds\ninside A and B subsets [2]. Apparently, the Ising AF on\nsuch lattice is non-frustrated having all spins up on sub-\nlattice A and down on sublattice B or vice versa in its\ntwo degenerate ground states. Their magnetizations are\nm=±NA−NB\nNA+NB=±1−η\n1+η, η≡NB\nNA<1\nHereNAandNBare the numbers of sites in A and B\nsublattices and we choose η <1 for definiteness.\nSeemingly, all known non-frustrated AF crystals with\njust one sort of magnetic ions have bipartite lattices that\nare the balanced ones, that is with η= 1 and purely AF\nground states, while Fig. 1(a) shows the imbalanced bi-\npartite lattice with η <1 (η= 0.5). Yet the dilution\nof balanced bipartite lattices results in appearance of a\nnumberofisolatedclusters,mostlywith η <1,thosewith\nη= 1being the rareexceptions. Figs. 1(b, c, d) show the\nimbalanced clusters on the square lattice. So at T= 0\nand arbitrarily small magnetic field dilute bipartite AF2\nmust show nonzero magnetization due to the presence of\nsuch imbalanced finite clusters. This circumstance was\nfirst noticed by Neel [3]. Still it stays unnoticed that for\nsome concentrations of magnetic ions pthe giant perco-\nlation cluster may also have the average imbalance ratio\nηp<1.\nIndeed, in finite sample the role of percolation cluster\nbelong to that with the largest number of sites and very\nprobably it is imbalanced, as most of them. However,\nin the thermodynamic limit ( N→ ∞)ηpwill tend to\nunity if there are only finite clusters of vacancies. Ap-\nparently, such finite clusters cannot make infinite lattice\nimbalanced as for every cluster deleting unequal number\nof sites from A and B sublattices there exists (with the\nsame probability) the shifted cluster of the same form\nwhich restores the balance. Thus at 1 −pc< p <1 the\nimbalanced percolation cluster can only exist as a finite-\nsize effect. Meanwhile, at pc< p <1−pcthere is infinite\npercolation cluster of vacancies to which this argument\ndoes not apply. Hence, here ηp<1 may also hold in the\nN→ ∞limit in some crystal lattices. Then the ground\nstate magnetization of dilute AF in this interval will be\nmp(H= +0) =1/integraldisplay\n01−η\n1+ηWp(η)dη+1−ηp\n1+ηp\nHereWp(η) is the imbalance distribution function of fi-\nnite clusters. Nowit seems that neither Wp(η) norηpare\nknown for the crystal lattices. So to find them is quite\nrelevant task for the physics of dilute short-range AF.\nThe magnetization of finite clusters vanishes at finite\ntemperatures, but that of the imbalanced percolation\ncluster would persist up to a finite TcandM∼Lat allT\ndue to its geometrical origin. Then macroscopic features\nof dilute AF would be those of ordinary ferromagnet in\nspite of the presence of antiparallel neighboring spins in\nthe ordered phase. In such a case, the mapping of this\nmodel onto random-field Ising magnet (RFIM) [4] would\nbe no longer valid as it suggests purely AF transition in\nDAFF. This possibility of AF order breaking is missed in\nRef. [4] which is a consequence of the mean-field treat-\nment of homogeneous magnetization.\nThe evidences in favorof DAFF ferrimagnetismcan be\nfound in experiments on several dilute Ising AF with p <\n1−pcshowingtheremanentmagnetizationwiththeusual\norder-parameter behavior [5]-[9] and prominent peak in\ntemperaturedependence ofmagneticsusceptibilitywhich\nappears in low fields as a result of dilution and becomes\ngradually smeared in higher fields [8],[9].\nWe should note that on the lattices having perfectly\nbalanced percolation cluster with ηp= 1 DAFF also have\na ferrimagnetic phase in its ground state. The differ-\nence with the imbalanced case is that it appears above\nsome finite critical field HAF(p) while this field is zero if\nηp<1. The schematic ground state phase diagrams are\nshown in Fig. 2. The validity of these pictures followsfromquitesimpleconsiderations. Letusconsidertheper-\nfectly balanced AF percolation cluster. As it necessary\nhas some imbalanced (magnetized) parts, the field will\ninduce the energy-reducing global flipping of their spins\nif the magnetic moment Mof the part points opposite to\nthe field and His greater thanB\nMJ. HereBis the num-\nber of AF bonds connecting the given part with the rest\nof percolation cluster, Jis AF exchange. First the large\nclusters with smallB\nMratio will be flipped in low fields\nwhile the field growth will induce the flipping of smaller\nand smaller ones. At last the remaining single spins flip\nalong the field at H=zJ(zis the lattice coordination\nnumber). The corresponding jumps of sublattice magne-\ntizations are seen in the numerical study of the ground\nstate of 3 d(simple cubic) DAFF with pas large as 0.9\nandH >2J[10].\n0 1pH\nFerromagnet\nImproper\nFerrimagnet\nAF\npc(a)\n0 1pH\nFerromagnet\nFerrimagnetAF\npc(b)zJzJ\n1-p cImproper\nFerrimagnet\nFIG.2. Schematicgroundstate H−pphasediagrams ofdilute\nAF. (a)ηp= 1 for all p, (b)ηp<1 forp <1−pc. The lines\nbetween phases are defined by HAF(p) andHF(p) discussed\nin text. In the improper ferrimagnetic regions sharp AF tran -\nsition is preserved at finite Tcwhile it becomes smeared fer-\nrimagnetic one at H >0 in genuine ferrimagnetic region in\n(b).\nApparently, this process results in appearance of a\nnonzero magnetization of percolation cluster in fields\nabove some HAF(p) and vanishing of its staggered mag-\nnetizationabovesomegreaterfield HF(p). Sothe ground\nstate at HAF(p)< H < H F(p) is ferrimagnetic. Yet in\nthis case Mappears at Tcas a secondary order param-\neterM∼L2and here sharp AF transition is preserved3\nas well as DAFF-RFIM mapping. So we may call this\nphase ’improper ferrimagnetic’ to distinguish it from the\ngenuine ferrimagnetic one in Fig 2b.\nStill the improper ferrimagnetic ground state would\ncause a drastic change in the dynamics of AF phase.\nThis is the consequence of huge degeneracy of the fer-\nrimagnetic ground state as at rational H/Jthere can be\na huge amount of parts of the percolation cluster with\nH/J=B/Mso their flipping does not change the en-\nergy. This degeneracy is explicitly demonstrated in nu-\nmerical studies of realistic DAFF systems [10], [11]. At\nfiniteTthis results in many (nearly) degenerate min-\nima of thermodynamic potential so the system can be\ntrapped in each of them, depending on the previous his-\ntory ofTandHvariations. The particular manifestation\nof these phenomena is the difference between field-cooled\nand zero-field-cooled thermodynamic parameters. Ap-\nparently, it would be also present in the ferrimagnetic\nphase of imbalanced DAFF right down to H= 0.\nConcerning the behavior of HAF(p) andHF(p) in Fig\n2 we can note that it is quite apparent that HAF(1) =\nHF(1) =zJwhile their diminishing to zero at p=pc\nin Fig 2a is the consequence of sparse structure of per-\ncolation cluster near pc. Here it is divided into loosely\nconnected parts with B/M→0 so their flipping fields\nalso go to zero resulting in HAF(pc) =HF(pc) = 0. In\nFig 2bHAF(p) seized to exist at p= 1−pcwhen, ac-\ncording to our surmise, ηpbecomes less than 1 in some\nlattices.\nThe notion of HAF(p) andHF(p) behavior one can get\nfrom the results of extensive numerical studies of DAFF\nground state on simple cubic and bcc lattices [11]. Here\nthe boundaries of the so called ”domain state” are de-\ntermined. In this state the percolation cluster of the\nflipped spins coexists with that of unflipped ones. Its\nupper boundary coincides with HF(p) while the lower\none can be somewhat higher than HAF(p), yet its behav-\nior for the simple cubic lattice [11] resembles that in Fig.\n2a. So, most probably, this lattice has ηp= 1 for all p.\nThe results for bcc lattice are less conclusive, here the\npercolation cluster of flipped spins can appear at rather\nlow fields, depending on the boundary conditions and\ndisorder realization [11]. This makes bcc lattice a valid\ncandidate for having ηp<1 (andHAF(p) = 0 ) at some\np > pc.\nTo get some notion of the DAFF thermodynamics in\nthe ferrimagnetic phase which may result from ηp<1\natp <1−pcwe consider here the nearest-neighbor AF\non the simplest hierarchical lattice, imitating the perco-\nlation cluster with fractal dimension d= 2 and η= 1/3.\nAs well, it may describe qualitatively large planar aggre-\ngates of AF particles or disordered AF thin films which\nmay have the imbalanced structure of a set of magnetic\nions. The model also exhibits a number of field-induced\nground state transitions marked by the magnetization\njumps which are discussed above.LOW-FIELD THERMODYNAMICS OF\nHIERARCHICAL ANTIFERROMAGNET\nWe consider the short-range Ising AF on the sim-\nplest ”diamond” hierarchical lattice [12]. Its building\nprocess is shown in Fig.3. On the n-th level of hier-\narchy the lattice has Nnsites,Nn=2\n3(4n+2), see\nRef. [13]. The coordination numbers of the sites are\nthe powers of 2: z= 2,4,8,.... At all levels of the hi-\nerarchy the lattice is bipartite and for n >0 the sites\nwith coordination number z= 2 constitutes the sub-\nlattice A (open circles in Fig.3) while the others be-\nlong to the sublattice B (filled circles), NA,n≥NB,n.\nAt then-th level NA,n= 2·4n−1forn >0 [13], so\nηn= (Nn−NA,n)/NA,n=/parenleftbig\n1+2·41−n/parenrightbig\n/3. We are in-\nn=0 n=1 n=2AA\nB B\nFIG. 3. Construction of hierarchical lattice. It is biparti te\nat all levels. Different circles designate its partitioning , open\ncircles sublattice A, filled circles sublattice B, NA≥NB.\nterested in the thermodynamic limit of infinite levels of\nhierarchy. In this limit η= 1/3 and fractal dimension\nd= 2 [13].\nFor the Ising spins Si=±1 placed on the sites of this\nlattice we consider the AF Hamiltonian\nH=J/summationdisplay\n<i∈A,j∈B>SiSj−HA/summationdisplay\ni∈ASi−HB/summationdisplay\nj∈BSj(1)\nwhere< i∈A,j∈B >means the summation over near-\nestneighborsanddifferentfieldsforthesublatticesarein-\ntroduced. This allows to find the averagemagnetizations\nof each sublattice and the order parameter for the transi-\ntion. Homogeneous field corresponds to HA=HB=H.\nThe usual way to get the partition function of the\nmodel is through the recursion relations for partial par-\ntition functions at different levels of hierarchy Zn(S,S′)\nhaving fixed values of the outmost left and right spins S\nandS′[12]. These relations read\nZn+1(S,S′) =/bracketleftBigg/summationdisplay\nS1=±1Zn(S,S1)ehnS1Zn(S1,S′)/bracketrightBigg2\n,(2)\nhn=Hn/T, H 0=HAandHn=HB, n≥1. The\ninitial condition for them is\nZ0(S,S′) =e−KSS′, K=J/T. (3)\nUsing the representation\nZn(S,S′) = exp1\n2[Cn+unSS′+(vn−hn)(S+S′)]4\nwe get from Eqs.(2,3)\nu0=−2K, v 0=hA, C 0= 0 (4)\nun+1= 2lncosh un+ln/parenleftbig\n1−tanh2untanh2vn/parenrightbig\n,\nvn+1= 2vn+2tanh−1(tanhuntanhvn)−2hn+hn+1,\nCn+1= 4Cn+un+1+4ln(2cosh vn).\nThe last of Eqs.(4) gives for n >0\nCn=un−4nu0+n−1/summationdisplay\nl=04n−l[ul+ln(2cosh vl)]\nso then-th level free energy per spin is\nFn=−T\nNnln/summationdisplay\nS,S′Zn(S,S′)ehB(S+S′)=\n−3\n4Tn−1/summationdisplay\nl=04−l[ul+ln(2cosh vl)]−3\n2J+O(1/Nn) (5)\nAtHn= 0 the model has phase transition at K=\nKc≈0.609 being the solution to the equation Kc=\nlncosh2Kc.uc= 2Kcis the stationary point of the\nzero-field equations, un+1= 2lncosh un,vn= 0. In the\nparamagneticphaseat K < K cun→0 forn→ ∞, while\nin the ordered phase at K > K cun→ ∞. According\nto above considerations the ground states of the model\nhave magnetizations ±(1/2), so we may expect that the\nordered phase is ferrimagnetic. To show this we consider\nEqs.(4) at\n0<(Tc−T)/Tc≡τ≪1,|hn| ≪τ.(6)\nIn this case unandvncan be found approximately in the\nthree regions of n:\n1) 1≤n≤λ,un−uc≤uc,|vn| ≪1\nun≈uc+κn−1(u1−uc),(7)\nvn≈hB+(2+κ)n−1κ˜h\n1+κ,˜h=hB−(κ+1)hA(8)\nu1= 2lncosh2 K, κ = 2tanh2 Kc≈1.68,\nThe value of λis defined by\nuλ= 2uc, κ−λ=u1−uc\nκuc≈τ (9)\nwhile|vn| ≪1 requires\n|˜h|(2+κ)λ=|˜h|τ−ln(2+κ)/lnκ≪1.(10)2)λ < n≤µ,1≪un≪ |vn|\nun≈2n−λ+1uc−n−1/summationdisplay\nk=λ2n−kln2coshvk,(11)\nvn≈hB\n3+4n−λ/parenleftbigg\nvλ−hB\n3/parenrightbigg\n,(12)\nvλ≈(2+κ)λ−1κ˜h\n(1+κ)=κ˜h\n(1+κ)(2+κ)τln(2+κ)/lnκ(13)\nThe value of µis defined by the equation uµ=\n|vµ|. Asuµ≈2µ−λ+1uc−µ−1/summationtext\nk=λ2µ−kln2cosh4k−λvλ≈\n2µ−λ2(uc−ln2)+4µ−λvλ, vµ≈4µ−λvλ, we get\n2µ=(uc−ln2)2λ\n|vλ|, uµ=|vµ|=(uc−ln2)2\n|vλ|.(14)\n3)µ > n,|vn| ≫1,un≈0,vn≈2n−µvµ.(15)\nNote that in the sums we consider the large numbers\nλandµas integers neglecting its fractional parts.\nUsing the above approximations for unandvnwe can\nfind from Eq.(5) free energy in the thermodynamic limit\nnear the transition point in a small field (cf. Eqs.(6),\n(10)). Thus, dividing the sum in (5) in three parts and\nn= 0 term,\n−4\n3F\nT= ln2cosh hA+Σλ+Σλµ+Σµ,\nwe get\nΣλ=λ/summationdisplay\nn=1/bracketleftBig\n4−n(uc+ln2)+ ucτ/parenleftBigκ\n4/parenrightBign/bracketrightBig\n+\n+λ/summationdisplay\nn=1˜h2\n2(1+κ)2(2+κ)2/parenleftBig\n1+κ\n2/parenrightBig2n\n≈1\n3(ln2+uc)+κ\n4−κucτ−4−λ\n3/parenleftbigg2+κ\n4−κ2uc+ln2/parenrightbigg\n+κ˜h2\n2(4+κ)(1+κ)2/parenleftBig\n1+κ\n2/parenrightBig2λ\nΣλµ=µ/summationdisplay\nn=λ+1/parenleftbig\n2−n−λ+1uc+4−nln2coshvn/parenrightbig\n−\n−µ/summationdisplay\nn=λ+14−nn−1/summationdisplay\nk=λ2n−kln2coshvk\n=µ/summationdisplay\nn=λ+1/parenleftbig\n2−n−λ+1uc+4−nln2coshvn/parenrightbig\n−\n−µ/summationdisplay\nk=λ/parenleftbig\n4−k−2−k−µ/parenrightbig\nln2coshvk\n= 4−λ2uc−4−µuµ+4−µln2coshvµ−4−λln2coshvλ\n≈4−λ(2uc−ln2)5\nΣµ≈∞/summationtext\nn=µ+14−n|vn| ≈4−µ|vµ|= 4−λ|vλ|.\nHere we used |vλ| ≪1,|vµ| ≫1, Eqs.(7, 8, 11, 12,\n15) and the relation following from Eq.(11),\nµ−1/summationdisplay\nk=λ2−k−µln2coshvk= 2−λ−µ+1uc−4−µuµ.\nFinally we have from Eqs.(5, 9, 13, 14)\nF/Tc≈ −Kc/2−ln2+τsc−aτ2−α−bτβ/vextendsingle/vextendsingle/vextendsingle˜h/vextendsingle/vextendsingle/vextendsingle−cτ−γ˜h2,\n(16)\nsc= ln2−2Kcκ−1\n4−κ≈0.34,a= 2Kc5−2κ\n4−κ−ln2≈0.17,\nb=3κ\n4(2+κ)(1+κ)≈0.13,c=3κ\n8(4+κ)(1+κ)2≈0.015.\nα= 2−ln4\nlnκ≈ −0.67, (17)\nβ=ln4−ln(2+κ)\nlnκ≈0.16,\nγ=2ln(2+ κ)−ln4\nlnκ≈2.35\nIn homogeneous field ˜h=−κh(cf. Eq.(8)) so Fin\nEq.(16) has the standard scaling form of a ferromag-\nnet with spontaneous magnetization m∼τβand diver-\ngent susceptibility χ∼τ−γ. This expression is valid at\n0< τ≪1,|h| ≪τβ+γ, cf. Eq.(10). Scaling indices\n(17) obey the usual relation α+2β+γ= 2. Negative α\nmeans that specific heat is finite at the transition point\nand has a cusp at Tc. Note also that scis the entropy at\nthe transition point. So this AF system looks like gen-\nuine ferromagnet, even featuring the absence (smearing)\nof transition in a finite field. The last is evident as the\nnontrivial stationary point of finite-field recursion rela-\ntions (4) cannot be reached from any initial conditions.\nYet the dependence of Fon˜hfrom Eq.(8) shows that\ntrue order parameter for the transition is a linear com-\nbination of MA=/summationtext\ni∈ASiandMB=/summationtext\ni∈BSiconjugate\nwith˜H=HB−(κ+1)HA. To distinguish the order\nparameter in the Hamiltonian (1) we perform a coordi-\nnate rotation in 2 dspace of vectors H= (HA,HB) and\nM= (MA,MB) to bring the term −MHin (1) to the\nform−MH=−˜M˜H−M′H′where\n˜M=MB−(κ+1)MA\n1+(κ+1)2,M′=(κ+1)MB+MA\n1+(κ+1)2,\nH′= (κ+1)HB+HA\nThus˜Mis the order parameter while M′andH′are\nnon-critical variables. Hence, /angbracketleftM′/angbracketright= 0 atH→0 so the\nspontaneous magnetic moments of the sublattices obey\nthe relation /angbracketleftMA/angbracketright=−(κ+1)/angbracketleftMB/angbracketright. Then for the spon-\ntaneous magnetizations mν=/angbracketleftMν/angbracketright/Nν(ν=A, B) we\nhave\nmA=−η(κ+1)mB≈ −0.9mBThis differs from the ground state relation mA=−mB.\nWe may suggestthat this is a consequenceof criticalfluc-\ntuations diminishing mAmore strongly than mBas all\nsites of sublattice A have the lowest coordination num-\nberzA= 2. To some extent this effect would be present\nin all dilute AF on imbalanced bipartite graphs since the\nsublattice A with larger amount of spins would necessary\nhave lower average coordination number ¯ zA=C/NA<\n¯zB=C/NB. HereCis the number of bonds and we used\nthe fact that all bonds are of A-B type. Thus ¯ zA=η¯zB\nsothe lower ηthestrongercanbe thefluctuation-induced\ndisbalance between mAandmBnearTc. Atη= 1 this\neffect vanishes so its observation in neutron-diffraction\nexperiments can certify the onset of imbalance in the\nmagnetic percolation cluster.\nGROUND STATE TRANSITIONS\nHere we assume HA=HB=H. AtT= 0 we define\n˜un= lim\nT→0Tun,˜vn= lim\nT→0Tvn, E= lim\nT→0Fto obtain\nfrom Eqs.(4, 5)\n˜u0=−2J˜v0=H\n˜un+1= 2(|˜un|−|˜vn|)ϑ(|˜un|−|˜vn|),(18)\n˜vn+1= 2˜vn+2min( |˜un|,|˜vn|)sgn(˜un˜vn)−H,(19)\n−(4/3)E=H+∞/summationdisplay\nn=14−n(˜un+|˜vn|)\nϑinEq.(18)isHeaviside’sstepfunction. SolvingEqs.(18,\n19), we get\n−(4/3)E=H+∞/summationdisplay\nn=14−n/vextendsingle/vextendsingleH−2n+1J/vextendsingle/vextendsingle,\nm=−∂E\n∂H=3\n4/bracketleftBigg\n1+∞/summationdisplay\nn=14−nsgn/parenleftbig\nH−2n+1J/parenrightbig/bracketrightBigg\n.\nSo atH/negationslash= 2kJ\nm= 2−1ϑ(2J−H)+/parenleftbig\n1−2·4−r/parenrightbig\nϑ(H−2J),\nwherer= [log2(H/J)] is an integer part of log2(H/J).\nAtHr= 2rJ,r/greaterorequalslant2, we have mr= 1−6·4−r. Field\ndependence of the ground state magnetization is shown\nin Fig.4. Due to the imbalance ratio η= 1/3 the system\nhas spontaneous magnetization m= 1/2 atH→+0.\nUnfortunately, the data on the percolation cluster mag-\nnetization on cubic and bcc lattices are totally absent in\nRef. [11] which deprives us of the opportunity to decide\nif there is the imbalance in real 3 dpercolation clusters at\npc< p <1−pc.\nThe jumps at Hr= 2rJresult from the flipping along\nthe field of single spins in sublattice B having the coordi-\nnation number 2r. As we discussed above in dilute crys-\ntalline lattices there are many more jumps appearing at6\n0.40.60.81m\nH/J 4 8 12 16 0\nFIG. 4. Field dependence of the ground state magnetization.\nrational values of H/Jwhere flipping of the magnetized\nparts of percolation cluster takes place [10]. Such jumps\nwere observed in low- Texperiments in FepZn1−pF2[14].\nDISCUSSION AND CONCLUSIONS\nThe decades of experimental investigations of dilute\nIsing AF have shown that DAFF - RFIM correspondence\nworks reasonably well at low dilution and low fields [15].\nMeanwhile the field-induced rounding of the transition\nappears at lower pwhich is impossible in the case of the\nAF ordered phase. One explanation assumes that this is\nnonequilibrium effect due to the pinning of AF domain\nwalls by the vacancies which results in very slow relax-\nation to the equilibrium AF structure [16]. Also one may\nsuggest that AF transition transforms at lower pinto a\nspin-glass one [15], [17].\nHere we argue that one more reason for the vanish-\ning of AF transition could be the imbalance of perco-\nlation cluster which makes transition ferrimagnetic at\nH= 0 and smeared at finite fields. Just these phe-\nnomena were found in FepZn1−pF2[5] - [7] and several\nother dilute AFs [8], [9]. Also the inspection of neutron-\ndiffraction data on metastability and domain formation\ninFepZn1−pF2family [18] makes authors to conclude\nthat AF order vanishes right at p= 1−pc. AF region\nin theH−pphase diagram of these compounds in Ref.\n[18] is quite similar to that in Fig.2b. Thus our surmise\nof possible imbalance of percolation cluster at p <1−pc\nseems to be true for bcc lattice.\nThis implies that qualitative features of the consid-\nered here model could apply to the thermodynamics\nofFepZn1−pF2compounds with vacancies’ percolation.\nThey are a small scaling index of remanent magnetiza-\ntion, ratherhigh index γ, largenegative αand disbalance\nin the sublattice magnetizations near Tc. But to observe\nthese features of ferrimagnetic transition the measure-ments in ultra-low fields (same as in Refs. [5]-[9]) are\nneeded to avoid its smearing. Also the irreversibility\nshould be taken into account as the theoretical results\nrefer only to the most stable state, which seemingly is a\nfield-cooled one in the ferrimagnetic phase.\nNow we do not know on which lattices DAFF would\nalso have the phase diagram of Fig.2b. Moreover, the\nexact form of HAF(p) andHF(p) is not known for the\nvariety of crystalline lattices of the known easy-axes an-\ntiferromagnets. Yet investigation of DAFF ground state\nin Ref. [11] shows that their determination is feasible\nwith modern numerical methods. Such studies and fur-\nther experiments revealing the details of low- Tand low-\nHbehavior of magnetization may help to elucidate the\nnature of transition in nearest-neighbor dilute AF.\nAuthorgratefullyacknowledgesuseful discussionswith\nV.P. Sakhnenko and M.P. Ivliev.\n∗pntim@live.ru\n[1] R.J. Baxter, Exactlysolvedmodels instatistical mechan-\nics, Academic Press, 1982.\n[2] G. Chatrand, Introductory Graph Theory , Dover, 1984.\n[3] L. Neel, C. R. Acad. Sci. Paris 252, 4075 (1961).\n[4] J.L. Cardy, Phys. Rev. B29, R2460 (1984).\n[5] C. Djurberg, J. Mattson and P. Nordblad, J. Appl. Phys.\n75, 5541 (1994); J. Mattson, C. Djurberg and P. Nord-\nblad, Phys. Rev. B61, 11274 (2000).\n[6] J. Kushauer, W. Kleemann, J. Mattsson et al.,\nPhys. Rev. B49, 6346 (1994).\n[7] M. Lederman, J. Hammann and R. Orbach, Physica\nB165-166 , 179 (1990).\n[8] H. Ikeda, J. Phys. C 16, L21 (1983); H. Ikeda, J. Phys. C\n16, L1033 (1983).\n[9] H. Ikeda and K. Kikuta, J. Phys. C 16, L445 (1983) H.\nIkeda and K. Kikuta, J. Phys. C 17, 1221 (1984).\n[10] S. Bastea and P.M. Duxbury, Phys. Rev. E58, 4261\n(1998)\n[11] A. Glaser, A.C. Jones and P.M. Duxbury, Phys. Rev.\nB71, 174423 (2005).\n[12] A.N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1979);\nR.B. Griffiths and M. Kaufman, Phys. Rev. B26, R5022\n(1982).\n[13] P.N. Timonin, Zh. Eksp. Teor. Fiz. 126, 1198 (2004).\n[14] A.R. King, V. Jaccarino, T. Sakakibara et al.,\nPhys. Rev. Lett. 47, 117 (1981).\n[15] D. P. Belanger in Spin Glasses and Random Fields , ed.\nA. P. Young, World Scientific (1997).\n[16] M. Staats, U.NowakandK.D.Usadel, Phase Transitions\n65, 159 (1998).\n[17] F.C. Montenegro, S.M. Rezende and M. D. Coutinho-\nFilho, J. Appl. Phys., 63, 3755 (1988).\n[18] W.C. Barber, F. Ye, D.P. Belanger et al., Phys. Rev.\nB69, 024409 (2004)."    },    {        "title": "0710.3592v1.Numerical_Study_of_a_Three_Dimensional_Mixed_Ising_Ferrimagnet_in_the_Presence_of_an_External_Field.pdf",        "content": "/C112/C104/C121/C115/C46/C115/C116/C97/C116/C46 /C115/C111/C108/C46/C40._/C98/C41/C50/C50/C48/C44/C57/C53/C57/C40._/C50/C48/C48/C48/C41\n/C83/C117/C98/C106/C101/C99/C116 /C99/C108/C97/C115/C115/C105/C102/C105/C99/C97/C116/C105/C111/C110/C58 /C55/C53/C46/C49/C48/C46/C72/C107/C59 /C55/C53/C46/C52/C48/C46/C77/C103/C59 /C55/C53/C46/C53/C48/C46/C71/C103\n/C78/C117/C109/C101/C114/C105/C99 /C97/C108/C83/C116/C117/C100/C121\n/C111/C102/C97/C84/C104/C114/C101/C101/C45/C68/C105/C109/C101/C110/C115/C105/C111/C110 /C97/C108/C77/C105/C120/C101/C100/C73/C115/C105/C110/C103 /C70/C101/C114/C114/C105/C109/C97/C103/C110 /C101/C116\n/C105/C110/C116/C104/C101/C80/C114/C101/C115/C101/C110/C99/C101 /C111/C102/C97/C110/C69/C120/C116/C101/C114/C110/C97/C108 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F. Correa1, N. Haberkorn1, G. Nieva1, D. J. Garc ´ıa1andB. Alascio1\n1Centro At´ omico Bariloche (CNEA) and Instituto Balseiro (U . N. Cuyo), 8400 Bariloche, R´ ıo Negro, Argentina\nPACS75.80.+q – Magnetomechanical effects, magnetostriction\nPACS75.50.Gg – Ferrimagnetics\nPACS75.47.Lx – Magnetic oxides\nAbstract – We report an unusual giant linear magnetostrictive effect i n the ferrimagnet\nGd2/3Ca1/3MnO3(Tc≈80 K). Remarkably, the magnetostriction, negative at high t emperature\n(T≈Tc), becomes positive below 15 K when the magnetization of the G d sublattice overcomes\nthe magnetization of the Mn sublattice. A rather simple mode l where the magnetic energy com-\npetes against the elastic energy gives a good account of the o bserved results and confirms that\nGd plays a crucial role in this unusual observation. Unlike p revious works in manganites where\nonly striction associated with 3 dMn orbitals is considered, our results show that the lanthan ide\n4forbitals related striction can be very important too and it c annot be disregarded.\nIntroduction. – Manganites are perovskites mostly\nknown for their spectacular colossal magnetoresistance\n(CMR): the electrical resistivity can change several orders\nof magnitude under a moderate applied magnetic field B\n[1]. They also show another impressive property called gi-\nant linear magnetostriction (MS): sample dimensions are\nstrongly affected by a magnetic field, either external or\nmolecular [2]. The effect is comparable in magnitude\n(∆L/L≥10−3at several Tesla) to the highest MS values\never reported. Both CMR and MS are particularly large\naround the Mn-ions ferromagnetic ordering temperature\n[3]. Associated with this order, and depending on the dop-\ning level, manganites can display a metal-insulator (MI)\ntransition, too. In this way, manganites offer a unique\ntesting ground to study the interplay between electronic,\nspin and lattice degrees of freedom.\nAs expected, structural distortion of the plain per-\novskite structure strongly affects the magnetic and\nelectronic properties of manganites. This is usually\nparametrized by the so called tolerance factor t, which\nquantizes the mismatch between the size of the different\nions in the formula. This mismatch primarily influences\nthe exchange interaction between Mn ions altering both\nthe length and the angle of the Mn-O-Mn bond. An ‘ uni-\nversal’ temperature Tversust= (dR/A−O)/√\n2(dMn−O)\nphase diagram has long been reported [4] for the hole\ndoped R 2/3A1/3MnO3manganites ( Ris a lanthanide andAis an alkaline-earth element). Slightly distorted struc-\ntures (t∼1) show the insulating-paramagnet (PMI) to\nmetallic-ferromagnet (FMM) transition. However, the\nmetallic state disappears at higher distortions ( t/lessorsimilar0.91)\neven though a transition to a insulating-ferromagnet\n(FMI) is observed.\nAn estimated value of t≈0.89 places Gd 2/3Ca1/3MnO3\nwell inside the insulating regime. Indeed, no MI transition\nis observed down to 5 K with the resistivity ρshowing a\ncharacteristic semiconducting behavior in the whole tem-\nperature range [5,6]. Nevertheless, magnetic properties\nare quite more interesting. Mn magnetic moments start\nordering ferromagnetically around Tc∼80 K. The Gd mo-\nments react to the internal field created by the Mn fer-\nromagnetic sublattice gradually aligning in the opposite\ndirection. The two sublattices compete each other giving\nrise first to a maximum in the magnetization around 50 K\nand finally to a full compensation at Tcomp∼15 K where\nthemagnetizationvanishes. Atlowertemperature, the Gd\nmagneticmoment overcomesthe Mn moment. The overall\ntemperaturedependenceofthemagnetizationcorresponds\nthen to a ferrimagnet created by the two opposite Mn and\nGd sublattices [5,7–9].\nIn this work we study the magnetostructural proper-\nties of Gd 2/3Ca1/3MnO3. The rather complex magnetic\nstructure clearly couples to the atomic lattice giving rise\nto a giant linear magnetostrictive effect [10]. Remarkably,\np-1V. F. Correa et al.\nthe negative field dependence of the MS observed at high\ntemperature changes its sign and becomes positive when\nT < T comp. We use a 4-site mean field approximation to\nmodel the experimental data. It demonstrates that the\ncompetition between Gd-Gd and Gd-Mn spin correlations\nis responsible of the sign change in the MS. This finding\nshowsthatthe usuallyunderestimatedMSassociatedwith\nthe lanthanide 4 forbitals in manganites can be compara-\nble to the usual giant striction given by the re-orientation\nof the Mn 3 dorbitals.\nExperimental details. – Pure single crystalline\nsamples of Gd 2/3Ca1/3MnO3were grown by the floating\nzone technique. Crystal quality and composition have\nbeen checked through XRD and EDS scans. A capaci-\ntive technique was used in the dilation experiments. The\nhigh resolution( ≤1˚A) dilatometer [11] is placed in a evac-\nuated environment with a low pressure ( P <10−1torr) of\nexchangeHe4gas. Magneticfieldisappliedalongthe [020]\ndirection of the orthorhombic Pnmacrystalline structure\n(a= 5.39˚A,b= 5.56˚Aandc= 7.5˚A) in all the experiments.\nDilation experiments are always performed in a longitudi-\nnal configuration with B/bardblL/bardbl[020]. Several samples of\ndifferent sizes have been measured with a perfect agree-\nment between them. Sample length Lis typically about\n200µm.\nResults and discussion. – Representative isother-\nmal linear magnetostriction results after a zero field cool-\ning procedure are shown in Fig. 1 (solid lines). The effect\nisgiantwith noevidenceofsaturationupto B=12T(the\nhighest applied field), reaching a maximum value around\nTc∼80 K (∆ L/L≈10−3). Two very distinctive regimes\nare found:\n(i) above Tcomp∼15 K the field dependence of Lis neg-\native and monotonic. Hysteresis and relaxation effects are\nimportant, mainly in the range 40 K /lessorsimilarT/lessorsimilarTc,\n(ii) below Tcompmagnetostriction becomes positive at\nlow fields B≤7 T (the initial negative slope at B <1 T\nis associated to magnetic domains and is absent if the ex-\nperiment is performed after a field cooling procedure). At\nhigher fields it turns negative again resulting in an overall\nnon-monotonic field dependence of the MS. On the other\nhand, around Tcomp(where the magnetizationalmost van-\nishes), the magnetostriction is negligible below B∼4 T.\nTcompmarks the onset of the Gd magnetic ordering\nwhich dominates the low temperature regime while Mn\nmoments prevail in the high temperature regime. In this\nsense, this unusual magnetostriction strongly points to-\nward the interplay of the different magnetic interactions:\nMn-Mn, Mn-Gd and Gd-Gd. We use a simple model to\nverify this hypothesis where the three different interac-\ntions are introduced in the Hamiltonian via Heisenberg-\nlike terms.\nWe consider an homogeneous network of Gd ions with\none Gd ion for each Mn, ignoring the random nature of\ntheir localizations. As there are 2/3 Gd ions for each Mn,\nwe rescale the Gd effective magnetic moment to J= 2 /3×/s52 /s56 /s49/s50/s40/s98/s41\n/s50 /s32/s120/s32 /s49/s48/s45/s52\n/s32/s76/s32/s47/s32/s76 /s32\n/s66/s32/s40/s84/s101/s115/s108/s97/s41/s52/s53/s32/s75\n/s55/s48/s32/s75\n/s49/s48/s53/s32/s75/s49 /s32/s120/s32 /s49/s48/s45/s52\n/s32/s32/s76/s32/s47/s32/s76/s32/s56/s32/s75\n/s126/s32/s49/s53/s32/s75/s40/s97/s41\nFig. 1: (color online) Experimental (solid) and calculated\n(dashed) magnetostriction. Upper (lower) panel shows resu lts\nin the low (high) temperature range. Curves are vertically\nshifted.\n7/2 = 7/3∼5/2. We choose the 5 /2 value for the Gd spin\nto retain the quantum nature of the spin without rescaling\nof theg-factor (gGd= 2). Manganese ions appears in a\nmixture of 1 /3 of S=3/2 and 2 /3 of S=2. As in both\ncases the orbital magnetic moment is quenched we take\ngMn= 2. To keep the experimental zero temperature net\nmagnetic moment of 1 µB(perfect ferrimagnetic ordering\ngiven by the two sublattices [5]) we take S=2 for the Mn\neffective magnetic moment.\nBased on these considerations we write a Hamilto-\nnian with a ferromagnetic (coupling KMn−Mn) network\nof manganese spins S(S=2) antiferromagnetically cou-\npled (KMn−Gd) to a network of ferromagnetic ( KGd−Gd)\ngadolinium spins J(J=5/2). The magnetic interaction is\ngiven by the Hamiltonian:\nHm=KMn−Mn/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj+KMn−Gd/summationdisplay\niSi·Ji\n+KGd−Gd/summationdisplay\n/angbracketlefti,j/angbracketrightJi·Jj+gµB/vectorB·/summationdisplay\ni(Si+Ji) (1)\nThe smaller values of the effective spins allow us also to\nuse a4-sites(2 Mn and 2Gd) cluster in the ConstantCou-\npling approximation (see Appendix) [12–14]. We consider\nsix neighbours ( z= 6). In the Constant Coupling approx-\nimation the interactions are isotropic, meaning that they\nrepresent averaged interactions.\np-2Unusual giant magnetostriction in the ferrimagnet Gd 2/3Ca1/3MnO3\nFollowing early works [14–16], we consider that the ex-\nchange parameters are strain dependent. If the lattice is\nundersomesmalldistortion δLallthecouplingparameters\nchange accordingly:\nKMn−Mn=K0\nMn−Mn+αδL\nKMn−Gd=K0\nMn−Gd+βδL (2)\nKGd−Gd=K0\nGd−Gd+γδL\nFrom fittings to magnetization experiments, we obtain:\nK0\nMn−Mn= -9 K, K0\nMn−Gd= 8 K and K0\nGd−Gd= 0 K.\nMn-Mn coupling K0\nMn−Mnis ferromagnetic with Tcclose\nto the experimental value of ∼80 K;K0\nMn−Gdis antiferro-\nmagnetic and gives Tcomp∼15 K together with an effec-\ntive null coupling between Gd ions (also expected due to\nthe dipolar origin of those interactions). Both, our exper-\nimental and calculated magnetization results are similar\nto those reported by Snyder et al. [5].\nThe presence of a distortion also increases the elastic\nenergy\nEe= 1/2Cδ2\nL (3)\nFor a state |G/an}bracketri}ht, the total energy Em+Eeis minimized\nfor\nδL(B) =−1\nC/an}bracketle{tG|α/summationdisplay\nSi·Sj+β/summationdisplay\nSi·Ji+γ/summationdisplay\nJi·Jj|G/an}bracketri}htB\nAs we are interested in the length distortion respect to\ntheB= 0 case, we compute\n∆L\nL=δL(B)−δL(0)\n=Λα(/an}bracketle{tG|/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj|G/an}bracketri}htB=0−/an}bracketle{tG|/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj|G/an}bracketri}htB)\n+ Λβ(/an}bracketle{tG|/summationdisplay\niSi·Ji|G/an}bracketri}htB=0−/an}bracketle{tG|/summationdisplay\niSi·Ji|G/an}bracketri}htB) (4)\n+ Λγ(/an}bracketle{tG|/summationdisplay\n/angbracketlefti,j/angbracketrightJi·Jj|G/an}bracketri}htB=0−/an}bracketle{tG|/summationdisplay\n/angbracketlefti,j/angbracketrightJi·Sj|G/an}bracketri}htB)\nwhere Λ χ=χ\nCandχ={α,β,γ}./an}bracketle{t/an}bracketri}htBdenotes the thermal\nexpectation value.\nThecorrelationsarecomputedusingthe 4-siteConstant\nCoupling approximation and correspondingly now i,j=\n1,2. Figure 2 shows the computed spin correlators as a\nfunction of magnetic field (∆ /an}bracketle{tO/an}bracketri}ht=/an}bracketle{tO/an}bracketri}htB−/an}bracketle{tO/an}bracketri}htB=0, where\nO=S1·S2,S1·J1orJ1·J2) in the different temperature\nranges: below, around and above Tcomp.\nThere are several issues to emphasize:\ni) the field dependence of the Mn-Mn correlations\n(∆/an}bracketle{tS1·S2/an}bracketri}ht) is monotonic and, except around Tc, it is\nalso very small (Fig. 2(a)), reflecting the fact that Mn\nsublattice is almost fully polarized at low temperature.\nii) Mn-Gd correlations (∆ /an}bracketle{tS1·J1/an}bracketri}ht) show the largest ef-\nfect below Tc, as seen in Fig. 2(b). This is reasonable\nsincetheappliedfieldtendstoalignbothsublatticesinthe\nsame direction gradually destroying the otherwise almost/s52 /s56 /s49/s50/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s66/s32/s40/s84/s101/s115/s108/s97/s41/s40/s99/s41/s32\n/s32/s60/s74\n/s49/s183/s74\n/s50/s62/s32/s32/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s40/s98/s41/s32\n/s32/s60/s83\n/s49/s183/s74\n/s49/s62/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32\n/s32/s60/s83\n/s49/s183/s83\n/s50/s62/s40/s97/s41\n/s32/s32/s32/s32/s56/s32/s75\n/s32/s32 /s49/s52/s32/s75\n/s32/s32 /s52/s53/s32/s75\nFig. 2: (color online) Computed correlations change respec t\nto the zero field situation for 8 K (below Tcomp), 14 K (close\ntoTcomp) and 45 K (above Tcomp). (a) Mn-Mn correlations,\n(b)Mn-Gd correlations and (c) Gd-Gd correlations.\nperfect low temperature ferrimagnet. Its contribution to\nthe low field total magnetostriction, however, is smaller\nthan that due to Gd-Gd correlationsby a factor1\n4approx-\nimately. At higher fields, on the other hand, ∆ /an}bracketle{tS1·J2/an}bracketri}ht\nbecomes the larger contribution.\niii) Gd-Gd correlations (∆ /an}bracketle{tJ1·J2/an}bracketri}ht) show the most rele-\nvantand distinct behavior. While ∆ /an}bracketle{tJ1·J2/an}bracketri}htis negativefor\nT > T comp, it vanishesaround Tcompand becomes positive\nwhenT < T comp. The physical origin of this sign change\nis related to the local field acting on Gd moments. This\nlocal field is made up of the molecular field created by the\nMn moment BMnplus the external field B. Above Tcomp,\nBMnpoints opposite to Bso an increasing B(B < B Mn)\nresults in a decreasing local field. This decreasing local\nfield reduces the Gd moment and consequently the Gd-Gd\ncorrelations. Below Tcomp, on the other hand, BMnpoints\nin the same direction of B, so an increasing B(B < B Mn)\nresultsinaincreasinglocalfield. Thisincreasinglocalfield\nraises Gd-Gd correlations. This sign change in the Gd-Gd\ncorrelations is indeed responsible for the sign change in\nthe magnetostriction observed at low temperature.\nComputed MS curves are also shown in Figure 1 (Λ α=\n12×10−4, Λβ= 0.8×10−4and Λ γ=−1.6×10−4). Λα\nis chosen as to get a good agreement at high temperature\n(T≥Tc) where the only non-negligiblecorrelatoris ∆ /an}bracketle{tS1·\np-3V. F. Correa et al.\nS2/an}bracketri}htwhile Λ γis chosen as to get a positive striction at low\ntemperature and field ( T < T comp,B≤5 T) where only\n∆/an}bracketle{tJ1·J2/an}bracketri}htis non-negligible. Λ βis then selected to get the\nbest agreement in the whole temperature and field range.\nThe model gives a good account for the non-monotonic\n∆L/LbelowTcomp(see curve at 8 K, Fig. 1(a)). It is\na consequence of two opposite contributions: a negative\nGd-Gd magnetostructural coupling (Λ γ<0) and a pos-\nitive Mn-Gd coupling (Λ β>0). As stated previously,\nMn-Mn correlations are almost saturated and they do not\ncontribute to the MS in this low temperature range. The\ncouplingparameters( Ks)used(fixedbythefitofthemag-\nnetic properties) allows us to reproduce only qualitatively\nthe field value of the MS maximum. The model also ac-\ncounts for the extinction of this maximum at Tcomp∼15\nK, where the striction becomes very small.\nAt higher T(Tcomp≤T≪Tc) the MS gets negative.\nAs before, Mn-Mn correlations almost do not change but\nnow both Gd-Mn and Gd-Gd correlation effects point in\nthe same direction. In the intermediate and high tempera-\nture range( T≫Tcomp), where the Mn-Mn contribution is\nthe morerelevantone, the agreementis fairlygood, except\naroundTc. Not only the magnitude of the magnetostric-\ntion is well accounted, also the curvature of the isotherms\nis very well reproduced (see Fig. 1(b)). It is interesting\nto stress that even though magnetization is an increasing\nfunction of field in the whole temperature range, the mag-\nnetic correlations and so magnetostriction shows two very\ndistinctive regimes: a monotonic magnetostriction at high\ntemperature that becomes non-monotonic below Tcomp.\nIn this isotropic model, Ccan be estimated as vBT,\nwherevis the volume of the perovskite unit cell and BT\nis the Bulk modulus. Mn-Mn parameter Λ αis positive,\nand so is α. Since the interaction is FM ( KMn−Mn<\n0), that implies that |KMn−Mn|increases as the lattice\ngets smaller. This is the expected behavior for exchange-\nlike interactions. There are no available pressure effects\non Gd 2/3Ca1/3MnO3to compare with. Nevertheless, a\nrough estimate can be done by replacing Gd by another\nlanthanide, i.e. by chemical pressure. This substitution\n(keeping the composition at R 2/3Ca1/3MnO3) does not\nmodify the Mn valence and so, its magnetic moment. So,\na changein Tccan in principle be associatedwith an inter-\nion distance dchange. In this isotropic approximation,\nd=v1/3. Taking BT= 150 Gpa [17], and v= 55.36˚A3, [7]\nwe getα= 722 K. For Dy 2/3Ca1/3MnO3,vDy≈55.10˚A3\n[18]. This results in a change of the exchange parameter\nKMn−Mngiven by ∆ K=αdDy−dGd\ndDy≈-2 K. This very\nrough estimate of a 20 percent increase in the exchange\nparameter ( KMn−Mn= -9 K) is of the same order of mag-\nnitude that the Tcincrease observed in Dy 2/3Ca1/3MnO3\n[19].\nGd-Gd parameter Λ γis negative and it may be related\nto the dipolar (anisotropic) origin of these interactions:\nfor zero distortion the net (average)interaction is zero but\nwhenthelatticeshrinksAFinteractionsprevails. Notably,themagnetostrictionassociatedwithGd(lowfieldpositive\nstriction below Tcomp) is of the same order of magnitude\nthan the observed striction in metallic gadolinium [14].\nOn the other hand, Mn-Gd parameter Λ βis positive and\nit is much more difficult to understand since Mn-Gd in-\nteractions are antiferromagnetic: this effective interaction\ndiminishes as the lattice shrinks. This counter-intuitive\nvalue of Λ βcould be related with the rotation of the oxy-\ngen tetrahedra that sourrounds Mn ions.\nConclusions. – An unusual non-monotonic giant\nmagnetostriction is observed in single crystals of the fer-\nrimagnet Gd 2/3Ca1/3MnO3at low temperature ( T <\nTcomp∼15 K) arising from the interplay between the Mn\nand Gd magnetic sublattices. A simple mean field ap-\nproximation where different magnetic interactions (Gd-\nGd, Mn-Mn and Mn-Gd) compete among them and with\nthe elastic energy gives a good account of the observed\nresults. Particularly, the change in the sign of the magne-\ntostrictive effect at low temperature is driven by the com-\npetitionbetweenMn-GdandGd-Gdmagneticcorrelations\nand it does not involve the Mn-Mn correlations. Unlike\nprevious works [20] in manganites where only striction as-\nsociated with dorbitals is considered, our results show\nthatforbitals related striction can be as important.\nAppendix. –\nConstant Coupling Approximation . The constant\ncoupling (CC) approximation [12–14] is an improvement\nover a classical mean field approximation. It allows to\nconsider correlations and to obtain a critical temperature\ncloser to the exact result. A classical mean field approxi-\nmation replacesall the interactions ofa site by an effective\nmagnetic field made up of the external field and its neigh-\nbours magnetization. This neighbours magnetization is\nassumed to be the same than that of the site. In the con-\nstant coupling approximation two systems must give iden-\ntical results for the magnetization. If the original problem\nis in a network with zneighbours per site, one system is\nmade with a single site and z“effective” neighbours, while\nthe other is made up with a cluster of 2 sites and z−1\n“effective” neighbours for each site. In a classical mean\nfield we have to search for an effective field proportional\nto the neighbours magnetization. The CC approximation\nconsists in searching for an effective field in both systems\nsuch that the same magnetization is obtained in the single\nsite and the cluster.\nTo illustrate the procedure we use a simple spin 1/2\nferromagnet. For a single site the Hamiltonian is written\nas\nHss=gµB(z/vectorBeff+/vectorB)·S\nwhereBeffis an effective field. In a classical mean field\napproximation Beff=KMn−Mn/an}bracketle{tS/an}bracketri}htand a self-consistent\n/an}bracketle{tS/an}bracketri}htis looked for. For a two sites cluster the Hamiltonian\nis\nH2s=KS1·S2+gµB/bracketleftBig\n(z−1)/vectorBeff+/vectorB/bracketrightBig\n·[S1+S2]\np-4Unusual giant magnetostriction in the ferrimagnet Gd 2/3Ca1/3MnO3\n/s45/s50/s48 /s48 /s50/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32\n/s50/s48/s75/s77/s32/s40\n/s66/s41\n/s66\n/s101/s102/s102/s32/s40/s84/s101/s115/s108/s97/s41/s32/s83/s105/s110/s103/s108/s101/s32/s83/s105/s116/s101\n/s32/s84/s119/s111/s32/s83/s105/s116/s101/s115\n/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s52/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s45/s50/s48 /s48 /s50/s48\n/s32/s32/s66\n/s101 /s102/s102/s32/s40/s84/s101/s115/s108/s97/s41\n/s32/s32/s77/s40/s83/s83/s41/s45/s77/s40/s50/s83/s41/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s32\n/s32/s32/s50/s55/s75\n/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s52/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s45/s50/s48 /s48 /s50/s48\n/s32/s32/s66\n/s101 /s102/s102/s32/s40/s84/s101/s115/s108/s97/s41\n/s32/s32/s77/s40/s83/s83/s41/s45/s77 /s40/s50/s83 /s41\nFig. 3: (color online) Magnetization as a function of the ef-\nfective field for a single site and a two site cluster (S=1/2,\nK= 30K,z= 6). Upper panel: 27 K. Lower panel: 20 K.\nInsets show the magnetization difference. The classical mea n\nfield transition temperature is TC,MF= 2/3zS(S+1)K= 90\nK.\nIn the upper panel of Fig. 3 we show the magnetization\nat high temperature (27 K, just above the transition) of\nboth a single site and a cluster of two sites as a function of\nBeffwith an applied externalfield B= 0.1 T. These mag-\nnetizationsarezerofor( z−1)Beff=−B(orzBeff=−B\nfor a single site) and show the expected paramagnetic-like\nbehaviour for small systems. At temperatures above the\ntransition temperature both magnetizations agree for a\nsingle field which is the searched effective field (around\n7 T for this temperature; the corresponding magnetiza-\ntion is less than 0 .1µB). As the temperatures lowers, this\nsolution moves toward higher fields with a corresponding\nlarger magnetization. Below the transition temperature,\ntwo new solutions (higher in energy) appears, just as in a\nregular mean field approximation (lower panel of Fig. 3).\nFor the ferrimagnet Gd 2/3Ca1/3MnO3we take as the\n“single” site an unit made up of an effective manganese\n(S=2,g= 2) and an effective Gd (J=5/2, g= 2) ion\nHss=gµB(z/vectorBeff,Mn+/vectorB)·S\n+gµB(z/vectorBeff,Gd+/vectorB)·J\n+KGd−MnS·J\nThe cluster is made with two Mn and two Gd ions and\nwe takez= 6. Each Mn (Gd) ion interacts with the other\nand with an external field made by the z−1 remaining\nneighbours. In each site, there is a Gd-Mn interaction.The Hamiltonian is\nH2s=KMn−MnS1·S2+KGd−GdJ1·J2\n+KGd−Mn(S1·J1+S2·J2)\n+gµB/bracketleftBig\n(z−1)/vectorBeff,Mn+/vectorB/bracketrightBig\n·[S1+S2]\n+gµB/bracketleftBig\n(z−1)/vectorBeff,Gd+/vectorB/bracketrightBig\n·[J1+J2]\nWhen the effective field is found, correlations between\nthe different ions can be computed in the 2-site (four ions)\ncluster.\n∗∗∗\nWe thank M. T. Causa, L. Manuel, C. Balseiro and A.\nAligia for fruitful discussions. The authors are member\nof CONICET, Argentina. Work partially supported by\nANPCyT PICT05-32900, PICT07-00812, PICT07-00819,\nPICT08-1043 and SeCTyP-UNCuyo 06/C326.\nREFERENCES\n[1]Jin S., Tiefel T. H., McCormack M., Fastnacht R.\nA., Ramesh R. andChen L. H. ,Science,264(1994) 413.\n[2]Ibarra M. R., Algarabel P. A., Marquina C.,\nBlasco J. andGarc´ıa J.,Phys. Rev. Lett. ,75(1995)\n3541.\n[3]Kimura T., Tomioka Y., Asamitsu A. andTokura Y. ,\nPhys. Rev. Lett. ,81(1998) 5920.\n[4]Hwang H. Y., Cheong S-W., Radaelli P. G., Marezio\nM. and Batlogg B. ,Phys. Rev. Lett. ,75(1995) 914.\n[5]Snyder G. Y., Booth C. H., Bridges F., Hiskes R.,\nDiCarolis S., Beasley M. R. andGeballe T. H. ,Phys.\nRev. B,55(1997) 6453.\n[6]Hueso L. E., Rivas J., Sande P., Fondado A., Ri-\nvadulla F. andL´opez-Quintella M. A. ,J. Magn.\nMagn. Mater. ,238(2002) 293.\n[7]Pe˜na O., Bahout M., Ghanimi K., Duran P., Gutier-\nrez D.andMoure C. ,J. Mater. 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A., Ritter C., Marquina C., Blasco\nJ., Garc ´ıa J., del Moral A. andArnold Z., Nature\n386, 256 (1997). ,Nature,386(1997) 256.\np-6"    },    {        "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi,  Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School  of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp  2 \n We investigate the Gilbert damping parameter  for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility,  was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that  should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role.  \n  3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme  of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment.  \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for  the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e  \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including  𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E  e l e m e n t ,  w h i c h  l i e s  f a r  b e l o w  t h e  F e r m i  l e v e l ,  d i d  n o t  p l a y  an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned  into 5-µm-wide and 500-µm-long  microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z  direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing  line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s  a n  \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t  𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows.  The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39]  \n          𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻,                                                                                ሺ1ሻ  \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s  t h e  \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s ,  b u t  f o r  s i m p l i c i t y ,  w e  \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular  momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s  𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n          𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0                                           ሺ2ሻ  \nUsing Eq. (2) to find the solution of 𝛼, we find \n          𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ.                                              ሺ3ሻ  \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s,  𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis.  7 \n  The energy dissipation (per unit cross section) through the DW  dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆  2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional  regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y  \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e  𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion:  \n          𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ .                                                                                         ሺ4ሻ  \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter,  𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions,  𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical:  𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion  𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e  e s t i m a t e d  b y  \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both  the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d  𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo  is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum  𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇  to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand,  𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇  ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the  damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t  \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a  governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c  r e s o n a n c e  ( F M R )  \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s  f o r  l o w - d i s s i p a t i o n  \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o  a l l o y s  o v e r  a  w i d e  r a n g e  o f  t e m p e r a t u r e s  a c r o s s  𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t  \nthe magnitudes of the quantities.  Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n   16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r  f i t s .  ( c )  T h e  D W  \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum  𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements  \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported  by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the  Korea Government \n(MSIP) (No. 2017R1C1B2009686).  \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n  202\n 222\n 242\n 262\n 282<v> [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6  M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]"    },    {        "title": "2312.15584v2.Controllable_magnon_frequency_comb_in_synthetic_ferrimagnets.pdf",        "content": "1 \n  \nDesign of c ontrollable magnon fr equency comb  in synthetic ferrimagnet s \n \nY . Liu1*, T. T. Liu1*, Q. Q. Yang1, G. Tian1, Z. P. Hou1, D. Y . Chen1, Z. Fan1, M. Zeng1,  \nX. B. Lu1, X. S. Gao1, M. H. Qin1,†, and J. –M. Liu1,2 \n1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials \nand Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China  \n \n  \n \n*These authors contributed equally to this work  \n†Email: qinmh@scnu.edu.cn  2 \n [Abstract]  Magnon frequency comb  provides opportunities for  exploring magnon nonlinear \neffects and measuring  the transmission  magnon frequency  in magnets , whose controllability \nbecomes vital for modulating the operating frequency and improving the measurement accuracy . \nNevertheless, such controllable frequency comb remains to be explored. In this work , we \ninvestigate theoretically and numerically the skyrmion -induced magnon frequency comb effect \ngenerated by interaction between the magnon excitation  mode and skyrmion breathing mode in \nsynthetic ferrimagnets. It is revealed that both the skyrmion breathing mode and the magnon \nfrequency gap  closely depend on  the net angular momentum δs, emphasizing the pivotal role of \nδs as an effective control parameter in governing the comb teeth.  With the increase of δs, the \nskyrmion size decreases, which results in the enlargement of the breathing frequency and the \ndistance between the comb teeth. Moreover, the dependences of the magnon  frequency  gap on \nδs and the inter-layer coupling allow one to modulat e the comb  lowest coherent frequency  via \nstructural control . Consequently,  the coherent modes generated by the comb may range from \ngigahertz to terahertz frequencies, serving as a bridge between microwave and terahertz waves. \nThus, this work represents a substantial advance in understanding the magnon frequency comb  \neffect  in ferrimagnets.   \n \nKeywords:  magnon, frequency comb, skyrmion, ferrimagnet s \n 3 \n Ⅰ. INTRODUCTION  \nThe development of the optical frequency comb provides a precise and direct link between \nmicrowave and optical frequencies, which represents  an established milestone for visible /near -\ninfrared metrology  [1]. This great breakthrough  has inspired scientists to search for frequency \ncombs in other physical systems  for potential applications . For example , as the quantum of \ncollective spin excitations in ordered magnets , magnons are drawing more and more attention  \nas information carriers for their low power consuming and convenient manipulation  [2–8]. \nActually, m any photon -like phenomena including  Kerr nonlinearities and Goos -Hänchen effect \nhave been reported in magnonic systems  [9–11], as well as the magnon frequency comb  effect . \nSpecifically , the magnon frequency combs have been predicted in earlier theoretical works , \nwhich  primarily relies on the nonlinear processes of magnons  attributing to the three -magnon \neffect in magnetic media  [12–14]. However, the three -magnon interaction of  uniform state s are \ntypically weak, which poses a significant challenge in creating magnon frequency combs  with \nsufficiently good performance . Interestingly, such nonlinear interaction can be significantly \nenhanced via the scattering between magnons and nonco llinear magnetic textures or so -called \nmagnetic solitons  [12,15 –17], which certainly benefits to the magnon frequency combs .  \nOf particular interest here, are magnetic skyrmions . As a type of vortex -like magnetic \ntextures , skyrmions  have shown great promise for both fundamental physics and potential \napplications in spintronics  [18–26]. Importantly , the nonlinear processes between the skyrmion \nbreathing modes and spin -wave modes can generate the magnon frequency combs in \nferromagnets  [12]. In detail , magnon s are scatter ed by the skyrmion and they simultaneously \ninteract with the skyrmion breathing mode. This dual interaction gives rise to a sequence of \nmagnon confluence and splitting processes , which ultimately results in  the magnon frequency \ncomb effect with the teeth distance corresponding to the skyrmion breathing mode. Furthermore , \nthe skyrmion -induced frequency combs  in antiferromagnets  can extend the frequency range to \nterahertz  [16,17] , significantly advanc ing terahertz technology applications in ultrafast \nmagnonics.  \nThese important reports  present an opportunity to realize frequency combs in magnonic \ndevices, which extensively  expand s the current understanding of fundamental physics. 4 \n However , no reliable  solution for regulating the comb teeth is available due to the difficulty in \ncontrolling the skyrmion breathing mode  in ferromagnetic and  antiferromagnetic media , as far \nas we know . Undoubtedly, modulating the comb teeth  is essential for future applications  \nbecause  it determines  the measurement accuracy.  Furthermore, the magnon frequency combs \nin ferromagnets and antiferromagnets operate in the microwave and terahertz bands, \nrespectively , while  the frequency gap between the two bands definitely deserves  to be bridged  \nto broaden the application range of the comb .  \nIn this regard , ferrimagnets  which combine the benefits of ultra -high working frequencies \nwith ease of detecting and modulating are  greatly potential in  solving this problem . As one of \nthe most important parameter s in ferrimagnets, the net angular momentum δs significantly \naffects the dynamics of magnons and skyrmion, and it  can be elaborately adjusted through \ntuning temperature or material composition  [27–33]. Generally, v ariations in δs are \naccompanied by alternations of the total magnetization, whi ch affects  the skyrmion  size and in \nturn changes the breathing mode  of the skyrmion . In addition , the critical frequency beyond \nwhich magnon excitation is possible  also depends on δs, allowing one to  modulat e the lowe st \nfrequency  within the comb . More importantly , magnon excitation frequency in ferrimagnets \nreaches into hundreds of gigahertz range, allowing the comb coherence mode to be extended \nup to terahertz and down to microwave frequencies, as depicted in Fig. 1(a) . This merit  will \nestablish  a pivotal connection between the microwave and terahertz frequency bands . Thus, \neffective control  of magnon frequency combs could be realized in ferrimagnets which urgent ly \ndeserves to be clarified  considering its importance for the development s of magnonics  and \nspintronics . \nIn this work , we investigate numerically and theoretically the magnon frequency comb \neffect in ferrimagnets induced by the interaction between the magnon excitation  mode and the \nskyrmion breathing mode . It will be demonstrated that both the skyrmion breathing mode and \nmagnon frequency gap depends on the net spin angular momentum  δs, allow ing one to control \nthe comb through tuning δs. In addition , the coupling between the two sublattices also affects \nthe magnon frequency  gap and modulates the lowest coherent mode  of the comb. The coherent \nmodes generated by the comb exist  from gigahertz to terahertz frequencies, serving as a bridge 5 \n between microwave and terahertz waves.  \n \n \nII. THEORETICAL ANALYSIS  \nA. Ferrimagnetic  skyrmion size  \nAs revealed in the earlier works,  skyrmion size mainly determines the energy to induce \nfluctuation and  affects the breathing mode of  skyrmion  [34–36]. Considering its important role \nin skyrmion dynamics , it would be of first priority to theoretically derive the skyrmion size . \nHere, the size is estimated based on the 360° domain wall  profil e which has been successfully  \nused in ferromagnets and antiferromagnets  [37,38] .  \nWithou t loss of generality , we study  a synthetic ferr imagnetic bilayer  film consisting of \ntwo diverse  ferromagnetic layers  with an antiferromagnetic inter -layer Ruderman -Kittel -\nKasuya -Yosida  exchange  coupling related to  the spacer , as depicted in  Fig. 1( b) [39]. The \ncontinuum mode l Hamiltonian  density  of the system can be written as  \n() ()2 2\ndemag interˆ 1 ( ) ,zz\ni i i i i i z H A D m m K HH     =  +  −  + −  + +   m m m m e\n (1) \nwhere mi represent s the local  unit magnetization vector  with the layer index i. A, D, and K are \nthe intra-layer ferromagnetic exchange, Dzyaloshinskii -Moriya (DM) interaction , and \nperpendicular magnetic anisotropy  constant s, respectively. Hdemag is the demagnetization  energy , \nand the last term is the inter-layer exchange coupling between the nearest neighbors which reads   \n \n() inter 1, ij Hc= −  mm               (2) \nwhere σ is the bilinear surface exchange coefficient between the two surface s, c is the  thickness  \nof ferrimagnetic layer . Here , the dipolar interaction is ignored considering that the net \nmagnetization  of ferrimagnet  could be  typically  several  order s of magnitude smaller than \nferromagnet  [29,40] . \nNext , we introduce the Néel vector n = (m1 − m2)/2 and the magnetization vector m = (m1 \n+ m2)/2 to deal with the dynamic equations of ferrimagnets . Therefore, t he Hamiltonian density  \ncan be expressed as   \n() ()() ()2 22\ndemag22 2 2 1 1z z z H A D n n K ncH=  +   −  + − + −  + n n n m.\n  (3) 6 \n It is convenient to use the polar coordinates  (r, ϕ), the polar  angle Θ = Θ(r, ϕ), and the azimuthal \nangle  Φ = Φ(r, ϕ), considering the rotational symmetry  of skyrmion . A skyrmion centered at r \n= 0 can be described by  \n(),,r  = = +\n              (4) \nwhere ν is the  skyrmion  vorticity, an d λ is the  skyrmion  helicity.  Then , the free energy  of the \nsystem  including a skyrmion in the polar coordinates can be written as  \n() ()22\n2 d4 sin d 4 sin d 4 dd\nd4 cos sin 2 cos dd2 .E Ac r r K c r r r rrr\nDc r rrr  \n      = +  +  −      \n− − +  −  \n\n    (5) \nHere, K´ = K − μ0Md/2 is the corrective easy -axis anisotropy  with Md = (M2 \n1+M2 \n2)/2 and the \nsaturation magnetization  M1 (GdCo)  and M2 (Co), in which the shape anisotropy of the \ndemagnetizing field  is taken into account . For a Néel  type skyrm ion (ν = 1 and λ = 0), Eq. ( 5) \nis simplified to \n22\n2 d14 sin d 4 sin dd\nd14 sin 2 dd2 .E Ac r r K c r rrr\nDc r rrr\n    = +  +       \n− + \n\n      (6) \nIt is noted that t he inter-layer nearest neighboring spins  keep antiparallel  with each other when \na strong inter-layer coupling is considered . Then, b ased on  the 360° domain wall  profil e [37,41] , \nthe skyrmion size can be analytically calculated by  \n2 2 216 .ARDAK D K=−\n             (7) \nIt is shown that the skyrmion size closely depends on the corrective anisotropy K´, in addition \nto the exchange and DM interactions. As a result, one may tune the net angular momentum δs \nand the tota l saturation magnetization  to control the skyrmion size, which in turn modulates the \nbreathing modes of the skyrmion  [34,42] .  \n \nB. The three -magnon processes  \nThe interaction between the breathing mode of skyrmion and magnons  should be clarified 7 \n to understand the three -magnon process which is responsible for the frequency comb  effect . We \nintroduce bosonic creation ( a† \nk) and annihilation ( ak) operators and reformulate the Néel vector \nn using the local coordinate ( ê1, ê2, ê3) and the Holstein -Primakoff transformation  [12,43]  , \n† ††\n†\n1\n† ††\n†\n2\n†\n31\n4 2\n4 2\n1 .a aa a a an a aS S\ni a aa a a an a aS S\nn a a + + −\n −− − −\n−,\n,\n           (8) \nBy substituting these connections back into Eq. (1), we derive the three -magnon terms as  \n() † † † †\n† †3\n†( ) ( ) ( ) ( ) . .\n      . .,x y x y\ne b d e b sa aa a aa a aa a a H a a a a h c\na a a a a a h c +  +  +  +  +\n + +\n     (9) \nHere, ae represents the magnon excitation mode with frequency ωe, while ab denotes the \nskyrmion breathing mode with frequency ωb. ad and as correspond to the difference -frequency \nmode with the frequency ωd = ωe − ωb and the sum -frequency mode with frequency ωs = ωe + \nωb, respectively. This occurrence of sum - and difference - frequency modes during the three -\nmagnon process results in the magnon frequency comb  effect . \n \nC. Manipulation of the magnon frequency co mb \nSubsequently, w e study  the magnon  excitation in the system . The dynamics of m and n can \nbe described by the continuum Landau -Lifshitz -Gilbert (LLG) equation  [6,28,29,40] , \n( )(), s m n ss+ =−  +  + m n m f n f n n\n          (10a) \n()()(), ms ss  =−  +  +  n n f n n n m\n          (10b) \nwhere s = (s1 + s2) and δs = (s1 − s2) with the  magnitude of the spin density si = Mi/γi, Mi is the \nmagnetization and γi = giμB/ħ is the gyromagnetic ratio of sublattice i. Simply, the damping \nconstant s of two layers are  considered  to be the same . fn = − δH / δn and fm = −δH / δm denote \nthe effective fields of n and m, respectively.  After safe and necessary simplification s of \ninsig nificant terms , we obtain  \n \n, . 44sc sc\n=  = m n n m n n\n              (11) \nIt is revealed  that the  dynamics of m is mainly determined by the inter -layer exchange coupling . 8 \n We derive the equation of motion for n by inserting Eq. ( 11) into [Eq. (10a)], and  obtain  \n \n(),s n s      + =− +  nf n n n n n\n             (12) \nwhere  = s2/a is the constant of inertia  with the homogeneous exchange constant  a = σ/c.  \nBy defining a complex field as ψ± = nx ± iny for the right - (+ sign) and left-handed  (− sign) \nmagnon s, and linearizing the above equation for nx and ny, the dispersion  and magnon frequency \ngap fgap are obtained ,  \n224 4 4\n, 2ss Ak K\nf  \n   + + +=\n           (13a) \ngap216\n2,ss Kf  \n  + +=\n             (13b) \nwhere k is the magnon wave vector , and +/− corresponds to the right -/left-handed magnons . It \nis demonstrate d that fgap closely depends on  δs and the inter-layer exchange coupling  σ. For non -\nzero δs, the left - and right -handed magnon s demonstrate different dispersion behaviors  and \nresonance frequencies. Take the  left-handed magnons  as an example , two δs terms in numerator  \nare with opposite signs, leading to the insensitiveness of the frequency gap to δs. Furthermore, \nbesides tuning δs, one may adjust the spacer thickness to control the inter -layer coupling and to \nmodulate the frequency gap.   \nUndoubtedly, t he theoretical analysis present ed above should be checked by numerical \nsimulations to ensure its reliability , as will be reported in the next section.  \n \n \nⅢ. MICROMAGNETIC SIMULAT ED RESULTS and DISCUSSION  \nIn this section, we perform the micromagnetic simulations to check  the validity of above  \ntheoretical analysis and  to further reveal the modulation of the magnon frequency comb in \nferrimagnets. Here, the simulations are performed on a ferrimagnetic bilayer GdCo/Co with the \nsize of 500 nm × 500 nm × 2 nm and  the cell size of 1 nm × 1 nm × 1 nm using MUMAX3  [44], \nnoting that  GdCo  and Co are widely used materials .  \nWe set the thickness of each ferromagnetic layer 1nm, the time step  of 5×10−15 s, the intra -\nsublattice exchange stiffness A = 10 pJ/m , the perpendicular magnetic anisotropy constant K = 9 \n 0.2 MJ/m3, DM coefficient D = 0.9 mJ/m2, the gyromagnetic ratio γi = giμB/ħ with the g-factors \ng1 = 2.2 and  g2 = 2, the damping constants α1 = α2 = 0.001 , the bilinear surface exchange \ncoefficient σ range s from −10 mJ/m2 to −30 mJ/m2, which are reasonable values for the realistic \nsystem . The magnetization s of nine various  cases are shown in Table 1, which correspond to \nnine different δs and Md. Here, the value of Md keeps decreasing as δs increases, consistent with \nthe experiments in which  elevating temperature to modulate δs reduces both the magnetizations \nof two sub -lattices [31,45] . The m agnon s are excited by applying a microwave driving field h(t) \n= (hsin(2 πft), 0, 0)  on a nanobelt , and they propagate along  the x-direction and are scattered by \nthe skyrmion , as depicted in Fig. 1( c). \n \nA. Effect of δs on the magnon frequency comb  \nIn ferrimagnet s, the demagnetization field and δs can be modulated through tuning the \nmagnetizations M1 and M2, which affect Md and the skyrmion size. Fig. 2(a) gives the Eq. (7) -\ncalculated and LLG simulated skyrmion radius R for various δs, which demonstrates a \nmonotonous decrease of R with the increas e of δs. It is noted that the skyrmion size is mainly \ndetermined by the competition between the magnetic anisotropy term and the DM interaction . \nThus, the corrective  anisotropy K´ is enhanced as δs increases  due to the decrease of Md, while \nthe DM interaction hardly be affected, resulting in the gradual  decrease of the skyrmion \nsize [31,45] .  \nSubsequently,  the effect of δs on the skyrmion breathing mode  is investigated through \nanalyzing the magnetic  excita tion spectrum. Here, a sinc-function  field h0(t) = (h0sinc(2 πf0t), 0, \n0) with the amplitude h0 = 50 mT and frequency f0 = 900 GHz is applied over the whole film to \ninduce  the system  response. Then, t he internal spectrum is obtained by performing a fast Fourier \ntransform (FFT) for each Néel vector n, and the corresponding results are shown in Fig. 2(b) . \nSpecifically, the spectra of the in-plane component nx or ny depict the ferrimagnetic resonance \nfrequency, representing the magnon frequency gap, whereas the spectrum of the out-of-plane \ncomponent nz corresponds to the skyrmion breathing mode. The primary peak  of the breathing \nmode  gradually shifts towards the high frequency side as δs increases, demonstrating the \neffective modulation of the breathing mode through tuning δs. Generally, it is expected that a 10 \n particle with small mass has a pronounced response to external fluctuations  [41]. As a result, \nthe breathing mode frequency increases when the radius and effective mass of the skyrmion are \ndecreased, as summarized in Fig. 2(a). Importantly , the frequency alternation reaches up to ~50 \nGHz, providing an opportunity to elaborately manipulate the magnon frequency comb teeth.  \nIt has be en established that generating  magnon frequency comb requires a substantial \ndriving force  [12], and a  microwave field h(t) = (hsin(2 πft), 0, 0) with a large amplitude h = \n100 mT and frequency f = 500 GHz is applied  to excite magnons . To investigate the nonlinear \nmagnon processes, we analyze the time -dependent Néel vector n around  the skyrmion region \nusing FFT. Fig . 3(a) illustrates the response distribution in the frequency space for various  δs. \nWithin the range  of δs from −1.24 × 10-7 Js/m3 to +1.24 × 10-7 Js/m3, a series of peaks arise  and \ntake shape as a comb. The spacing between the peaks  increases with the increasing δs, while the \nnumber of magnon coherent modes decreases. This method of tuning the coherent frequency \nspacing is feasible since δs can be stably adjusted. Furthermore , the magnitude of the frequency \npeak depends on the discrepancy between the driving frequency and the ferrim agnetic \nresonance frequen cy. For example , as δs increases, the magnon energy decreases and more \nmagnons are excited during the magnon nonlinear process.  Thus, th e magnitudes of the \nfrequency peaks in combs are generally enhanced  as the ferrimagnetic  resonance frequency \n(dashed lines)  approaches to  the driving frequency , as shown in Fig. 3(a) .  \nThe magnon dispersion are simulated using FFT for n in the magnon transmission region  \nto estimate  the magnon  frequency gap . Fig. 4 presents the simulated dispersion of the left- and \nright -handed magnons for various δs for σ = −10 mJ/m2, which coincides well with the \ntheor etical  prediction in Eq. (1 3). For compensated ferrimagnet with zero δs, the dispersions of \nthe left - and right -handed magnons overlap with each other with a frequency gap  around ~255 \nGHz [29], as shown in Fig. 4(b) . The overlap of the dispersions is broken for non -zero δs. For \na positive δs, the frequency gap  of the right -handed magnons decreases, while that of the left -\nhanded magnons is enhanced, as shown in Fig. 4(a). One notes that the  right -handed magnons \nare with an energy generally lower than the left-handed magnons  for a positive δs, resulting in  \na smaller frequency gap . The same mechanism also works for negative δs, as demonstrated in \nFig. 4(c) which shows that the left -handed magnons are with a frequency gap  lower that the 11 \n right -handed magnons.  \nThe Eq. (13) -calculated and LLG simulated frequency gap s fgap for various δs are \nsummarized in Fig. 3(b ). The analytical results are well consistent with the simulations, while \nthe quantitative deviation for large | δs| may attribute to the ignorance  of some related higher -\norder terms of m in theory.  It is clearly shown that fgap of the left -handed magnon gradually \ndecreases with the increase of  δs, while that of the right -handed magnons increases rather \nquickly , resulting in the fgap discrepancy between the left - and right -handed magnons  for non -\nzero δs. \nAs a matter of fact , the δs-induced fgap discrepancy affects the frequency domain of the \ncomb  because  the lowest coherent frequency is limited by  fgap. In other words, the magnons can \nbe excited and transferred only when its frequency is larger than fgap, while only the skyrmion \nbreathing mode fb exist s below the frequency gap. To investigate the effect of δs on the \ncoherence modes , we calculated the spectra of magnons and present  the results  in Fig.  5. Taking \nδs = −1.24 × 10-7 Js/m3 as an example  shown in Fig. 5(a) , the lowest coherent frequency of the \nleft-handed magnons is ~276 GHz, while that of the right -handed magnons is ~190 GHz.  Thus, \nδs impacts  not only the frequency comb teeth  but also the coherence modes  of the  right - and \nleft-handed magnon s, as also shown in Fig. 5(b) . \nFurthermore, it is worth noting that  there is a threshold value of the exciting field to \ngenerate the frequency comb. The simulated response frequency ( fr) of the system as a function \nof the driving amplitude  for various δs are presented  in Fig. 6. The frequency comb can be \nobviously observed when the exciting field is larger than 41.1 mT for δs = −1.24 × 10-7 Js/m3. \nFurthermore, the threshold value significantly decreases with the increase of δs, which attributes \nto the fact that the resonant frequency of ferrimagnets gradually approaches to the driving \nfrequency. Thus, the excitation of magnon s is enhanced as δs increases,  and the frequency comb \ncould be generated  even  under a weak exciting field.  \n \nB. Modulation  of the frequency gap  \nThe δs-dependent  skyrmion size  and frequency gap  of magnons have been clarified, \nallowing one to modulate the teeth and coherent  mode  of the comb through  tuning δs. Moreover , 12 \n the frequency gap  also depends on the inter -layer coupling, as predicted in Eq. (1 3), which \ndeserves to be further revealed considering the profound significance of extending the lowest \ncoherent mode  of the comb  for future applications  and the controllability of  the inter -layer \ncoupling through tuning  the spacer  depth .  \nHere,  the effect of the inter -layer coupling on fgap is investigated through analyzing  the \nsurface exchange coefficient σ-dependent magnon dispersion. For simplicity, a sinc -function \nfield with a frequency ~900GHz is applied to excite the magnon modes in compensated \nferrimagnet δs = 0. Fig. 7 presen ts the simulated and calculated  magnon dispersion for various \nσ, which exhibits well consistence between the simulations and theory. With the enhancement \nof the inter -layer coupling, the frequency gap  is significantly widened . This phenomenon \nattributes to the fact that the enhanced inter -layer exchange field suppresses excit ation of low \nenergy magnons, similar to the case of antiferromagnets  [46]. Thus , one may control the spacer \ndepth  and in turn tune  the inter -layer coupling  to manipulate the lowest coherent mode  of the \ncomb.  Additionally, both the frequency gap s of the left - and right -handed magnons  are \nenhanced as the inter -layer coupling increases , which could be used to alleviat e possible \ninterference of unwanted low -frequency magnon modes.    \nAt last, we  investigate the dependence of the magnon frequency comb on the inter-layer \ncoupling and the driving frequency . The simulated response frequency as a function of driving \nfrequency for various σ are summarized  in Fig. 8. For every frequency , the magnon frequency \ncomb can be realized. With the enhancement of the inter -layer coupling, the skyrmion  breathing \nfrequency  fb slightly shifts towards the high frequency side, which enlarges the distance \nbetween the comb teeth. Moreover, the resonance frequency also increases  and approaches to \nthe driving frequency. Thus, the response of the system is strongly enhanced , as shown in Fig s. \n8(a)-(c). However, the number of the comb teeth could be reduced, noting that magnons cannot \nexist below the resonance frequency.  \n \nC. Discussion  \nSo far, we have elucidated the important role of the net angular momentum δs and the inter -\nlayer coupling  in modulating the magnon frequency comb in ferrimagnets , which provides 13 \n valuable insights for material selection and serves as a guide for future experiments. For \nexample, the coherent modes generated by the comb in ferrimagnets could exist from  gigahertz \nto terahertz frequencies, serving as a bridge between microwave and terahertz waves. More \nimportantly, t he controllable comb teeth provide a solution for precise synthesis of high  \nfrequency magnon modes.  \nSimilar to the earlier report , δs emerges as a essential  control parameter in modulating \nferrimagnetic dynamics, offering better control through temperature or material composition \nadjustments  [27–31]. Simultaneously, the inter -layer coupling can be modulated by tuning  the \nspacer thickness  [39]. Notably, most parameters chosen in this work  are comparable to those in \nGdCo/Co  [20,39,45,47] , and the actualization of magnon frequency combs in ferrimagnets \nawaits validation in forthcoming experiments . Furthermore, t he magnon frequency comb  effect  \nuncovered  here exhibits universality over all chiral magnetic models, and the system can be \nextend ed from  thin films to bulk materials.  \n \n \nⅣ. CONCLUSION  \nIn conclusion, we have studied  the magnon frequency comb  effect  in ferrimagnets using \nanalytical methods and numerical simulations. The coherent modes generated by the comb \ncould exist from gigahertz to terahertz frequencies, serving as a bridge between microwave and \nterahertz waves.  As δs increases, the total magnetization a nd skyrmion size decreases, which \nenlarges the skyrmion breathing frequency and the distance between the comb teeth. The close \ndependence of the skyrmion breathing mode on δs allow s one to modulate the teeth of the comb \nthrough  tuning temperature or material composition. Furthermore , the magnon frequency gap \nis also related to  δs and the inter-layer coupling, providing the opportunity to modulate the \nlowest coherent frequency of the comb through structural control.  Thus, this work demonstrates \nthe controllable magnon frequency comb in ferrimagnets and holds significance for broader \napplication of ferrimagnets and magnon frequency comb.  \n  \n 14 \n Acknowledgment  \nThe work is supported by the Natural Science Foundatio n of China (Grants No. U22A20117, \nNo. 52371243, No. 51971096, No. 92163210, and No. 51721001), the Guangdong Basic and \nApplied Basic Research Foundation (Grant No. 2022A1515011727).  \n 15 \n References:  \n[1] S. T. Cundiff and J. Ye, Colloquium: Femtosecond Optical Frequency Combs , Rev. Mod. \nPhys. 75, 325 (2003).  \n[2] P. Yan, X. S. Wang, and X. R. 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Parameters used in the micromagnetic  simulation.  \nIndex  1 2 3 4 5 6 7 8 9 \nM1 (kA/m)  460 455 450 445 440 435 430 425 420 \nM2 (kA/m)  440 430 420 410 400 390 380 370 360 \nδs (×10-7 J∙s/m3) −1.24 −0.93 −0.62 −0.31 0 0.31 0.62 0.93 1.24 \nMd (×1011 A2/m2) 2.026 1.960 1.895 1.831 1.768 1.707 1.647 1.588 1.530 \n 20 \n  \nFig. 1 . (a) Schematic illustration of interactions between the ferrimagnetic  skyrmion  and \nmagnons , and the resul ted magnon frequency comb , and  (b) sketch of a  synthetic  ferrimagnetic  \nsystem with a Néel skyrmion , and  (c) snapshot of the interaction between  propagating magnons \nand skyrmion in ferrimagnet .  \n \n21 \n  \nFig. 2. (a) The analytically calculated (lines) and simulated (symbols) skyrmion radius  (R) and \nthe skyrmion breathing frequency ( fb) as functions of the  net angular momentum δs, and (b) t he \nskyrmion breathing mode  spectrum  for ny and nz for δs = −1.24 × 10-7 Js/m3 (top), 0 (middle), \nand 1 .24 × 10-7 Js/m3 (bottom) .  \n \n22 \n  \nFig. 3. (a) The magnon spectrum for δs = −1.24  ×10-7 Js/m3 (top), 0 (middle), and 1.24  ×10-7 \nJs/m3 (bottom), respectivel y. The black (red) da shed line  indicates the frequency gap of the left - \n(right -) handed mag nons. (b) The calculated (lines) and simulated (symbols) frequency ga p (fgap) \nas a function of δs with the interlayer coupling σ = −10 mJ/m2 for different magnon mode \nhandedness.  \n \n23 \n  \nFig. 4. Ferrimagnetic magnon  dispersion for (a ) δs = −1.24 ×10-7 Js/m3, (b) δs = 0 and ( c) δs = \n−1.24 ×10-7 Js/m3 with the interlayer coupling σ = −10 mJ/m2. Dash  lines  show the analytical \nresults.  \n  \n24 \n  \nFig. 5. The magnon spectrum for (a) δs = −1.24 ×10-7 Js/m3 and (b) 1.24 ×10-7 Js/m3 with \ndifferent magnon handednesses . The black (red) dashed line indicates the frequency gap of the \nleft- (right -) handed magnons.  \n25 \n  \nFig. 6. The response frequency ( fr) of the system as a function of the microwave driving \namplitude ( h) for (a) δs = −1.24 × 10-7 Js/m3, (b) δs = 0 Js/m3, and (c) δs = +1.24 × 10-7 Js/m3. \nThe driving frequency is fixed at 500 GHz.  \n26 \n  \nFig. 7. Ferrimagnetic magnon  dispersion for (a)  σ = −10 mJ/m2, (b) σ = −20 mJ/m2 and ( c) σ = \n−30 mJ/m2 for δs = 0 Js/m3. \n \n27 \n  \nFig. 8. The respo nse frequency ( fr) of the system as a function of the microwave driving \nfrequency ( f) for (a) σ = −10 mJ/m2, (b) σ = −20 mJ/m2, and ( c) σ = −30 mJ/m2. \n \n"    },    {        "title": "1711.10790v1.Thermal_contribution_to_the_spin_orbit_torque_in_metallic_ferrimagnetic_systems.pdf",        "content": "1 \n Thermal contribution to the spin -orbit torque in metallic/ferrimagnetic systems  \nThai Ha Pham1, S.-G. Je1,2, P. Vallobra1, T. Fache1, D. Lacour1, G. Malinowski1, M. C. Cyrille3, G. Gaudin2, O. \nBoulle2, M. Hehn1, J.-C. Rojas -Sánchez1* and S. Mangin1 \n1 . Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, F-54011 Nancy, France  \n2 . CNRS, SPINTEC, F -38000 Grenoble \n3. Leti, technology research institute, CEA, F-38000 Grenoble  \n*juan -carlos.rojas -sanchez@univ -lorraine.fr  \n \nAbstract  \nWe report a systematic study of current -induced perpendicular magnetization switching in \nW/Co xTb1-x/Al thin films with strong perpendicular  magnetic anisotropy . Various Co xTb1-x \nferrimagnetic alloys with different magnetic compensation temperatures are presented. The \nsystem s are characterized using MOKE , SQUID and  anomalous Hall resistance at  different \ncryostat temperature ranging from 10K to 350 K. The current -switching experiments  are \nperformed in the spin–orbit torque geometry where the current pulses are  injected in plane and the \nmagnetization reversal is detected by measuring the Hall resistance. The full reversal magnetization has \nbeen observed in all samples . Some experimental results could only be explained by the strong  sample  \nheating effect during the current pulse s injection . We have found that, for a given composition  x \nand switching polarity , the devices always reach the same temperature Tswitch (x) before \nswitching independently of the cryostat  temperature. Tswitch  seems to scale with the Curie \ntemperature of the CoxTb1-x ferrimagnetic alloys. This explain s the evolution of the critical \ncurrent (and critical current density) as a function of the alloy  concentration.  Future application \ncould take advantages of this heatin g effect which allows reducing  the in -plane external field.  \nUnexpected double magnetization  switching has been  observed when the heat generated by \nthe current allow s cross es the compen sation temperature .  \n \nI. Introduction  \nSpin Orbit Torque switching with perpendicularly magnetized material in Hall bar based devices \noffers  a simple and powerful geometry to probe current induced magnetization reversal  and \nhad opened a new way to manipulating magnetization at the nanoscale . The underlying physics \nis quite rich and complex including origin of spin-orbit torque (SOT), interfacial effects and \nthermal contributions.  Magnetization switching by SOT was first observed  in heavy \nmetal/ferromagnetic, HM/FM, ultrathin films  [1–3]. The torque is mainly related to the spin Hall \neffect (SHE)  [4-10], where the charge current flowing in the heavy metal is converted into a 2 \n vertical spin current due to the large spin -orbit coupling. This spin current is then transferr ed to \nthe FM magnetization, which leads to a torque, namely the spin orbit torque.  It has been  \ndemonstrated for instance using  Pt [4–9], Ta [9–12], W [13–16] as HM and FM layers  with \nperpendicular magnetization like C oFeB [7,9–12,14,15] , Co [4–6], CoFeAl  [16] or  (Co/Ni)  [8] \nmultilayers. Interface effects can play a key role in the SOT i n particular, interfacial spin memory  \nloss  [17] and spin transparency  [18] which affects the transmitted spin . Furthermore, \nadditional charge to spin current conversion can also occur due to Edelstein effect   [19] in \nRashba  [20] and topological insulator interfaces  [19]. There is some attempts to unify a \nmodel  [21–23] including the aforementioned effects.  It might be also an interfacial DMI \n(Dzyaloshinskii -Moriya interaction ) which  favors  formation of  chiral Neel domain wall  [24]. It \nwas shown  in FM/ HM systems that the reversal of the magnetization occurs  first by a magnetic \ndomain nucleation followed by a domain wall prop agation thanks to SHE and iDMI  [8,25 –28]. \nThe thermal contribution  [6,29]  is usually neglected in those experiments.  \nFor possible application s the critical switching current need s to be reduced while maintaining a \nsufficient thermal stability. In the literature the critical  current density to reverse the \nmagnetization , Jcc, is typically of the order of ~1010 to 1012 A/m2 depending on the applied \ncurrent pulse duration and on the in -plane external magnetic field  [4,5,8,30] . Jcc is proportional \nto the m agnetization times the thickness of the FM layer ( Jcc  Mt F). Recently , transition metal -\nrare earth TM -RE ferrimagnetic materials start ed to attract large attention for spin -orbitronics \napplications  [31–36]. In these  ferrimagnetic alloys  the net magnetization  is given by the sum of \nthe magnetization of the two magnetic sub-lattices (r are earth and transition metal)  which are \nantiferromagnetically coupled . The most advantage of ferrimagnetic materials is that its net \nmagnetization M can be tuned by changing its composition or temperature  [37]. As a result , a \nmagnetic compensation point with zero magnetization can occur for a certain alloy \nconcentration, xMcomp , or temperature, TMcomp , where the magnetization of both sub -lattices \ncompensates . Moreover TM -RE thin f ilms are characterized by a large bulk magnetic anisotropy \nperpendicular to the film plane (PMA) which make easier to integrate TM -RE with different NM \nmaterials while keeping large thermal stability   [38]. Furthermore, the control of magnetization \nswitching using ultrafast femtosecond laser pulse has been demonstrated recently for various \nTM-RE materials  [39,40] . Those features are encouraging  to combine the control of \nmagnetization by both optical and electrical means. Concerning the Spin Orbit Torque  (SOT) \nswitching, reports on  experiments with TM-RE alloys claim  that the spin -orbit torque efficiency \nreaches a maximum at the magnetic compensation point  [31–36], however the critical current is \nnot minimum  at this point  [33,35,36] . In this study we address the SOT-switching experimen ts \non well characterized  //W/Co xTb1-x/Al systems  for various concentrations . We demonstrate that \nthermal effects are keys to explain the current induced magnetization reversal in this system . \nWhen the current is injected in the bilayer the Joule heating leads to a large increase of the \nsample temperature . Using systematic SOT measurement s at different temperatures and alloy 3 \n compositions, we establish that for each concentration x the current induced magnetization \nswitching occurs for a unique sample tempe rature Tswitch (x). Tswitch  scales with the Curie \ntemperature ( TC) of the alloy. Those new f indings  open new rooms to explore combination of \nSOT and thermal contribution towards reducing critical current density to reverse M and \nconsequently  low power consu mption  applications . In the specific case where Tswitch  is close to \nTMcomp an unexpected “double switching ” is observed.  \n \nII. Basic characterization  \nTo study SOT magnetization switching in RE -TM alloys, a model system composed of CoxTb1-X \nferrimagnetic alloy s deposited  on a tungsten heavy metal  with high charge to spin conversion \nefficiency  [13] was considered .  The samples were grown by dc magnetron sputtering on \nthermally oxide Si  substrates (Si -SiO 2). The full stacks of the samples are Si -SiO 2//W(3  \nnm)/Co xTb1-x(3.5 nm)/Al(3  nm) with  0.71  x  0.86 . The 3 nm thick Al (naturally oxidized and \npassivated  after the deposition ) is used to cap the ferrimagnetic layer. The W and CoTb lay ers \nhave amorphous structure.  As described in the introduction , ferrimagnetic alloys like CoTb can \nshow a compensation point at which the Co and Tb moments  cancel  each other, resulting in \nzero net magnetization . When the net magnetization of the alloy is parallel (resp. antiparallel ) to \nthe magnetization of the Terbium sub -lattice the alloy will be call Terbium rich  (resp. Cobalt \nrich) . The samples were characterized by a SQUID -VSM  magnetometer and Magneto -optic ally \nKerr effect (MOKE)  at room temperature . The SQUID measurements obtained at room \ntemperature are presented in Fig 1a . Magnetization compensation is observed  for a \nconcentration x Mcomp =0.77 where  the coercivity Hc diverge s and the net saturation \nmagneti zation M s tends to zero . This value is close to  the one report ed for bulk and thicker \nCoxTb1-x films  [37,41]  at room temperature . Additionally to the divergence of Hc, MOKE \nmeasurements show that  the Kerr angle rotation changes its sign between Co-rich and Tb-rich \nsamples , which can be explained by the fact that  Kerr rotation is mainly sensitive to the Cobalt \nsub-lattice  (see for instance fig . S1 in supplementary material  [42]).  Both SQUID and MOKE \nresults clearly show that all CoTb films studied have a strong out of plane magnetic  anisotropy.  \nTo study Spin Orbit Torque switching,  the stacks were patterned by standard UV lithography \ninto micro -sized Hall crosses with a channel of 2, 4, 10 and 20 m. The results  shown ar e \nobtained for  a width of 20 m unless other wise  specified. Ti(5)/Au(100) ohmic contacts were \ndefined  by evaporation deposition and lift-off method  on top of W layers . By measuring the \nanomalous Hall resistance RAHE of the hall crosses while sweeping the external perpendicular \nmagnetic field Hz at different temperatures , we could determin e the magnetic compensa tion \ntemperature of the samples . Fig. 1b shows  the temperature dependence of Hc for 78% of Co.  \nThe coercive  field  Hc diverge s around  280 K which determines TMcomp  for this composition. 4 \n Moreover, we can observe in the insets that the RAHE(Hz) cycle is reversed for Tb -rich (T<280 K) \nand Co -rich( T>280 K) phases , namely change of field switching polarity (Field -SP). The latter  is \ndue to the fact the Anomalous hall resistance is sensitive to the cobalt sub -lattice . Van der Pauw \nresistivity measurements leads to a resistivity of W in Si-SiO 2//W(3 nm)MgO(3 nm) of W = 162 \n.cm. Then  we could deduce the Co 0.72Tb0.28 resistivity CoTb = 200 .cm which  decrease s to \n135 .cm when the Cobalt concentration reaches Co0.86Tb0.14 in accord with previous \nresults  [43]. Despite this trend,  the a mplitude of  RAHE(Hz), RAHE, increases as a function of the \nCo concentration verifying that RAHE is mainly sensitive  to the Cobalt sub -lattice  (see also Fig \nS2  [42]). \n \nIII. Thermal ly assisted and spin -orbit torque switching  \nFig. 2a shows a scheme of a Hall bar along with the convention s used  for current injection , \nvoltage probe  and directions  axes .  Typical RAHE(Hz) cycles obtained at room temperature with a \nlow in -plane d c current of 400 A (charge current density of about 2.4 109A/m2 flowing in each \nlayer) for a Tb-rich ( resp. Co-rich)  sample  is shown in Fig. 2b ( resp. Fig. 2e) . As expected , a \nchange of  Field-SP is observed since the alloy net magnetization is parallel to  the magnetization \nof the Cobalt sub -lattice in one case and antiparallel  in the other . For the same samples the \ncurrent -induced switching cycles are shown in Fig. 2c -d (Tb -rich) and 2f -g (Co -rich) with a n in-\nplane bias field of Hx=100 mT  (Fig. 2c and 2f) and -100 mT  (Fig. 2d and 2g). The current injection  \nwas performed with pulse duration  of 100 s using a K6221 source coupled to a K2182  Keithley  \nnanovoltmeter. The Hall voltage is measured during the pulse. We have observed the current \ninduced magnetization switching in all the samples for 0.72  x  0.86. The Hall resistance \namplitudes  are the same  for the current -switching and the field -switching cycles indicating that \nthe reversal  of magnetization is fully achieved  in both cases . The da ta of the full series are \nshown in Fig. S3   [42]. Sharp current switching  are observed and  the critical current reduce s \nwhen Hx increases  following similar trends that for ferromagnetic materials  [30] as shown in \nFig. S4 . Remarkably, we observe a full magnetization reversal even for a n in-plane field  Hx as low \nas 2 mT. The role of the in -plane field can be understood as the field to balance the iDMI to \npropagate domain  walls which have in -plane magnetization after nucleation of magnetic \ndomains or the  field to break the symmetry and to allow for a deterministic switching  [8,30] . If \nthe SOT depends on the Co moment , the SOT acts as an effective field HSHE  m   [24,44]  \nwhere  m is the magnetic moment and  the spin polarization of the spin current Js injected from \nthe W layer into the CoTb layer.  is along the y direction in our measurement geometry (it \nchanges between +y and –y when the direction of the injected current is inverted). m changes \nits sign upon  the change of the in -plane field  direction . Then the sign of the Hall cycle vs current, \nRAHE(i), is reversed when Hx is reversed  as observed in Fig 2 c-f. Additionally in ferrimagnetic 5 \n alloys, the effective field HSHE can be reversed if a Co -rich s ample is replaced by a Tb-rich one (a \nschematic is shown in Fig. 2h for Hx>0 and i>0). The identical effect will be observed if the same \nsample is kept and the magnetic compensation temperature is crossed. The fact that the \nsample s which have been identified as Co-rich and Tb -rich at room temperature are showing \nthe same current -switching polarity (Fig 2c and 2f) can only be understood if the so call Tb -rich \nsample  has cross ed compensation to become Co -Rich. This compensation crossing is due to the \nJoule heating effect.  This assumpt ion was tested by measuring  RAHE(Hz) cycles for different \napplied current pulses on the “Tb-rich” sample shown in Figure 3. We observe that for applied \ncurrent s i< 19.5 mA the sign of the cycle demonstrate a “ Tb-rich” nature, however  for current \ni>19.5 mA the RAHE(Hz) cycles are reversed and demonstrate a  “Co-rich” nature . This is clear \nexperimental evidence that  for current close to 19.5 mA the device reach es the sample \ncompensation temperature  (TMcomp  ~320 K). This demonstrates  that the sample is strongly \nheated during the current -switching experiments. The temperature can be determined by the \nresistance value as explained in the next section. In Figure 3 the corresponding temperature is \nshown using color code.  We can determine  that a temperature of 460 K is reached  for 24mA \nwhich is the critical current to switch M (Fig. 2c) . Th is current -switching  of 24 mA is then \nobtained for a temperature above TMcomp which explain why the sign of the RAHE(i) cycle  is the \none expected for a Co -rich sample.  We have performed RAHE(Hz) cycles with intensity current \npulse as high as 34 mA  (~525 K)  where we can observe that the device shows a  ferromagnetic \nhysteresis loop and remain perpendicular ly magneti zed. TC is then higher than 525 K . \n \nIV. Characteristic temperatures of switching  \nSince we have addressed the reason of the observed switching polarity  several questions  are \narising :  i) how much are the devices heated when the switching  occurs ? ii)  Does the \ntemperature  at which  the switching occurs changes with the initial temperature (temperature \nat which the experiment is carried out, T cryostat ) ?  iii) How d oes the switching current and \nswitching temperature depend on composition? And iv ) What is the physical meaning of this \nswitching temperature: Angular compensation temperature T Acomp ? In order to address all those  \nquestions we have performed a series of temperature dependence experiments for various  \nsamples.  \nFig. 4 a shows the RAHE(ipulse) cycles for Hx>0 at different cryostat temperature for W/Co0.73Tb0.27 \n(Tb-rich at room temperature). We have observed the Down -Up current -switching polarity  at \n300 K . For 150 K Tcryostat  250 K a double  current switching  loop is observed. This type of \ndouble current switching can be explain ed when the switching temperat ure is close to TMcomp  \nand its origin will be discussed later in the paper (Fig. 4a show s only the case of 150 K for \nclarity ). For  10 K T 100 K we obser ve only Down -Up current -switching polarity  (we didn’t 6 \n increase too much the pulse current to avoid burn ing the device). One can calibrated the real \nsample temperature at different pulse -current performing the following protocol: i) measuring \nthe resistance of the current channel Rchannel (ipulse) as function of pulse current intensity as \nshown in Fig. 4b for  different cryostat temperatures, and ii) measuring the temperature \ndependence of the current channel Rchannel (T) as shown in F ig 4c (for which we use a very low dc \nbias current of only 400 A). Interesting ly, we observe that for Co -rich current -switching polarity \n(Down -Up) the device reaches the same resistance (1.373 k ) and consequently the same \nswitching temperature Tswitch  = 435 K  25 K  for this W/Co 0.73Tb0.27. We note that the resistance \ndecreases when T increases which is a feature and confirmation of amorphous materials  [45]. \nWe have performed the same protocol for various compositions and different devices. An \nexam ple for Co -rich sample at room temperature is shown in Fig. 4 d-f (W/Co 0.79Tb0.21) where \nwe also observe that the critical current heat s up the device to the same channel resistance  (Fig. \n4e), so the same T switch  (~485 K for this Co xTb1-x sample)  irrespective  of the initial temperature . \nMoreover , on this particular sample TMcomp  is about 500 K  and we observe no change of C urrent -\nSP even for T as low as 10 K  which is well below its TMcomp .  \nAdditionally to the characteristic Tswitch  we have just disc ussed , one can also investigate  the \ntemperature dependence of the critical current as shown in Fig . 5a for a Co0.78Tb0.22 sample ( Co-\nrich at room temperature ).  The extrapolation of the linear dependence to zero current is \ndefined as T*. In Fig. 5b is found out that T*~470 K for W/ Co 0.78Tb0.22.  \nFig. 6a show s RAHE(ipulse) for Co 0.72Tb0.28 performed for a cryostat temperature of 150 K to \nhighl ight the observation of the two switching. At lower current  (37.5 mA) , i.e. lower Joule \nheating effect,  we observed Up-Down current -switching polarity  while  the second one which \noccurs  at higher current  (43 mA)  is Down -Up. This can be  understood considering that to \nachieve the first switching the device reaches a temperature below its TMcomp  so the sample is  \nstill Tb -rich and the Up -Down current -switching polarity observed is as expected (i.e Fig 2h) . If \nwe continue  increas ing the intensity of the applied in -plane pulse current we overcome TMcomp  \nand the n the  perpendicular component of effective torque field HSHE now changes its sign as \ndiscussed previous ly. Consequently, the second observed switching agree well for Co -rich phase  \n(Down -Up). In Fig. 6b is shown the temperature dependence of both switching current s.  As \ndiscussed, the first  (second) switching  agrees with a Tb -rich (Co -rich) current -switching polarity \nand happens for T<TMcomp  (T>TMcomp ). The linear extrapolation of both switching currents  \nroughly tends to  350 K (Tb-rich switching ) and T*~455 K (Co-rich switching ). The value of T* \nseems to be slightly higher than Tswitch  (~435 K   20 K  as determined in Fig 4c).  \n \n 7 \n V. T-x switching  phase diagram  and conclusions  \nIn figure 7a the different characteristic temperature s of our W/CoTb systems can be plotted in  \nthe (T,xCo) phase diagram. The determine d TMcomp  decreases linearly with the Co  concentration \nas reported for bulk CoTb and thick CoGd films (300 nm)  [41,46] . However Tswitch  and T* \nincrease l inearly with the Co-concentration  and scale with the Curie temperature Tc thus  \ndepend ing on composition , and independent of initial temperature. It is remarkable that the \nTswitch  and T* are nearly the same, indicating that, to achieve the switching, one has to reach a \nspecific temperature. The three first questions in section IV  are answered . Now let’s discuss the  \nphysical meaning of these switching temperatures. It is clear that for Co -Current SP the \ntemperature of switching is above TMcomp  and below Tc. Fig. 7a also shows Tc in bulk CoTb after \nHans et al.   [41]. The angular composition temperature , TAcomp , scales with  TMco mp. Indeed, \ntypically TAcomp  ~(TMcomp +30 K) for Co0.775Gd 0.225 thick films  (300 nm)   [46]. This is explained by \nthe relationship between the angular moment  L, magnetic moment and the gyromagnetic ratio \n or Landé g -factor ( LTM, RE =MTM, RE /TM, RE  and =gB/hb). Therefore  in CoTb it is expected that the \ntrends of TAcomp  and TMcomp  are similar and they decreas e with increasing Co concentration . \nHowever Tswitch  increases with Co concentration which indicates  that Tswitch  is not scaling with \nTAcomp  or TMcomp. For sake of comparison we plotted the switching -current for experiment \nperformed at room temperature  together with the temperature increase T = TswitchTcryostat ,  \ndue to Joule heating effect, Fig. 7b . We can observe that the temperature is increased between \n100 K and 300 K. This variation will increase when we reduce Tcryostat .  Considering that resistivity \nof both, CoTb  and W layers, change similarly with temperature and using the resistivity \nmeasured at room temperature we can estimate the critical current density JCC flowing on each \nlayer as displayed in Fig. 7c. We observe that Jcc on W is reduce d by a factor of ~2 wh ile varying \nthe composition of CoTb . We observe the minimum of Jcc at the lower Co concentration \nmeasured. This c an be explained by the fact that TC and Tswitch  decrease with de creas ing Co-\nconcentration  but a relation ship between Tswitch  and TC will need additional studies . \nConclusions  \nIn conclusion w e have fully characterized the current -induced switching experiment in a series \nof //W(3nm)/Co xTb1-x(3.5nm)/AlO x(3nm) samples. In addition to the SOT effect we demonstrate \na strong thermal contribut ion to achieve the magnetization reversal.  For the Co -rich current -\nswitching polarity the device needs  to reach the same temperature  Tswitch  to achieve the \nswitching. This Tswitch  increase s with Co -concentration which then scale with Curie temperature  \nTc. It is then unlikely that Tswitch  corresponds to angular momentum compensation temperature \n(which scales with TMcomp  decreasing with Co -concentration) . Those results highl ight the \nimportance of considering thermal contributions in SOT switching experiment s and the fact that  \nthe spin Hall angle determination might be overestimated when thermal contribution is 8 \n neglected . The use of resistive W layer i ncrease the heating of the device, reducing strongly the \nexternal in -plane needed to assist the SOT . Those results are important for the full \nunderstanding of current -induced magne tization switching and may lead the way to  new \ntechnological applications  taking advantages of the rather strong heating  \nAcknowledgements  \nThis work was supported partly by the french PIA project “Lorraine Université d’Excellence”, \nreference ANR -15-IDEX -04-LUE. by the ANR -NSF Project, ANR -13-IS04 -0008 - 01, “COMAG” by \nthe ANR -Labcom Project LSTNM, Experiments were  performed using equipment fro m the \nTUBE —Daum funded by FEDER (EU), ANR, the Region Lorraine and Grand Nancy.  We thank  J. \nSampaio and S.  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Malinowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton,  “Engineered materials \nfor all -optical helicity -dependent magnetic switching.” Nat. Mater.  13, 286 (2014).  \n[41] P. Hansen, C. Clausen, G. Much, M. Rosenkranz, and K. Witter, “Magnetic and magneto -optical \nproperties of rare -earth transition -metal alloys con taining Gd, Tb, Fe, Co” 756, (1989).  \n[42] T. H. Pham, S. G. Je, P. Vallobra, F. Thibaud, D. Lacour, G. Malinowski, M. C. Cyrille, G. Gaudin, O. \nBoulle, M. Hehn, J. C. Rojas -Sanchez, and S. Mangin, “Thermal contribution to the spin -orbit \ntorque in metallic/ ferrimagnetic systems” Suppl. Inf.  (2017).  \n[43] T. W. Kim and R. J. Gambino, “Composition dependence of the Hall effect in amorphous \nComposition dependence of the Hall effect in amorphous Tb x Co 1 À x thin films” J. Appl. Phys.  \n87, 1869 (2000).  \n[44]  a. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. a. Zvezdin, A. Anane, J. Grollier, \nand A. Fert, “Matching domain -wall configuration and spin -orbit torques for efficient domain -wall \nmotion” Phys. Rev. B  87, 20402 (2013).  \n[45] A. Fert, R. Aso moza, A. Fert, R. Asomoza, S. Universite, and P. O. France, “Transport properties of \nmagnetic amorphous alloys Transport properties of magnetic amorphous alloys” 1886 , (1979).  \n[46] M. Binder,  a. Weber, O. Mosendz, G. Woltersdorf, M. Izquierdo, I. Neudecker, J. Dahn, T. \nHatchard, J. -U. Thiele, C. Back, and M. Scheinfein, “Magnetization dynamics of the ferrimagnet \nCoGd near the compensation of magnetization and angular momentum” Phys.  Rev. B  74, 134404 \n(2006).  \n \n \n  12 \n  \n \nFigure 1. //W(3)/Co xTb1-x (3.5)  /Al(3) : a) Coercive field Hc and saturation magnetization Ms obtained by \nSQUID measurements vs. cobalt concentration at room temperature. Magnetic compensation is \nobserved for xMcomp  ~0.77. b) Temperature dependence of coercitivity on Hall bar for x=0.78 showing that \nTMcomp ~280 K  for x=0.78. Insets show Hall resistance cycle for a temperature below (left) and above \n(right) TMcomp . The change of field - switching polarity (Field -SP) evi dences that RAHE is mainly sensitive to \nthe magnetization of the Cobalt sub -lattice.  \n  \n13 \n  \n \nFigure 2. Anomalous Hall effect and Current -induced magnetization reversal at room temperature on Si -\nSiO 2//W(3)/Co xTb1-x (3.5)  /Al(3). a) Scheme of the Hall bar along with geometry used. The spin \npolarization  is along y axes. (b and g ) Sweeping perpendicular field (H ||z) with a low dc bias of 400 A. \n(c,d,e and f ) Sweeping in -plane current (i pulse ||x) with an in -plane field Hx=±100 mT .The width of the \nchannel current is 20 m. The current switching polarity (current -SP) for Hx>0 is Down -Up for Co -rich and \nTb-rich samples (c,f). The Current -SP in (c) is opposite than predicted for Tb -rich samples as shown in the \nschematic in (h). The p erpendicular effective torque field HSHE is proportional to m and should lead to \na change of the sign in the RAHE(i) cycle  when changing from Co -rich to Tb -rich phase.  \n  \n14 \n  \n \nFigure 3. R AHE(Hz, ipulse)  cycles on Si -SiO 2//W(3)/Co 0.76Tb0.24(3.5)/Al(3) Hall bar measured at room \ntemperature. The cycles are vertically offset for clarity. It is observed that for i <  19.5 mA the cycle has a \nsignature corresponding to Tb -rich phase according to our convention. However for i> 19.5 mA the cycle \nchanges their sign and now corresponds to Co -rich phase.  It is an evidence of the Joule heating effect \nwhen high pulse current is applied. 19.5 mA roughly corresponds to TMcomp . The critical current for this \ndevice is about 24 mA. Thus during the electrical switching T device>TMcomp >300 K for this Co 0.76 system.  \n  \n15 \n  \n \n \nFig4. a) RAHE(ipulse)  at different cryostat temperatures. Cycles for x=0.73 (Tb -rich at room temperature ). b) \nRChannel (ipulse)  and c) RChannel (T)  at different cryostat temperatures. The vertical dashed line points out the \ncritical current to reverse M. It is observed that independently of the initial temperature, the device \nalways reaches the same value of  longitudinal  resistance (1373 ) which means it reaches the same \ntemperature .  The linear extrapolation of RChannel (T)  allows us to know the temperature corresponding to \nthe current -induced magnetization reversal . Such a temperature is defined as Tswitch. RChannel (T)  is \nperformed with  a low bias current of 400 A. For Co 0.73 we found that Tswitch= 435 K  25 K.  d) RAHE(ipulse)  \ncycles for x=0.79 (Co -rich at room temperature)  at different cryostat temperatures. e) RChannel (ipulse)  and f) \nRChannel (T)  at different cryostat temperatures. It is also observed that independently of the initial \ntemperature, the device always reaches the same resistance , thus is the same temperature . In this case \nit corresponds to 1291  and Tswitch ~ 485 K.  \n  \n16 \n  \n \nFigure 5 . a) RAHE(ipulse)  at different cryostat temperatures. Cycles for x=0.78 (Co -rich at r oom \ntemperature ). b) The critical current to reverse M increase s linearly when T decrease and saturate for T< \n50 K. The extrapolation of th e linear behavior at higher temperature for zero current is defined as T*. \n  \n17 \n  \n \nFig6. a) RAHE(ipulse)  at 150K for x=0.73 (Tb -rich at room temperature ). There are two switching s: i) at lower \ncurrent it agrees with a Tb-rich switching polarity (Up-Down) . ii) The reversal with hi gher current agrees \nwith a Co -rich switching polarity (Down -Up).  b) Temperature dependence o f the critical current s for this \ncomposition. TMcomp  would be between 350 K and 455 K.  \n  \n18 \n  \n \nFigure 7 . a) Characteristic temperatures as function of Co concentration: T*, Tswitch and TMcomp  \nindependently measured in Hall bar patterned devices (lines are guides for the eyes). It is observed that \nTMcomp  follows the same behavior reported for bulk Co xTb1-x alloys. T* and Tswitch have the same trend \nthan that of the Curie temperature Tc. The gre en dashed line stands for Tc in bulk CoTb after Hans et \nal.  [41]. b) The total critical current injected to reverse M when the experiment is performed at room \ntemperature (= Tcryostat ), and the variation of temperature TswitchTcryostat  to reach the switching. c) The \ncritical current density, calculated from b, flowing in W and CoTb layers, respectively.  \n \n1 \n Supplementary Material  \nThermal contribution to the spin -orbit torque in metallic/ferrimagnetic systems  \nThai Ha Pham1, S.-G. Je1,2, P. Vallobra1, T. Fache1, D. Lacour1, G. Malinowski1, M. C. Cyrille3, G. Gaudin2, O. \nBoulle2, M. Hehn1, J.-C. Rojas -Sánchez1* and S. Mangin1 \n1 . Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, F-54011 Nancy, France  \n2 . CNRS, SPINTEC, F -38000 Grenoble \n3. Leti, technology research institute, CEA, F -38000 Grenoble  \n*juan -carlos.rojas -sanchez@univ -lorraine.fr  \n \nS1- Magneto -optically Kerr effect measurements at room temperature  \nFigure  S1 show s the coercitivity obtained by MOKE of W/Co xTb1-x thin film as a function of x (the Co-\nconcentration ). The  results show that coercive field Hc diverges about Co -concentration of 7 7 % in \nagreement with SQUID results (Fig. 1a). The insets highlight the  opposite sign of Kerr angle  rotation \nwhen the CoTb alloy change from Tb -rich to Co -rich phase.  \n \n \nFigure S1 . //W(3)/Co xTb1-x (3.5)  /Al(3) : Coercive field Hc obtained by MOKE measurements at room \ntemperature  as a function of the  Cobalt concentration . Insets show raw data of MOKE cycles for a Tb -\nrich (left) and Co -rich (right) samples.  \n \n \n2 \n S2- Hall resistance amplitude | RAHE|, channel resistance Rchannel , and Co xTb1-x \nresistivity CoxTb1 -x at room temperature  \nFigure S2 present the evolution of the Hall resistance amplitude |RAHE|, the channel resistance Rchannel , \nand the CoxTb1-x resistivity CoTb at room temperature  as a function of the Co -concentration . Despite the \ndecreas e of the channel resistance  with increasing Co -concentration, we observe that | RAHE| increase s \nwith Co-concentration . Thus corroborate  that the Hall resistance is mostly sensitive to Co magnetic \nmoment.  The W resistivity, W = 162 .cm, is of similar order than CoTb alloys.  \n \n \nFigure S2 . Si-SiO 2//W(3)/Co xTb1-x (3.5)  /Al(3) : The change of Hall resistance amplitude  |RAHE|, channel \nresistance  Rchannel , and Co xTb1-x resistivity as a function of the Cobalt concentration at room temperature . \nThe vertical blue dashed line points correspond to the magnetic compensation point at room \ntemperature. The horizontal red dashed line shows the value of W resistivity (ρ W).  \n \n \n \n3 \n S3- Current -switching in the full W/CoxTb1 -x series for different in -plane field at \nroom temperature  \nFigure S3 (S4) presents the c urrent -induced magnetization reversal performed at room temperature on \nSi-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm) with an in -plane field Hx=100 mT (5 mT).   \n \n \nFigure S3 . Current -induced magnetizati on reversal at room temperature : Hall resistance as a function of \nthe injected current measured for various  ipulse Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm), with an \nin-plane field of Hx= 100 mT.  \n4 \n  \nFigure S4 . Current -induced magnetization reversal at  room temperature : Hall resistance as a function of \nthe injected current measured for various  ipulse Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm), with an \nin-plane field of Hx= 5 mT. \n \n  \n5 \n S4- Power consumption  in the W/Co xTb1-x Hall bar with 20 m of width  at room \ntemperature  \nFigure S 5 shows the electrical power consumption to  revers e the magnetization i n Hall bar of length L= \n100 m and width  of w  = 20 m at room temperature on  various  Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) \n/AlOx (3nm) with an in -plane field  of 100 mT.  \n \n \nFigure S5 . Si-SiO 2//W(3)/Co xTb1-x (3.5)  /AlOx(3): The electrical power consumption to switch  the \nmagnetization at room temperature of a Si-SiO 2//W(3)/Co xTb1-x (3.5)  /AlOx(3) hall bar as a function of  \nCobalt concentration. The current channe l dimensions are length L= 100 m, and width of w = 20 m. \n  \n6 \n S5- Hx-I switching phase diagram in the W/Co xTb1-x  \nFigure S6 (S7) presents a  2D plot summarizing the c urrent –switching cycles performed under different \nexternal in -plane field Hx (phase di agram ) at room temperature  for a channel width of w= 20 m (10 \nm). Figure S6  are the results obtained for   Si-SiO 2//W(3  nm)/Co 0.86Tb14 (3.5 nm) /AlOx(3 nm) and fig. S7 \nfor Si-SiO 2//W(3  nm)/Co 0.84Tb0.16  (3.5 nm) /AlOx(3 nm). \n \n \n \nFigure S6 . 2D-plot of c urrent –switching cycles performed at room temperature on Si -SiO 2//W(3  \nnm)/Co 0.86Tb14 (3.5 nm) /AlOx(3 nm) for a channel width of w  = 20 m. The R(ipulse, H x) cycles were carried \nout with different  applied in plane  field between –3 kG and + 3 kG. The red (blue) color region stand for \nUp (Down) magnetic configuration according the schematic Hall bar shown in Fig. 2a.  \n \n7 \n  \nFigure S 7. 2D-plot of c urrent –switching cycles performed at room temperature on Si -SiO 2//W(3  \nnm)/Co 0.84Tb16 (3.5 nm) /AlOx(3 nm) for a channel width of w  = 10 m. The R(ipulse, H x) cycles were carried \nout with different  applied in plane  field between –6.5 kG and +6.5 kG. The red (blue) color region stand \nfor Up (Down) magnetic configuration according the schematic Hall bar shown in Fig. 2a. \n \n"    },    {        "title": "1210.3706v2.Promising_ferrimagnetic_double_perovskite_oxides_towards_high_spin_polarization_at_high_temperature.pdf",        "content": "arXiv:1210.3706v2  [cond-mat.mtrl-sci]  27 Oct 2012Promising ferrimagnetic double perovskite oxides towards high spin\npolarization at high temperature\nSi-Da Li,1,2Peng Chen,1and Bang-Gui Liu1,a)\n1)Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,\nBeijing 100190, China\n2)Department of Physics & School of Gifted Youth, University o f Science and Technology of China, Hefei 230026,\nChina\n(Dated: 19 April 2019)\nWe predict through our first-principles calculations that four doub le perovskite oxides of Bi 2ABO6(AB\n= FeMo, MnMo, MnOs, CrOs) are half-metallic ferrimagnets. Our calc ulated results shows that the four\noptimized structures have negative formation energy, from -0.42 to -0.26 eV per formula unit, which implies\nthat they could probably be realized. In the case of Bi 2FeMoO 6, the half-metallic gap and Curie temperature\nare predicted to reach to 0.71 eV and 650 K, respectively, which indic ates that high spin polarization could be\nkept athigh temperaturesfarbeyond roomtemperature. It isbe lieved that someofthem could be synthesized\nsoon and would prove useful for spintronic applications.\nPACS numbers: 75.30.-m, 75.50.-y, 71.20.-b, 75.10.-b\nI. INTRODUCTION\nMagnetic materials that have high spin polarization at\nroom temperature or higher are highly desirable for spin-\ntronic applications1,2. Half-metallic materials are good\ncandidates because they can have high spin polarization\nat high temperature3. In the case of CrO 2, 96% spin\npolarization has been achieved experimentally2. Since\n1998, double perovskite oxides have been explored ex-\ntensively for this purpose because both half-metallicity\nand high Curie temperature can be achieved in such\nmaterials4,5. It has been shown experimentally that sev-\neral double perovskite oxides, such as Sr 2FeMoO 6and\nSr2CrReO 6materials5, can keep their ferromagnetic or\nferrimagnetic phases far beyond room temperature, and\nmore importantly, high-quality materials have been real-\nized recently6–11.\nHere, we present our first-principles exploration on\ndouble perovskite oxides of Bi 2ABO6with A being 3d\ntransition metals and B 4d/5d ones. We optimize their\nstructures fully and then investigate their stability, elec-\ntronic structures, and magnetic properties. We find four\nhalf-metallic ferimagnetic materials with negative forma-\ntion energies. For the best case of Bi 2FeMoO 6, the half-\nmetallic gap and Curie temperature reach to 0.71 eV and\n650 K, respectively. This means that high spin polariza-\ntion could be realized at high temperatures well above\nroom temperature. More detailed results will be pre-\nsented in the following.\nII. COMPUTATIONAL DETAILS\nWe use the pseudo-potential and plane wave methods\nwithin the density functional theory (DFT)12, as imple-\na)Email: bgliu@iphy.ac.cnmented in package VASP13. We use generalized gradient\napproximation (GGA)14for the exchange-correlationpo-\ntential. In addition to usual valence states, the semicore\nd states are considered for Bi and the semicore p states\nforCr, Mn, Fe, Mo,andOs. Scalarapproximationisused\nfor relativistic effect15, and the spin-orbit coupling is ne-\nglected because it has little effect on our main conclusion\n(to be detailed in the following). We use Monkhorst-\nPack method to generates the K-point mesh16, choosing\n6×6×6 (6×6×4) for structure optimizations and total\nenergy calculations of 10-atom (20-atom) unit cells and\n12×12×12for electronic structure calculations. The cut-\noffenergyissetto500eVandthecriteriaforconvergence\nis 10−6eV for electronic steps and 0.005 eV/ ˚A on atoms\nfor ionic steps.\nMetropolis algorithm and variants are used for our\nMonte Carlo simulations17,18. Phase transition tempera-\nturesTcaredeterminedthroughinvestigatingtheaverage\nmagnetization, magnetic susceptibility, and fourth-order\nBinder’scumulantasfunctions oftemperature18. Several\nthree-dimensional lattices of upto 30 ×30×30 magnetic\nunit cells with periodic boundary condition are used in\nthese calculations. The first 90,000 Monte Carlo steps\n(MCS) of total 150,000 MCS are used for the thermal\nequilibrium, and the remaining 60,000 MCS are used to\ncalculate the average magnetization for a given temper-\nature. The Curie temperature Tcvalue is determined\nthrough investigating the magnetization as a function of\ntemperature.\nIII. MAIN CALCULATED RESULTS AND ANALYSIS\nComparing Sr 2FeMoO 65, Bi2FeCrO 66,7,19,20and oth-\ners similar8–11, we consider double perovskite structure\nof formula Bi 2ABO6, taking some 3d transition-metal\nelements for A and some 4d/5d for B. We optimize\nfully their crystal structures in terms of the unit cell2\nof the 10 atoms. The optimized Bi 2ABO6has space\ngroup Rc (#146). This crystal structure, similar to R3c\n(#161), is distorted from cubic double perovskite struc-\nture, has rhombohedral symmetry, and includes 10 in-\nternal parameters5,10,11. We shall present four of the\nBi2ABO6compounds, for AB = FeMo, MnMo, MnOs,\nand CrOs, because they havenegative formation energies\nso that their experimental realization should be prob-\nable. Our optimized structural parameters of the four\nBi2ABO6compounds are summarized in Table I. The\ntotal magnetic moments per formula unit and the partial\nmoments in the spheres of the magnetic A and B atoms\nare summarized in Table II. The magnetic moments in\nthe spheres of other atoms are much smaller. The to-\ntal moments are integers in unit of Bohr magneton µB,\nshowing a feature of half-metallicity3. The magnetic mo-\nment at the A atom is antiparallel to that at the B atom,\nwhichmeansthatthemagneticorderinthesecompounds\nis ferrimagnetic.\nTABLE I. Optimized structural parameters of double per-\novskite Bi 2ABO6with the Rc (#146) crystal structure for\nAB = FeMo, MnMo, MnOs, and CrOs.\nAB FeMo MnMo MnOs CrOs\na(˚A) 5.725 5.779 5.761 5.771\nc(˚A) 14.054 14.093 13.607 12.816\nα(◦) 59.91 60.20 61.61 64.37\nBiz1 0.9881 0.9859 0.9917 0.0294\nBiz′\n1 0.4837 0.4851 0.4923 0.5071\nAz2 0.2589 0.2572 0.2593 0.2682\nBz′\n2 0.7675 0.7657 0.7647 0.2681\nOx3 0.5504 0.5612 0.5534 0.4811\nOx′\n3 0.0506 0.0471 0.0520 0.0558\nOy3 0.9357 0.9220 0.9273 0.9353\nOy′\n3 0.4350 0.4397 0.4348 0.4287\nOz3 0.1022 0.1036 0.0981 0.1075\nOz′\n3 0.6099 0.6145 0.6085 0.6009\nIn Fig. 1 we present spin-resolved density of states\n(DOS, in states/eV per formula unit) between -7.7\nand 3 eV of the double perovskite Bi 2FeMoO 6and\nBi2MnMoO 6. The total DOS in majority-spin channel\nis zero at the Fermi level in both of the cases. This in-\ndicates that the two double perovskite compounds are\nhalf-metallic, in agreement with the integral magnetic\nmoments in unit of µB. In Fig. 2 we present spin-\nresolved density of states between -8 and 3 eV of the\ndoubleperovskiteBi 2MnOsO 6andBi 2CrOsO 6. Theyare\nboth half-metallic, too, but it is in minority-spin channel\nthat the totalDOS atthe Fermilevelis equivalentto zero\nin these two cases. The filled electronic states near the\nFermileveloriginatemainlyfromtheBatom(MoorOs).\nWe can use half-metallic gap Egas the key parameter to\ndescribe the half-metallic property3,21–23. TheEgvalues\nof the four compounds, from 0.25 to 0.71 eV, are summa-\nrized in Table II. For the Bi 2FeMoO 6,Egis equivalent to\n0.71 eV, which implies that high spin polarization could\nbe robust even after the spin-orbit coupling is taken intoaccount.\u0000\n✁\u0000 ✂✄\u0000✆☎✝\u0000 ✞✟\u0000✆✠✝\u0000 ✡✄\u0000☞☛✍✌✎☛✏✡✑✠\n\u0000✆✒\u0000 ✂\n\u0000 ✞\n\u0000 ✡\n✌\n✡\n✞\n✂\n✒\n✓✕✔✗✖✘\n✙✚✛✢✜✤✣✦✥★✧✪✩✓\n✣✬✫✖\n✭ ✮✆✭ ✯✕✰✱✳✲✴★✵✶✮✷\u0000\n✁\u0000 ✂✄\u0000✆☎✝\u0000 ✞✄\u0000✆✟✝\u0000 ✠✄\u0000☛✡✌☞✍✡✎✠✏✟\n\u0000✆✑\u0000 ✂\n\u0000 ✞\n\u0000 ✠\n☞\n✠\n✞\n✂\n✑\n✒✔✓✖✕✗\n✘✙✚✜✛✣✢✥✤✧✦✩★✒\n✢✖✪✕\n✫ ✬✆✫ ✭✯✮✰✔✱✲✴✳✲✬✵\nFIG. 1. (color online) Spin-resolved density of states (DOS ,\nin state/eV per formula unit) of double perovskite Bi 2ABO6\nfor AB=FeMo (a) and AB=MnMo (b). The solid line is total\nDOS, and short-dashed, dot-dashed, and dotted lines refer t o\npartial DOS projected in the atomic spheres of Bi, A, B, and\nO, respectively. The upper part in each panel is majority-sp in\nDOS result, and the lower the minority-spin one.\nWe investigate their formation energies to determine\nthe stability of these materials towards experimental re-\nalization. For achieving reasonable reliability, we choose\nstable and reachable compounds as our references, and\ntry to use those reference compounds whose valence\nstates concerned are close to those of our compounds.\nTherefore, we use Bi 2O3, Cr3O4, Mn3O4, Fe3O4, MoO 2,\nand OsO 2for our reference compounds for calculating\nthe formation energies. The formation energy is defined\nas\nEf=E(Bi2ABO6)−Eref, (1)\nwhereE(X) isthetotalenergyofX,and Erefisdefinedas\nE(Bi2O3)+1\n3E(A3O4)+2E(BO2)−7\n6E(O2). This crite-\nria is much more severe than merely using AO compound\nor bulk A materials because O atom in the gas state has\nhigher energy than in compounds such as Fe 3O4. This\nshould be more precise because the bonds in our mate-\nrials are almost formed between metal atom and O, not\nbetween metal atoms. The formation energies for the3\u0000\n✁\u0000✄✂☎\u0000 ✆✝\u0000✄✞☎\u0000 ✟✠\u0000✄✡☛\u0000 ☞✝\u0000✍✌✏✎✑✌✒☞✓✡\n\u0000 ✆\n\u0000 ✟\n\u0000 ☞\n✎\n☞\n✟\n✆✔✖✕✘✗\n✙\n✚✛✜✣✢✥✤✧✦✩★✫✪✬✔✖✤✮✭✯✗\n✰ ✱✄✰ ✲✖✳✴✶✵✷✹✸✺✼✻✺\u0000\n✁\u0000✄✂☎\u0000 ✆✝\u0000✄✞☎\u0000 ✟✠\u0000✄✡☛\u0000 ☞✝\u0000✍✌✏✎✑✌✒☞✓✡\n\u0000 ✆\n\u0000 ✟\n\u0000 ☞\n✎\n☞\n✟\n✆✔\n✕✖✗✙✘✛✚✢✜✤✣✦✥★✧✩✚✫✪✭✬\n✮ ✯✄✮ ✰✩✱✲✴✳✵✷✶✸✺✹✸✧✴✻✫✬\nFIG. 2. (color online) Spin-resolved density of states (DOS ,\nin state/eV per formula unit) of double perovskite Bi 2ABO6\nfor AB=MnOs (a) and AB=CrOs (b). The solid line is total\nDOS, and short-dashed, dot-dashed, and dotted lines refer t o\npartial DOS projected in the atomic spheres of Bi, A, B, and\nO, respectively. The upper part in each panel is majority-sp in\nDOS result, and the lower the minority-spin one.\nfour compounds are summarized in Table II. The nega-\ntive values means that they should probably be realized.\nTABLE II. Calculated values of formation energy ( Efin eV),\nmagnetic moment of atom A ( MAinµB), magnetic moment\nof atom B ( MBinµB), total magnetic moment ( MinµB\nper formula unit), half-metallic gap ( Egin eV), and Curie\ntemperature ( Tcin K) of double perovskite Bi 2ABO6for AB\n= FeMo, MnMo, MnOs, and CrOs.\nAB FeMo MnMo MnOs CrOs\nMA 3.638 4.279 4.066 2.636\nMB -1.755 -1.391 -1.041 -0.613\nM 2.000 3.000 3.000 2.000\nEg 0.71 0.47 0.46 0.25\nEf -0.41 -0.26 -0.42 -0.29\nTc 650 255 174 201\nIn order to estimate the Curie temperatures ( Tc) of\nthe materials, we calculate the spin exchange interac-\ntions between the nearest and next nearest neighboring\nmagnetic atoms (A and B) in terms of the 20-atom unitcells. Rigorously speaking, there are some induced spin\ndensity in the spheres of the Bi and O atoms, less than\n0.05µB. Because they are very small compared to those\nin the spheres of the magnetic atoms, we shall consider\nonly the magnetic atoms in the following. Actually, A\nand B atoms form a lattice of magnetic unit cells (cu-\nbic unit cells of a NaCl crystal structure)5,10,11. In these\ncalculations, we fix the structures and change the mag-\nnetic orders of A and B atoms. In order to make the\nelectronic steps converge for a magnetic order, the linear\nmixing parameter should be decreased to an small value,\n0.1 or smaller. Through comparing the total energies, we\nobtain the spin exchange interaction constants: JABfor\nthe nearest A and B pair, JAAandJBBfor the A-A and\nB-B next nearest pairs. JABis dominant over the others.\nThe resultant spin Hamiltonian reads:\nH=/summationdisplay\n/angbracketleftij/angbracketrightJij/vectorSi·/vectorSj (2)\nwhere/vectorSiis spin operator at site i(in both of the A and\nB sublattices), the summation is over spin pairs, and the\nspin interaction constant Jijis limited to the nearest and\nthe next nearest neighboring spins.\u0001 \u0002 \u0000 \u0003 \u0004 \u0005 \u0006 \u0007 \b \t \n \u000b \f \r \u000e \u000f \u0010 \u0011 \u0012 \u0013 \u0014 \u0015\n\u0016 \u0017 \u0018\u0019 \u001a \u001b\u001c \u001d \u001e\u001f  !\" # $% & '( ) *+ , -. / 01 2 34 5 6\n789:;<=>?@ABCD E F G H IJ K L MN OP Q\nR S T UV W X YZ [ \\ ]^ _ ` a\nFIG. 3. (color online) Average normalized magnetizations a s\nfunctions of temperature for double perovskite Bi 2FeMoO 6\nfor four different Lvalues. Monte Carlo simulations are done\nwithL×L×Lmagnetic unit cells.\nWe carry out Monte Carlo simulations to estimate the\nTcof the materials17,18. It is well known that Curie\ntemperature will be a little underestimated if classical\napproximation to the Heisenberg model (2) is used in\nthe Monte Carlo simulation, but it must be much over-\nestimated if the model (2) is reduced to Ising model.\nFor comparison, we do our Monte carlo simulations with\nboth of the approximate models. We present the aver-\nage normalized magnetization of Bi 2FeMoO 6, from clas-\nsical Heisenberg model, as a representative in Fig. 3.\nTheTcvalue can be estimated to be 650 K. The others\ncan be done in the same way. The calculated Tcval-\nues are summarized in Table II. In contrast, the Ising\nmodel results are 1010, 396, 264, and 270 K for AB =4\nFeMo, MnMo, MnOs, and CrOs, respectively. There-\nfore, the Curie temperatures for the four half-metallic\nferrimagnets are, at least, 650, 255, 174, and 201 K for\nBi2FeMoO 6, Bi2MnMoO 6, Bi2MnOsO 6, andBi 2CrOsO 6,\nrespectively. High Curie temperature well above room\ntemperature could be realized in Bi 2FeMoO 6.\nIV. DISCUSSIONS AND CONCLUSION\nOur calculated results shows that the spin exchange\ninteraction between the nearest A and B atoms is posi-\ntive, and the A-A and B-B interactions are either weak\nor negative depending on specific A and B atoms10,11.\nIn the case of Bi 2FeMoO 6, our calculations show that\nthe nearest A-B spin exchange energy is 39.2 meV, and\nthe nearest A-A and B-B spin exchange energies are 0.13\nand -0.71 meV, respectively. The main spin interaction\nis intermediated by the O atom in between the magnetic\nA and B atoms, with the A-O-B bond angle being al-\nmost 180◦, and therefore, it is an antiferromagnetic su-\nperexchange. The A atom contributes a different mag-\nnetic moment from the B atom so that ferrimagnetism is\nformed in these double perovskite compounds. Possible\noverlapping of the nearest O wave functions should play\nsome roles in these compounds, but the main mechanism\nfor the ferrimagnetism must be the antiferromagnetic su-\nperexchange between the nearest A and B atoms.\nIn summary, our first-principles calculations show that\nfour double perovskite oxides, Bi 2ABO6(AB = FeMo,\nMnMo, MnOs, and CrOs), have negative formation en-\nergy, from -0.42 to -0.26 eV per formula unit. In the case\nof Bi2FeMoO 6, our calculated results uncover that its\nhalf-metallic gap and Curie temperature reachto 0.71eV\nand 650 K, respectively. These indicates that they could\nprobably be realized and high spin polarization could be\nachieved at high temperature. We believe that at least\nsome of them could be synthesized soon and would prove\nuseful for spintronic applications.\nACKNOWLEDGMENTS\nThis work is supported by Chinese Department of Sci-\nence and Technology (Grant No. 2012CB932302) andby Nature Science Foundation of China (Grant Nos.\n11174359 and 10874232).\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton,\nS. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M.\nTreger, Science 294, 1488 (2001).\n2G. M.Muller, J.Walowski, M.Djordjevic, G.-X.Miao, A.Gupt a,\nA. V. Ramos, K. Gehrke, V. Moshnyaga, K. Samwer, J. Schmal-\nhorst, A. Thomas, A. Hutten, G. Reiss, J. S. Moodera, and M.\nMunzenberg, Nat. Mater. 8, 56 (2009).\n3R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J.\nBuschow, Phys. Rev. Lett. 50, 2024 (1983).\n4J.-H. Park, E. Vescovo, H.-J. Kim, C. Kwon, R. Ramesh, and T.\nVenkatesan, Nature (London) 392, 794 (1998).\n5K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y.\nTokura, Nature (London) 395, 677 (1998).\n6R. Nechache, C. Harnagea, A. Pignolet, F. Normandin, T. Vere s,\nL.-P. Carignan, and D. Menard, Appl. Phys. Lett. 89, 102902\n(2006).\n7B. Aissa, R. Nechache, D. Therriault, F. Rosei, and M. Nedil,\nAppl. Phys. Lett. 99, 183505 (2011).\n8H. Boschker, J. Kautz, E. P. Houwman, W. Siemons, D. H. A.\nBlank, M. Huijben, G. Koster, A. Vailionis, and G. Rijnders,\nPhys. Rev. Lett. 109, 157207 (2012).\n9A. J. Hauser, J. R. Soliz, M. Dixit, R. E. A. Williams, M. A.\nSusner, B. Peters, L. M. Mier, T. L. Gustafson, M. D. Sumption ,\nH. L. Fraser, P. M. Woodward, and F. Y. Yang, Phys. Rev. B\n85, 161201(R) (2012).\n10D. Serrate, J. M. de Teresa, and M. R. Ibarra, J. Phys. Conden.\nMatter19, 023201 (2007); and references therein.\n11M. Opel, J. Phys. D 45, 033001 (2012); and references therein.\n12P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W.\nKohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n13G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); G. Kresse\nand J. Furthmuller, Phys. Rev. B 54, 11169 (1996).\n14J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,\n3865 (1996).\n15A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J. Phys.\nC13, 2675 (1980).\n16H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).\n17N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. M.\nTeller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).\n18K. Binder and D. W. Heermann, Monte Carlo Simulation in\nStatistical Physics (Springer, Berlin, 2002).\n19P. Baettig and N. A. Spaldin, Appl. Phys. Lett. 86, 012505\n(2005).\n20P. Baettig, C. Ederer, and N. A. Spaldin, Phys. Rev. B 72,\n214105 (2005).\n21B.-G. Liu, Phys. Rev. B 67, 172411 (2003).\n22W.-H. Xie, Y.-Q. Xu, B.-G. Liu, D. G. Pettifor, Phys. Rev. Let t.\n91, 037204 (2003).\n23B.-G. Liu, in Lecture Notes in Physics Vol.676(I. Galanakis\nand P. H. Dederichs, Eds, Half-metallic Alloys - Fundamentals\nand Applications , Springer Berlin 2005), pp. 267-291."    },    {        "title": "1703.07515v1.Fast_domain_wall_motion_induced_by_antiferromagnetic_spin_dynamics_at_the_angular_momentum_compensation_temperature_of_ferrimagnets.pdf",        "content": "Fast domain wall motion induced by antiferromagnetic spin dynamics at the angular momentum \ncompensation temperature of ferrimagnets  \n \nKab-Jin Kim1,2†★, Se Kwon Kim3†, Takayuki Tono1, Se-Hyeok Oh4,Takaya Okuno1, Woo Seung Ham1, \nYuushou Hirata1, Sanghoon Kim1, Gyoungchoon Go5, Yaroslav Tserkovnyak3, Arata Tsukamoto6, \nTakahiro Moriyama1, Kyung -Jin Lee4,5,7★, and Teruo Ono1★ \n \n \n \n1Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611 -0011, Japan  \n2 Department of Physics, Korea Advanced Institu te of Science and Technology, Daejeon 34141, Korea  \n3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA  \n4Department of Nano -Semiconductor and Engineering, Korea University, Seoul 02841, Korea  \n5Department of Mat erials Science & Engineering, Korea University, Seoul 02841, South Korea  \n6College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan  \n7KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, \nSouth Korea  \n \n★ Correspondence to: kabjin@scl.kyoto -u.ac.jp, kj_lee@korea.ac.kr , ono@scl.kyoto -u.ac.jp  \n  Antiferromagnetic spintronics is an emerging research field which aims to utilize \nantiferromagnets as core elements in spintronic devices1,2. A central mo tivation toward this \ndirection is that antiferromagnetic spin dynamics is expected to be much faster than ferromagnetic \ncounterpart because antiferromagnets have higher resonance frequencies than ferromagnets3. \nRecent theories indeed predicted faster dynam ics of antiferromagnetic domain walls (DWs) than \nferromagnetic DWs4-6. However, experimental investigations of antiferromagnetic spin dynamics \nhave remained unexplored mainly because of the immunity of antiferromagnets to magnetic fields. \nFurthermore, this  immunity makes field -driven antiferromagnetic DW motion impossible despite \nrich physics of field -driven DW dynamics as proven in ferromagnetic DW studies. Here we show \nthat fast field -driven antiferromagnetic spin dynamics is realized in ferrimagnets at t he angular \nmomentum compensation point TA. Using rare -earth–3d-transition metal ferrimagnetic compounds \nwhere net magnetic moment is nonzero at TA, the field -driven DW mobility remarkably enhance s \nup to 20 km s−1T−1. The collective coordinate approach generalized for ferrimagnets7 and atomistic \nspin model simulations6,8 show that this remarkable enhancement is a consequence of \nantiferromagnetic spin dynamics at TA. Our finding allows us to inve stigate the physic s of \nantiferromagnetic spin dynamics and highlights the importance of tuning of the angular \nmomentum compensation point of ferrimagnets, which could be a key towards ferrimagnetic \nspintronics.  Encoding information using magnetic DW motion  is essential for future magnetic memory \ndevices , such as racetrack memor ies9,10 . High-speed DW motion is a key prerequisite for making the \nracetrack feasible. However, velocity breakdown due to the angular precession  of DW , referred to as the \nWalker breakdown11, generall y limits the functional performance in ferromagnet -based DW devices. \nRecently, it was reported  that the DW speed boosts up  significantly in antiferromagnets  due to the  \nsuppression of  the angular precession4-6. However, the immunity  of antiferromag nets to m agnetic fields \nyields  notorious difficulties  in creating, manipulating , and detecting  antiferromagnetic DW s, com pared to \nferromagnetic one s. One possibility  to avoid these difficulties  is offered by the synthetic \nantiferromagnets12, where the net magnetic moment can be controlled by tuning the thickness of two \nferromagnetic layers coupled antiferromagnetically. However, they still suffer from the field -immunity  \nwhen the net magnetic moment approaches zero , preventing the study of antiferromagnetic DW \ndynami cs. Here we  show that magnetic field -controlled antiferromagnetic spin dynamics can be achieved  \nby employing ferrimagnets.  \nThere  are a class of ferrimagnet s, rare earth ( RE)–transition metal ( TM) compound s, where the \nspins of two inequivalent  sublattices  are coupled antiferromagnetically. Because of different Land é g-\nfactors  between RE and TM elements , these ferrimagnet s have two special temperatures below the Curie \ntemperature : the magneti sation compensation temperature, TM, at which the two magnetic momen ts \ncancel each other , and the angular momentum compensa tion temperature , TA, at which the net angular \nmomentum vanishes13-15. In particular, the existence of TA in ferrimagnets provides a framework to \ninvestigate antiferromagnetic spin dynamics. It is beca use the time evolution of the state of a magnet is \ngoverned by the commutation relation of the angular momentum, not of the magnetic moment . As a \nresult,  the nature of the dynamics of the ferrimagnets is expected to change  from ferromagnetic to \nantiferroma gnetic as approaching the angular momentum compensation point  TA. Furthermore, the net \nmagnetic moment of ferrimagnets is nonzero at TA and can thus  couple to an external magnetic field , \nopen ing a new possibility of field -driven antiferromagnetic spin dyna mics.  In order to pursuit this possibility, w e first describe distinguishing  features of ferrimagnetic DWs \nnear TA based on the collective coordinate approach7. A ferrimagnetic DW effectively acts as an \nantiferromagnetic DW  around TA. The low -energy dynami cs of a  DW in quasi one -dimensional magnets \nis generally described by two collective coordinates, its position X and angle Φ, which capture  the \ntranslational  and spin -rotational degrees of freedom  of the DW , respectively.  In ferromagnets, they are \ngyrotropically coupled by the Berry phase  that is proportional to the net spin density , and the motion of a  \nDW slows down  severely above a certain critical stimulus, engendering the phenomenon of the Walker \nbreakdown11. In antiferromagnets, on the other hand,  the dynamics of  X and Φ are independent owing to \nvanishing the net spin density7. The DW dynamics in antiferromagnets  is thus free from  the Walker \nbreakdown  and can be fast in a broad range of external driving forces compar ed to that in ferromagnets.   \nIn the following, we explain  a theory for the field-driven DW dynamics  in ferrimagnets  in high \nfields , which agrees with  our experi ment al results  as discussed  later. The DW velocity  can be derived  as \nfollows  by invoking the energy conservation and the gyrotropic coupling between the two collective \ncoordinates  X and Φ (see Supplementary  Information for microscopic derivations) . For a sufficiently \nstrong external field , the anisotropy  energy can be neglected and the time derivatives of X and Φ can be \nconsidered to approach  constant values16, \nV X and \n . The energy -dissipation rate caused by \nthese dynamics is given by \n2 2 2\n22 112   V sαsα P Α , where \nΑ is the cross sectional area of \nthe magnet, \n is the domain -wall width, \n1α and \n1sare the Gilbert damping constant and the spin angular \nmomentum density of one sublattice, respectively, and \n2α and \n2s are for the other sublattice. Invoking the \nconservation of the total energy, the rate of energy dissipation can be equated to the decreasing rate of the \nZeeman energy induced by the translation motion of the domain wall, which yields the equation, \nVHM M V sαsα Α Α2 12 2 2\n22 11 2 2   \n, where \n1M  and \n2M are th e magnetization of the \ntwo sublattices, and \nH is an external field. Note that the net magnetization, \n2 1M M , does not vanish at \nTA due to the difference in the Land é g-factors  of two sublattice atoms , which is essential to drive a DW with an external field.  In addition, when there is finite net spin angular momentum \n2 1ss , e.g., away \nfrom TA, the angular and linear velocities are related by the gyrotropic coupling whose strength is \nproportional to the net  angular momentum \n2 1ss17. Balancing the  gyrotropic force on Φ with the  \ndissipative force yields  \nVss sαsα2 1 22 11  . Solving the two aforementioned equations for \nV\nand \n , we obtain  \n \n H\nss s s)M M(s sV2\n2 12\n22 112 1 22 11\n \n\n,      \n\n H\nss s s)M M(ss\n2\n2 12\n22 112 1 2 1\n\n   (1) \nAs the system approaches the angu lar momentum compensation point \n02 1ss , the domain wall \nspeed  \nV  increases, whereas the precessional frequency \n  decreases. At T = TA, X and Φ are completely \ndecoupled  and the pure translational  dynamics  of the DW  is obtained , implying that the ferrimagnetic DW \neffectively acts as antiferromagnetic DW  and its motion is driven by a magnetic field at T = TA.  \nIn order to prove the above theoretical prediction, we investigate DW dynamics  in ferrimagneti c \nGdFeCo compounds. Figure 1  shows a schematic illustration of our sample. SiN(5 nm)/Gd 23Fe67.4Co9.6 \n(30 nm)/SiN(5 nm) films are  deposited on intrinsic Si substrate by magnetron sputtering . GdFeCo is a \nwell-known RE– TM ferrimagnet ic compound , in which RE and TM moment s are coupled \nantiferromagnetically18. The relative mag netic moment s of RE and TM can be easily controlled by \nvarying the composition or temperature, so that TM and TA can be easily designed  in RE –TM \nferrimagnet s. The GdFeCo film is then patte rned into micro wires  with 5 μm wi dth and 65  μm length  \nusing electron beam lithography and Ar ion milling . A Hall bar is designed to detect the DW motion via \nthe anomalous Hall effect (AHE) voltage , VH.  \nWe first characterise the magnetic properties of the GdFeCo  microstrips . Figure 2 a shows the \nhysteresis loop s of GdFeCo microstrips  at various temperatures . The  AHE resistance , RH (RH = VH /I), is \nmeasured by sweeping the  out-of-plane magnetic field , BZ. Square hysteresis loops are clearly observed, \nindicating that GdFeCo has a perpend icular magnetic anisotropy. The coercivity field, BC, and the \nmagnitude of the Hall resistance  change , ΔRH \n  Z Z B R B R R H H H , are extracted from the hysteresis loop s and summarised in Fig. 2b. BC increases with increasing temperature , but a sudden drop  \nis observed at T = 220 K.  A sign change of RH is observed at the same temperature.  This is a typical \nbehaviour of ferrimagnet s at the magnetisation  compensation temperature TM19. As T approaches TM, the \nnet magnetic moment converges to zero , and thus , a large r magnetic field is required to obtain  a \nsufficiently high Zeeman energy to switch the magnetisati on. Thus, BC diverges at TM. The s ign change of \nRH represents additional  evidence of TM. The magneto -transport properties of GdFeCo are known to be \ndominated by FeCo mome nts because  the 4 f shell, which is respons ible for the magnetic properties of Gd, \nis located far below the Fermi energy level20. Thus, the sign change of RH indicates a change in  the \nrelative direction of the FeCo  moments  with respect t o the magnetic field, which occur s at TM. At T < TM, \nthe Gd moment dominates over the FeCo moment so that the Gd moment  aligns alo ng the magnetic field \ndirection . However,  at T > TM, the FeCo moment  is dominant and thus  aligns along the magnetic field \ndirection. Therefore, Fig. 2b allows us to identify TM for our GdFeCo sample , which  is approximately 220 \nK. \nAlthough TM can be easily determined by magnetisation  or magneto -transport  measurement s, it is \ngenerally not easy to determine  TA because TA is not related to the net magnetisation  but rather to the  \nangular momentum of the system. For GdFeCo, the net magneti sation  \nM and angular momentum \nA  are \nwritten as follows13,14,21. \nFeCo Gd M M M\n  and \nFeCo FeCo Gd Gd FeCo Gd   / M / M A AA \n  , where \n FeCoGdM\n and \n FeCoGdA are the magnetic moment and angular momentum  of the Gd (FeCo) sub -lattices , \nrespectively , and \nB g FeCoGd FeCoGd  is the gyromagnetic ratio of Gd (FeCo ), where \nB  is the Bohr \nmagneton and \n  is the reduced Plank constant. According to the literature s22–24, \nFeCog  (~2.2) is slightly \nlarger than  \nGdg  (~2) owing to the spin -orbit coupling of FeCo and zero orbital angular momentum of the \nhalf-filled 4 f shell of Gd ; therefore,  TA is expected to be higher than TM in GdFeCo.  \nBased  on above consideration , we measure the field-driven  DW speed  at T > TM using a real-time \nDW detection method25–27. We first saturate the magnetisation  by applying a large  negative field (| B| > \n|BC|) and then switch the field to the positive direction . Thi s positive field is a DW driving field , Bd, and should be  smaller than  BC (|Bd| < |BC|). Next , we inject a current pulse into the electrode to create a DW by \na current -induced Oersted field , as shown in Fig.  1. The created DW propagate s along the wire due to the  \npresence of  Bd, and the DW motion is detected at the Hall bar by monitoring the change s in VH. Here , the \nchange s in VH are recorded by an oscilloscope  such that nanosecond time -resolution can be achieved . The \nDW speed  can be calculated from the arri val time and the travel distance (60 μm) of the DW.  The details \nof the measureme nt scheme are explained in the Method  section .  \nFigure 3 a shows the DW speed  as a function of Bd at several temperatures  above TM. The DW \nvelocity increases linearly with field for all temperatures. Such a linear behaviour can be described by\n0 dB B v\n. Here , μ is referred to as the DW mobility and B0 is the correction  field, which  generally \narises from imperfection s of the sample  or complexit ies of the internal DW structure28,29. Figure 3b  shows  \nthe DW  velocity as a function of temperature for several bias fields. The DW velocity shows a sharp peak \nas expected near T = TA based on Eq. (1). The DW mobility μ is estimat ed from the linear fit in Fig. 3 a \n(dashed lines) and is plotted as a fu nction of  the temperature in Fig. 3 c. Starting from T = 260  K, which is \nslightly higher than TM, μ increases steeply , reach ing its maximum at T = 310  K, and then decreases with \na further  increase in  temperature. The peak mobility is as high as 20 km ·s−1·T−1 at T = 310  K. These \nexperimental results are in agreement  with the analytical expression Eq. (1), which  predicts a Lorentz ian \nshape of the DW velocity near T = TA with the width \n22 11 sαsα~ . Such a consistency between \nexperiment and theory manifests that the ferrimagnetic DW indeed acts as an antiferromagnetic DW  at T \n= TA.  \nWe next perform atomic spin model simulations  based on the atomistic Landau -Lifshitz -Gilbert \n(LLG) equation5,8 (see Method section for details)  to verify the experimental result and theor etical \nprediction . We employ a set of the reduced magnetic moments around s1−𝑠2=0 as shown in Table 1. \nThe total number of set is 9  in which the index  5 corresponds to  the temperature at TA. We assume that a \nDW is of Bloch -type with perpendicular mag netic anisotropy along the z axis. Fig ure 4a shows the DW \nvelocity as a function of the external field BZ, applied along the z axis. Velocities increase  linearly with BZ for BZ  > 10 mT . The numerical results (circular symbols) are in excellent agreement w ith the analytic \nsolution (solid lines) for the DW velocity for high fields in Eq. (1). The inset of Fig.  4a shows the DW \nvelocity in low field regime s (BZ  < 10 mT) . The Walker breakdown occurs in this regime  except for the \ncase of TA. The vertical dashed  lines represent the Walker breakdown field. Fig ure 4b shows the DW \nvelocity as a  function of  δs=s1−𝑠2. At δs=0, the DW velocity  is the highest , which agrees with the \nexperimental result  and theory . This good agreement also supports that field -driven antiferromagnetic \nspin dynamics is realized in ferrimagnets at T A. \n To date , the angular momentum compensation  point and its effect on the magnetisation  dynamics  \nhave  often been overlooked in studies of ferrimagnet s. Laser-induced magnetisation  switching18,30,31 and \nmagnetic DW motion32–33, which have bee n major research themes  of ferrima gnets, have mostly been \nstudied without identifying  TA. It has been investigated in the context of magnetic resonance  or \nmagnetisation  switching by current  around TA 14,15,34, but a clear identification of spin dynamics at T = TA \nhave  been remained elusive15. On the other hand, our results  clearly show that the antiferromagnetic spin \ndynamics is achievable at T = TA. Moreo ver, such antiferromagne tic spin dynamics  can be controlled by \nmagnetic field due to the finite ma gnetic moment at T = TA, which opens a way of studying field-driven \nantiferromagnetic spin dynamics. Furthermore, the fact that field-driven DW speed  exhibits a sharp and \nnarrow peak at TA provides  a sim ple but accurate  method to determine TA, which has not been possible.  \nWe also achieve a fast DW speed near room temperature , opening a possibility for ultra -high speed \ndevice operation at room temperature. We expect that s uch findings are also advantageous  for current -\ninduced DW motion in ferrimagnets. A low threshold current density , more than one order of magnitude \nsmaller than that of ferromagnets,  has already been demonstrated  in ferrimagnets32–33. Therefore, by \ntuning TA, one could obtain  high-speed an d low power consumed  spintronic  device s using ferrimagnet s, \nwhich could even be superior to ferromagnetic system s. To conclude, o ur work suggests that  revealing \nand tailoring TA, which has not been paid much attention , is crucial for controlling ferrimagne tic \nmagnetisation  dynamics, and therefore could be a key for realising ferrimagnet ic spintronics .  Method  \nFilm  preparation and device fabrication.  The studied samples are amorphous thin films of \nGd 23Fe67.4Co9.6 of 30 nm thickness which have been deposited  by magnetron sputtering. To avoid \noxidation of the GdFeCo layer, 5 nm Si 3N4 were used as buffer and capping layers, respectively. The \nfilms  exhibit an out -of-plane magnetic anisotropy . GdFeCo micro strips  with a 100 nm -wide Hall bar \nstructure were fabricat ed using electron beam lithography and Ar ion milling process. A negative tone \nelectron beam resist (maN -2403) was used for lithography at a fine resolution (~5 nm). For current \ninjection, Ti(5 nm)/Au(100 nm) electrodes were stacked on the wire. To make an  Ohmic contact, the \nSi3N4 capping layer was removed  by weak ion milling before electrode deposition.  \nExperimental setup  for field -driven domain wall motion . A pulse generator (Picosecond 10, 300B) \nwas used to generate a current pulse to  create the DW. 100m A and 10ns current pulse is used to create the \nDW. For field -driven DW motion, 1mA dc current (corresponding current density is 7  109 A·m−2) was \nflowed along the wire to generate anomalous Hall voltage, VH. Yokogawa 7651  was used as a current \nsource. The VH at the Hall cross was recorded by the oscilloscope  (Textronix 7354)  through the 46 dB \ndifferent ial amplifier. Low temperature probe sta tion was used for measuring the DW motion in a wide \nrange of temperature.  \nDW detection technique . We used a time -of-flight measurement of DW propagation to obtain a DW \nspeed  in a flow regime. The procedure for measuring the DW speed  is as follows . First, a large out -of-\nplane magnetic field Bsat = −200 mT is applied to reset the magnetisation . Next, a drive field Bd, in the \nrange of | BP| < |Bd| < |BC|, is applied in the opposite direction. Here, BP is the pinning field of DW motion \nand BC is the coercive field of  the sample . Since the  Bd is smaller than the BC, the drive field does not \nreverse the magnetisation  or create DWs. N ext, a current pulse (100 mA, 1 0 ns) is injected by a pulse \ngenerator  to create  a DW next to the contact line  through current -induced Oersted field . As soon as the \nDW is create d, the  Bd pushes the DW because the Bd is larger than the BP. Then  the DW propagates along \nthe wire and passes through the Hall cross  region . When the DW passes through the Hall cross, the Hall \nvoltage changes abruptly because the magnetisation  state of th e Hall cross reverses as a result of the DW passage. This Hall signal change is recorded by the oscilloscope through the 46dB differential amplifier. \nWe refer to this as a ‘signal trace’. Since the detected Hall voltage change includes a large background \nsignal, we subtract the background from the ‘signal trace’ by measuring a ‘reference trace’. The reference \ntrace is obtained in the same manner as the signal trace, except that the saturation field direction is \nreversed ( Bsat = +200 mT). In this reference t race, no DW is nucleated , so that only the electronic noise \ncan be detected in the oscilloscope  in the reference trace . To obtain a sufficiently high signal -to-noise \nratio, we averaged the data from 5 repeated measurements .  \nAtomic spin model simulation . We adopt the atomistic model simulation s ince the ferrimagnet consists \nof two magnetic components, i.e., RE and TM on an  atomic scale . The one -dimensional Hamiltonian of \nferrimagnet is described by ℋ=𝐴𝑠𝑖𝑚∑𝑺𝒊∙𝑺𝒊+𝟏 𝑖 −𝐾𝑠𝑖𝑚∑(𝑺𝒊∙𝒛̂)2𝑖 +𝜅𝑠𝑖𝑚∑(𝑺𝒊∙𝒚̂)2𝑖 , where 𝑺𝒊 is the \nnormalized magnetic moment at lattice site  𝑖. The odd number of 𝑖 represents a site for TM, and the even \nnumber of 𝑖 represents a site for RE. 𝐴𝑠𝑖𝑚,𝐾𝑠𝑖𝑚,𝜅𝑠𝑖𝑚 denote the exchange, easy -axis anisotropy along the \nz axis, and hard -axis anisotropy, respectively. We sol ve the atomistic LLG equation  𝜕𝑺𝒊\n𝜕𝑡=−𝛾𝑖𝑺𝒊×\n𝑯𝒆𝒇𝒇 ,𝒊+𝛼𝑖𝑺𝒊×𝜕𝑺𝒊\n𝜕𝑡, where   𝑯𝒆𝒇𝒇 ,𝒊=−1\n𝑀𝑖𝜕ℋ\n𝜕𝑺𝒊  is the effective field, γi=gi 𝜇𝐵ℏ⁄ is the gyromagnetic \nratio, and 𝑀𝑖 is the magnetic moment for site 𝑖. We use parameters as 𝐴𝑠𝑖𝑚=7.5𝑚𝑒𝑉 ,𝐾𝑠𝑖𝑚=\n0.3𝑚𝑒𝑉,𝜅𝑠𝑖𝑚=−0.8𝜇𝑒𝑉, damping constant  𝛼𝑇𝑀=𝛼𝑅𝐸=0.004, the lattice constant is 0.4 nm, and \nLand é g-factors for each site are gTM=2.2 and gRE=2. References  \n1. MacDonald, A. H. & Tsoi, M. Antiferromagnetic metal spintronics. Phil. Tra ns. R. Soc. A  369, 3098 –\n3114 (2011).  \n2. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics . Nat. Nanotech . \n11, 231-241 (2016 ). \n3. Keffer, F. & Kittel, C. Theory of antiferromagnetic resonance. Phys. Rev . 85, 329 –337 (1952).  \n4. Gomon ay, O. , Jungwirth, T.  & Sinova , J. High Antiferromagnetic Domain Wall Velocity Induced by \nNéel Spin -Orbit Torques . Phys. Rev. Lett.  117, 017202  (2016).  \n5. Shiino, T. et al. Antiferromagnetic Domain Wall Motion Driven by Spin -Orbit Torques. Phys. Rev. \nLett. 117, 087203 (2016).  \n6. Tveten, E.G.,  Qaiumzadeh, A., &  Brataas, A.  Antiferromagnetic Domain Wall Motion Induced by \nSpin Waves . Phys. Rev. Lett . 112, 147204 (2014 ). \n7. Tveten, E.G.,  Qaiumzadeh, A., Tretiakov,  O. A. & Brataas, A.  Staggered Dynamics in \nAntiferromagnets  by Collective Coordinates. Phys. Rev. Lett . 110, 127208 (2013) . \n8. Evans , R.F.L. et al. Atomistic spin model simulations of magnetic nanomaterials. J. Phys.: \nCondensed Matter , 26, 103202 (2014) . \n9. Yamaguchi, A. et al. Real -space observation of current -driven domain wall motion in submicron \nmagnetic wires. Phys. Rev. Lett . 92, 077205 (2004).  \n10. Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain -wall racetrack memory. Science  320, \n190–194 (2008).  \n11. Schryer, N. L. &Wa lker, L. R. The motion of 180 ° domain walls in uniform dc magnetic fields. J. \nAppl. Phys . 45, 5406 -5421 (1974).  \n12. Yang, S. -H., Ryu, K. -S. & Parkin, S. Domain -wall velocities of up to 750  m s−1 driven by exchange -\ncoupling torque in synthetic antiferromagnets . Nature Nanotech . 10, 221 –226 (2015) . \n13. Wangness, R. K. Sublattice effects in magnetic resonance. Phys. Rev . 91, 1085 -1091 (1953).  14. Stanciu, C. D. et al. Ultrafast spin dynamics across compensation points in ferrimagnetic GdFeCo: \nThe role of angular momentum compensation. Phys. Rev. B  73, 220402(R) (2006).  \n15. Binder, M. et al. Magnetization dynamics of the ferrimagnet CoGd near the compensation of \nmagnetizat ion and angular momentum. Phys. Rev. B  74, 134404 (2006).  \n16. Clarke, D. J. , Tretiakov, O. A. , Chern, G. -W., Bazaliy, Ya. B. & Tchernyshyov , O. Dynamics of a \nvortex domain wall in a magnetic nanostrip: Application of the collective -coordinate approach. Phys. \nRev. B  78, 134412 (2008) . \n17. Thiele, A.A. Steady -State Motion of Magnetic Domains . Phys. Rev. Lett . 30, 230  (1973).  \n18. Radu, I.  et al. Transient ferromagnetic -like state mediating ultrafast reversal of antiferromagnetically \ncoupled spins, Nature , 472, 205 -208 (20 11). \n19. Okuno, T. et al. Temperature dependence of magnetoresistance in GdFeCo/Pt heterostructure, Appl. \nPhys. Express  9, 073001 (2016).  \n20. Tanaka, H., Takayama, S. & Fujiwara, T. Electronic -structure calculations for amorphous and \ncrystalline Gd 33Fe67alloys. Phys. Rev. B  46, 7390 -7394 (1992).  \n21. Tsuya, N. 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Propagation of a domain wall in a submicrometer magnetic wire. Science  284, 468 –470 \n(1999).  \n29. Volkov, V. V. & B okov, V. A.  Domain Wall Dynamics in Ferromagnets.  Physics of the Solid State  \n50, 199 -228 (2008).  \n30. Alebrand, S. et al. Light -induced magnetization reversal of high -anisotropy TbCo alloy films. Appl. \nPhys. Lett.  101, 162408 (2012).  \n31. Stanciu, C. D. et al. Subpi cosecond Magnetization Reversal across Ferrimagnetic Compensation \nPoints . Phys. Rev. Lett.  99, 217204  (2007). \n32. Ngo, D -T., Ikeda, K. & Awano, H. Direct Observation of Domain Wall Motion Induced by Low -\nCurrent Density in TbFeCo Wires. Appl. Phys. Express  4, 093002 (2011).  \n33. Awano, H., Investigation of domain wall motion in RE -TM magnetic wire towards a current driven \nmemory and logic. J. Magn. Magn. Mater . 383, 50-55 (2015).  \n34. Jiang, X., Gao., L. Sun J. Z. & Parkin S. S. P. Temperature Dependence of Current -Induce d \nMagnetization Switching in Spin Valves with a Ferrimagnetic CoGd Free Layer. Phys. Rev. Lett.  97, \n217202 (2006).  Acknowledgements  \nThis work was partly supported by JSPS KAKENHI Grant Numbers 15H05702, 26870300, 26870304, \n26103002, 25220604 , 2604316  Collaborative Research Program of the Institute for Chemical Research, \nKyoto University, and R & D project for ICT Key Technology of MEXT from the Japan Society for the \nPromotion of Science (JSPS). KJK acknowledges support from the KAIST start -up funding.  SKK and YT \nacknowledge the support from the Army Research Office under Contract No. 911NF -14-1-0016 . K.-J.L. \nacknowledges support from Creative Materials  Discovery Program through the National Research \nFoundation of Korea  (NRF -2015M3D1A1070465) . \n \nAuthor contri butions  \nK.-J.K., T.M. , and T.O. planed  the study. A.T. grew and optimi sed the GdFeCo film. T.T. fabricated the \ndevice  and performed  the experiment  with the guide of K. -J.K.. T.Okuno , W.-S.H., Y.H. , and  S.K. helped \nthe experiment . S.-K.K., K.-J.L., and Y.T.  provide theory.  S.-H.O., G.G., and K. -J.L. performed the \nnumerical  simulation. K.-J.K., S.-K.K., K. -J.L., T.M., and T.O analysed the result s. K.-J.K., S.-K.K., K. -\nJ.L., T.M.,  and T.O. wrote the manuscript.  \n \nAdditional  Information  \nSupplementary Information  is available in the online version of the paper. Reprints and permissions \ninformation is available at www.nature.com/reprints . Correspondence and requests for materials should \nbe addressed to K.-J.K, K.-J.L. and T. O.  \nCompeting financial interests  \nThe authors declare no competing financial interests.  \n  Figure Legends  \nFigure 1| Schematic illustration of Device structure . Schematic illustration of GdFeCo \nmicrowire . The inset  shows  schematic illustration of two spin sub-lattices  below and above the \nmagnetisation  compensation temperature, TM. Blue and red arrows indicate Gd and FeCo moments, \nrespectively.  \nFigure 2| Identification of magnetisation  compensation temperature TM. a, Anomalous Hall \neffect resistance RH as a function of perpendicular magnetic field BZ for several temperatures as denoted \nin the figure. b,. Coercive field, BC, and the magnitude of the Hall resistance change , ΔRH \n  Z Z B R B R R H H H\nwith respective to the temperature. The region shaded in red indicates the \nmagnetisation  compensation temperature TM \n \nFigure 3| Field -driven domain wall  (DW)  dynamics across the angular momentum \ncompensation temperature TA. a. DW speed  v as a function of driving field Bd for several \ntemperatures as denoted in the figure. Dashed lines are best fits based on \n0 dB B v . b. DW speed  v \nas a function of temperature T for several driving fields as denoted in the figure.  c. DW mobility μ as a \nfunction of temperature T. The red and blue shaded regions in b and c indicate the magnetisation  \ncompensation temperature, TM, and angular momentum compensation temperature, TA, respectively.  \n \nFigure 4| Simulation results of ferri magnetic domain wal l (DW) a. DW speed as a function of \nthe out -of-plane field BZ for various indices (see Table 1). Symbols are numerical results whereas solid \nlines are Eq. (1). Inset shows low field regimes, where vertical dotted lines indicate the Walker \nbreakdown fields.  b. Computed DW speed as a function of of δs=s1−𝑠2 at various values of BZ.  \n \n   \n \n \nFig.1  \n  \n \n \n \n \n \nFig.2  \n  \n \n \n \n \nFig.3  \n  \n \n \n \n \n \n \n \nFig.4  \n \n \n \n  \n \nTABLE 1. Parameters used in the numerical simulation.  \nIndex  1 2 3 4 5 6 7 8 9 \n𝑴𝑭𝒆𝑪𝒐 (𝒌𝑨/𝒎) 1120  1115  1110  1105  1100  1095  1090  1085  1080  \n𝑴𝑮𝒅(𝒌𝑨/𝒎) 1040  1030  1020  1010  1000  990 980 970 960 \n𝜹𝒔(𝟏𝟎−𝟕𝑱∙𝒔 𝒎𝟑⁄ ) -1.24 -0.93 -0.62 -0.31 0 0.31 0.62 0.93 1.24 \n \n "    },    {        "title": "2009.05742v1.Ferrimagnetic_States_of_Na_K_Alloy_Clusters_in_Zeolite_Low_Silica_X.pdf",        "content": "arXiv:2009.05742v1  [cond-mat.soft]  12 Sep 2020Ferrimagnetic States of Na-K Alloy Clusters in Zeolite Low- Silica X\nTakehito Nakano,1,2,∗Shingo Araki,3,†Luu Manh Kien,4,2Nguyen Hoang Nam,5\nDuong Thi Hanh,2Akihiro Owaki,2Ken Goto,2Akira Matsuo,6Koichi Kindo,6and Yasuo Nozue2,‡\n1Institute of Quantum Beam Science, Graduate School of Scien ce and Engineering, Ibaraki University,\n2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan\n2Department of Physics, Graduate School of Science, Osaka Un iversity,\n1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan\n3Department of Physics, Okayama University, Okayama 700-85 30, Japan\n4Nano and Energy Center, Hanoi University of Science,\nVietnam National University, 334 Nguyen Trai, Thanh Xuan, H anoi, Vietnam\n5Center for Materials Science, Faculty of Physics,\nHanoi University of Science, Vietnam National University,\n334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam\n6Institute for Solid State Physics, University of Tokyo,\n5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan\n(Dated: September 15, 2020)\nIn zeolite low-silica X (LSX), β-cages with the inside diameter of ≈7˚A are arrayed in a diamond\nstructure. Among them, supercages with the inside diameter of≈13˚A are formed and arrayed in a\ndiamondstructurebythesharingofwindows withtheinsided iameterof ≈8˚A.Thechemicalformula\nof zeolite LSX used in the present study is given by Na xK12−xAl12Si12O48per supercage (or β-cage),\nwhereNa xK12−xandAl 12Si12O48aretheexchangeablecationsofzeolite LSXandthealuminos ilicate\nframework, respectively. Na-K alloy clusters are incorpor ated in these cages by the loading of guest\nK metal at nK atoms per supercage (or β-cage). A N´ eel’s N-type ferrimagnetism has been observed\natn= 7.8 forx= 4. In the present paper, optical, magnetic and electrical p roperties are studied\nin detail mainly for x= 4. Ferrimagnetic properties are observed at 6 .5< n <8.5. At the same\ntime, the Curie constant suddenly increases. An optical refl ection band of β-cage clusters at 2.8 eV\nis observed at n >6.5 in accordance with the sudden increase in the Curie constan t. An electrical\nresistivity indicates metallic values at n/greaterorapproxeql6, because a metallic state is realized in the energy\nband of supercage clusters. The ferrimagnetism is explaine d by the antiferromagnetic interaction\nbetween the magnetic sublattice of itinerant electron ferr omagnetism at supercage clusters and that\nof localized moments at β-cage clusters. The electrical resistivity in ferrimagnet ic samples at n= 8.2\nforx= 4 increases extraordinarily at very low temperatures, suc h as≈106times larger than the\nvalue at higher temperatures. Observed anomalies in the ele ctrical resistivity resembles the Kondo\ninsulator, but itinerant electrons of narrow energy band of supercage clusters are ferromagnetic\ndifferently from the Kondo insulator.\nPACS numbers: 82.75.Vx, 71.28.+d, 75.30.Mb, 75.50.Xx, 75.75.-c, 36.40. -c\nI. INTRODUCTION\nZeolite crystals have free spaces of regular cages for\nguest materials [1]. There are many different types of ze-\nolite structures [2]. Alkali metal clusters incorporated in\ncages of zeolites have a wide variety in electronic proper-\nties, such as a ferrimagnetism, a ferromagnetism, an an-\ntiferromagnetism, and an insulator-to-metal transition,\ndepending on the kind of alkali metals, their loading den-\nsity, and the structure type of zeolite frameworks [1, 3].\nIn zeolite low-silica X (LSX), supercages and β-cages\nwith the inside diameters of ≈13and≈7˚A, respectively,\narearrayedinadiamondstructure,namelythedoubledi-\namond structure. Up to now, detailed studies have made\n[1, 3–14]. A N´ eel’s N-type ferrimagnetism has been ob-\nserved in Na-K alloy clusters incorporated into zeolite\n∗takehito.nakano.phys@vc.ibaraki.ac.jp\n†araki@science.okayama-u.ac.jp\n‡nozue@phys.sci.osaka-u.ac.jpLSX, where an antiferromagnetic interaction works be-\ntween nonequivalent magnetic sublattices of supercages\nandβ-cages [1, 3, 6, 9]. In the present paper, their op-\ntical, electrical and magnetic properties are studied in\ndetail.\nBesides the N´ eel’s N-type ferrimagnetism, a ferromag-\nnetism has been observed in Na-rich Na-K alloy clusters\nin zeolite LSX [12]. In pure Na clusters in zeolite LSX,\na metallic phase has been observed with the increase in\nNa loading density [1, 5, 8, 10, 13]. In pure K clusters in\nzeolite LSX, a ferrimagnetic property at higher K load-\ning densities has been observed in a metallic phase [1, 9].\nUnder the pressure loading of K-metal into zeolite LSX,\nan itinerant electron ferromagnetism has been newly ob-\nserved at the loading pressure of ≈0.9 GPa [14].\nAfter the discovery of ferromagnetic properties in K\nclusters in zeolite A [15], detailed studies have been\nmade [1, 3, 16–38]. In zeolite A, α-cages with the in-\nside diameter of ≈11˚A are arrayed in a simple cubic\nstructure. A spin-cant model of Mott-insulator antifer-\nromagnetism of K cluster array in α-cages is proposed2\n[1, 29, 32, 38]. In Rb clusters in zeolite A, a ferrimag-\nnetism has been observed [25, 39, 40]. An antiferromag-\nnetism of Mott insulator in alkali metal clusters in so-\ndalite has been clearly observed [41], and detailed stud-\nies have been made [42–55]. In sodalite, β-cages are ar-\nrayed in a body centered cubic structure. Alkali met-\nals in quasi-low-dimensional systems, such as the quasi-\none-dimensional metallic system in channel-type zeolite\nL [56–59], has been studied.\nA. Zeolite LSX\nZeolite X is one of the most typical aluminosilicate\nzeolites, and is nonmagnetic insulator unless guest mate-\nrials are loaded. Zeolite LSX is the zeolite X with Si/Al\n= 1 in aluminosilicate framework. The framework of ze-\nolite LSX is negatively charged and illustrated in Fig. 1\ntogether with typical sites of exchangeable monovalent\ncations ( As). Al and Si atoms are alternately connected\nby the sharing of O atoms. The space group is Fd¯3 with\nthe lattice constant of 25 ˚A. The chemical formula per\nunit cell is given by A96Al96Si96O384before the loading\nof guest materials. The number of cations is the same\nas that of aluminium atoms in framework. The frame-\nworkstructure type ofzeolite LSX is called FAU (IUPAC\nnomenclature [2]). The framework of FAU is constructed\nofβ-cages arrayed in a diamond structure. Among β-\ncages, “supercages (cavities) of FAU” are formed and\nalso arrayed in a diamond structure. The distance be-\ntween adjoining β-cages (or supercages of FAU) is 10.8\n˚A. Hereafter, we call “supercage of FAU” simply by “su-\npercage”. There are eight supercages (or eight β-cages)\nin the unit cell, and the chemical formula per supercage\n(orβ-cage) is given by A12Al12Si12O48. Zeolite LSX\nused in the present study contains Na and K cations,\nand the chemical formula per supercage (or β-cage) is\ngiven by Na xK12−xAl12Si12O48. Hereafter, we call it by\nNaxK12−x-LSX.\nInordertoacquireanintuitiveunderstandingofframe-\nwork structure, a polyhedral form is illustrated in Fig. 2.\nEachβ-cageis connectedto fouradjoining β-cagesbythe\nsharing of hexagonal prisms (double 6-membered rings,\nD6Rs). Supercages share windows of twelve-membered\nrings (12Rs) with adjoining supercages. The inside di-\nameters of 12R and 6R are ≈8 and≈3˚A, respectively.\nEachβ-cage shares 6-membered rings (6Rs) with four\nadjoining supercages.\nB. Alkali metal loading into zeolite Na xK12−x-LSX\nAlkali metals are easily loaded into zeolite by the va-\npor phase for unsaturated condition or by the direct con-\ntact with alkali metal for the saturated condition. In\nthe present paper, we loaded guest K metal at natoms\nper supercage (or β-cage) into Na xK12−x-LSX, and dis-\ncribe it as K n/NaxK12−x-LSX. The average number of!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nAl\nO\nSi\nβ-cage A\nsupercage of FAU 10.8 /uni00C5.169\nFIG. 1. (Color online) Aluminosilicate framework structur e\nof zeolite LSX and typical sites of exchangeable Acations\nwithout guest materials. β-cages are arrayed in a diamond\nstructure. Among them, supercages of FAU are formed and\narrayed in a diamond structure. The distance between ad-\njoiningβ-cages (or supercages of FAU) is 10.8 ˚A. See also the\npolyhedral illustration of the structure in Fig. 2.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nβ-cage \nsupercage \nof FAU \n6R 12R D6R \nFIG. 2. (Color online) Schematic illustrations of framewor k\npolyhedra of zeolite LSX. Each β-cage is connected to four\nadjoining β-cages by the sharing of double 6-membered rings\n(D6Rs), and arrayed in a diamond structure. Supercages of\nFAU are arrayed in a diamond structure by the sharing of\ntwelve-memberedrings(12Rs)withfouradjoiningsupercag es.\ns-electrons provided by the loading of alkali metal is also\nnper supercage (or β-cage).\nAn outermost s-electron of an alkali atom has a large\nsize and a small ionization energy, so that s-electrons in\nbulk alkali metals are well described by the free-electron\nmodel.s-electrons introduced in zeolite by the loading of\nguest alkali atoms move freely over cations distributed in\ncages. The aluminosilicate framework, however, is neg-\natively charged and has high-energy conduction bands.\nTherefore, s-electrons are repulsed by the framework.\nThes-electrons successively occupy quantum states of\nclusters formed in cages. If we assume a spherical quan-\ntum well (SQW) potential for cage, quantum states, such\nas 1s, 1pand 1dstates, are formed in the increasing or-\nderofenergy, andtwo, six andten s-electronscanoccupy\nrespective quantum states successively [1]. Schematic il-\nlustrations of cluster in supercage and quantum states3\nofs-electron in the SQW potential with the diameter of\n13˚A are given in Fig. 3. A large sphere in supercage\nis a schematic image of s-electron wave function. 1 s, 1p\nand 1dquantum states have energies 0.9, 1.8 and 3.0 eV\nfrom the bottom ofthe SQWpotential, respectively. The\nnumber in each parentheses indicates the degeneracy in-\ncluding spin. The optical excitations (dipole transitions)\nare allowed between 1 s-and-1pand between 1 p-and-1d\nstates. That between 1 s-and-1dis forbidden.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n3.0 eV \n1.8 eV \n0.9 eV \nsupercage of FAU 1s(2) 1d(10) \n1p(6) \n13 Å ≈13 Å \nFIG. 3. (Color online) Schematic illustrations of alkali me tal\ncluster in supercage of FAU and the quantum states of s-\nelectron in the SQW potential with the diameter of 13 ˚A.\nThe SQW potential, however, is primitive for the su-\npercage cluster, because of large 12R windows. The\nspheres of s-electron wave functions in adjoining su-\npercages largely overlap with each other, because the\ndistance between adjoining supercages is 10.8 ˚A which\nis shorter than the inside diameter of supercage ≈13˚A.\nNevertheless, we use 1 s, 1pand 1dquantum states of the\nSQW potential, because of a convenient model to think\nabout quantum states localized in supercage. In zeolite\nA, K clusters are well localized in α-cages with the inside\ndiameter of ≈11˚A, and the SQW model well explains ex-\nperimental results, because of rather narrow windows of\nα-cages [1, 16, 17, 38]. Electrons in regular supercages of\nzeolite LSX are expected to construct the energy band, if\nthe contributions of the electron-phonon interaction and\nthe electron correlation are not significant. Because the\nsupercage has the Tdsymmetry which has no inversion\nsymmetry at the cage center, 1 s, 1pand 1dstates hy-\nbridize with each other. The electronic states of energy\nband are constructed of these hybridized states depend-\ning on the positions in the Brillouin zone. For example,\nthe electronic states at the bottom of the lowest band are\nmainly constructed of 1 sstates.\nSchematic illustrations of cluster in β-cage and quan-\ntumstatesof s-electronintheSQWpotentialwiththedi-\nameterof7 ˚AaregiveninFig.4. Alargespherein β-cage\nis a schematic image of s-electron wave function. 1 sand\n1pquantum states have energies 3.1 and 6.3 eV from the\nbottom of the SQW potential, respectively. These ener-\ngies are much higher than respective states in supercage,\nbecause of a narrow size of β-cage. As adjoining β-cages\nare well separated by D6Rs as shown in Fig. 2, s-electron\nwave functions in adjoining β-cages scarcely overlapwith\neach other, but a finite overlap occurs through 6Rs be-tween supercages and β-cages.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n6.3 eV \n3.1 eV \nβ-cage 1s(2) 1p(6) \n7 Å ≈7  Å \nFIG. 4. (Color online) Schematic illustrations of alkali me tal\ncluster in β-cage and the quantum states of s-electron in the\nSQW potential with the diameter of 7 ˚A.\nC. Electronic properties of Na-K alloy clusters in\nKn/NaxK12−x-LSX\nElectronic properties of Na-K alloy clusters in\nKn/NaxK12−x-LSXlargelydependon xaswellas n. The\ncontributions of Na atoms are the larger ionization en-\nergy and the smaller cation size, compared with those\nof K atoms. In K n/K12-LSX (namely x= 0), pure K\nclusters show a metallic phase at n/greaterorapproxeql6 and a ferrimag-\nnetic property at the saturation loading density n≈9 at\nambient pressure [1, 9]. Under the pressure loading of K-\nmetal into zeolite K 12-LSX, the disappearance of the fer-\nrimagnetism has occurred and an itinerant electron fer-\nromagnetism have been newly observed at n≈15 at the\nloading pressure ≈0.9 GPa [14]. In Na-K alloy clusters\nin Kn/Na4K8-LSX (namely x= 4), the N´ eel’s N-type\nferrimagnetism has been observed [1, 3, 4, 6]. Under the\npressure loading of K-metal into zeolite Na 4K8-LSX, a\nnew ferrimagnetism have been observed at the loading\npressure ≈0.5 GPa [7]. In K n/Na7.3K4.7-LSX (namely\nx= 7.3), a nearly pure ferromagnetism in an insulating\nphase has been observed at n≈9 [12]. The origin of the\nferromagnetismis assigned to the ferromagnetic superex-\nchange coupling between magnetic moments at β-cage\nclusters via sp3closed-shell clusters at supercages.\nPure Na clusters are generated by the Na metal load-\ning into zeolite Na 12-LSX (namely x= 12). Insulating\nand non-magnetic states of pure Na clusters have been\nobserved in Na n/Na12-LSX for n/lessorapproxeql11. A metallic phase\nhas been observed with the increase in n. A thermally\nactivated paramagnetic susceptibility has been observed\nsignificantly at n≈16, and is assigned to the thermal\ndistribution of metastable small polarons [1, 5, 8]. The\ntemperaturedependenceofthe paramagneticsusceptibil-\nity has been observed in the shift of23Na NMR narrow\nline [10, 13], although there are many nonequivalent Na\nsites in Na n/Na12-LSX [1, 11]. This result indicates that\nNa cations are hopping thermally over many Na sites at\nhigher temperatures during the NMR time window, and4\nnuclei of relevant Na cations feel average paramagnetic\nfield of thermally metastable small polarons.\nIn the present paper, optical, magnetic and electri-\ncal properties in K n/NaxK12−x-LSX are studied in de-\ntail mainly for x= 4. Ferrimagnetic properties are ob-\nserved at 6 .5< n <8.5 in K n/Na4K8-LSX. At the same\ntime, the Curie constant suddenly increases, and a re-\nflection band of β-cage clusters at 2.8 eV is observed at\nn >6.5. An electrical resistivity indicates metallic value\natn/greaterorapproxeql6. The electrical resistivity increases extraor-\ndinarily at very low temperatures in ferrimagnetic sam-\nples, such as ≈106times larger than the value at higher\ntemperatures. The ferrimagnetism is explained by the\nantiferromagnetic interaction between the magnetic sub-\nlattice of itinerant electron ferromagnetism at supercage\nclusters and that of localized moments at β-cageclusters.\nWe try to explain these anomalies of electrical resistivity\nby the analogy of the Kondo insulator, where itinerant\nelectron spins of supercage clusters interact with local-\nized electron spins of β-cage clusters. Itinerant electrons\nof narrow energy band of supercage clusters, however, is\nferromagnetic, differently from the Kondo insulator.\nII. EXPERIMENTAL PROCEDURES\nZeolites are crystalline powder of few microns in grain\nsize. The as-synthesized zeolite LSX was x= 9. Na\ncations were fully exchanged to K 12-LSX in KCl aqueous\nsolution. K 12-LSX was partly ion-exchanged in aque-\nous NaCl solution in order to get Na xK12−x-LSX. The\nvalue ofxwas estimated by means of inductively coupled\nplasma (ICP) spectroscopy. Zeolite Na xK12−x-LSX was\nfully dehydrated in vacuum at 500◦C for one day. Dis-\ntilled potassium metal was set into a quartz glass tube\ntogether with the dehydrated Na xK12−x-LSX in a glove-\nbox filled with a pure He gas containing less than 1 ppm\nof O2and H 2O. The potassium metal in the quartz glass\ntube was adsorbed into the Na xK12−x-LSX at 150◦C.\nThe thermal annealing was made for enough time to get\nthe homogeneous K-loading. The value of nwas esti-\nmated from the weight ratio of K-metal to Na xK12−x-\nLSX powder.\nThe optical diffuse reflectivity rwas measured at room\ntemperaturebythe useofanFTIRspectrometer(Nicolet\nMagna 550) and a double monochromator-type UV-vis-\nNIR spectrometer (Varian Cary 5G). KBr powder was\nused for the reference ofwhite powder. Since samples are\nextremely air-sensitive, optical measurements were per-\nformed on samples sealed in quartz glass tubes. The dif-\nfuse reflectivity rwas transformed to the optical absorp-\ntion spectrum by the Kubelka-Munk function (1 −r)2/2r\nwhich gives the ratio of the absorption coefficient to the\nreciprocal of powder size. The sum of the normal reflec-\ntivityRand the transmission coefficient Trwas obtained\nby the transformation R+Tr= 4r/(1+r)2[16]. The nor-\nmalreflectivityspectrumwasobtainedas R= 4r/(1+r)2\nat the spectral region for Tr≪R.A SQUID magnetometer (MPMS-XL, Quantum De-\nsign) was used for magnetic measurements in the tem-\nperature range 1.8-300 K. A diamagnetic signal from the\nquartz glass tube is included in the SQUID signal as the\ntemperature-independent background, and is subtracted\nfrom measured magnetization.\nFor an electrical resistivity measurement, powder sam-\nples were put between two gold electrodes, and an ade-\nquate compression force ≈1 MPa was applied during the\nmeasurements. Because of the extreme air-sensitivity\nof samples, they were kept in a handmade air-proof\ncell. These setting procedures were completed inside\nthe glovebox. The cell was set into Physical Property\nMeasurement System (PPMS, Quantum Design), and\nthe temperature was changed between 2 and 300 K.\nThe electrical resistivity of the cell was measured by the\nfour-terminal method with the use of Agilent E4980A\nLCR meter at the frequency range from 20 Hz to 2\nMHz and DC. The frequency dependence of the complex\nimpedance was analyzed by the Cole-Cole plot, and the\nDC or 20 Hz electrical resistivity ρwas obtained by the\nmultiplication of the dimensional factor (area/thickness)\nofcompressed powder. Due to the constrictionresistance\n[60] at connections between powder particles as well as\nthe low filling density of powder particles, the observed\nresistivity is about two orders of magnitude larger than\nthe true value. The relative values in different samples,\nhowever, can be compared with each other within an am-\nbiguity of factor, because of the constant compression\nforce. Fortunately, values in the present study change\nin the several orders of magnitude. Detailed experimen-\ntal procedures are explained elsewhere [8]. The upper\nlimit of the present resistivity measurement was ≈109\nΩcm, and obtained values for ρ/greaterorapproxeql109Ωcm are unreli-\nable. The ionic conductance of dehydrated zeolites un-\nder the low compression force is expected in the order of\n10−9Ω−1cm−1at room temperature [61], and is negligi-\nble at lower temperatures in the present study. A small\nresistivity of the short circuit in the cell ( <0.1 Ωcm) is\nincluded in the measured value, but is negligible in the\npresent study.\nThe high-field magnetization was measured by using\nan induction method with a multilayer pulse magnet at\nthe Institute for Solid State Physics, the University of\nTokyo. A non-destructive pulsed magnet for 70 T was\nused for this measurement. Sample sealed in a high-\nquality quartz glass tube with a diameter of 2 mm was\nset in the pickup coils. The observed magnetization is\nnormalized by the results obtained by the SQUID mag-\nnetometer at H <5×104Oe.\nIII. EXPERIMENTAL RESULTS\nA. Optical properties\nOpticalresonantabsorptionand reflectionspectrapro-\nvide an important information on the dipole transition of5\nelectronicstatesincludingnonmagneticones. Absorption\nspectra of dilutely K-loaded K n/NaxK12−x-LSX (n≪1)\nat room temperature (RT) are shown in Fig. 5 for x= 0,\n1.5, 4 and 7.3. Spectra in K n/K12-LSX, K n/Na1.5K10.5-\nLSX and K n/Na4K8-LSX have continuous peaks above\n≈0.6eV. These peaksare assignedto the excitation from\n1s-like states to the empty energy bands of supercage\nnetwork [3]. A new band appears at ≈2.6 eV with mark\nin Kn/Na7.3K4.7-LSX, in addition to above mentioned\ncontinuous peaks. This new band is assigned to the ex-\ncitation from 1 s-like states to 1 p-like ones of clusters in\nβ-cages [12].\nFIG. 5. (Color online) Absorption spectra of dilutely K-\nloaded K n/K12-LSX (x= 0), K n/Na1.5K10.5-LSX (x= 1.5),\nKn/Na4K8-LSX (x= 4) and K n/Na7.3K4.7-LSX (x= 7.3) at\nroom temperature, where n≪1.\nIf we assume strict SQW potentials shown in Figs. 3\nand 4, the 1 s–1pexcitation energies are expected at 0.9\nand 3.2 eV in clusters localized in supercage and β-cage,\nrespectively. Because of the lack of the inversion symme-\ntry at the center of supercage, 1 s, 1pand 1dstates hy-\nbridize partly with each other in the energy band. Con-\ntinuous DOS of the hybridized energy band of supercage\nnetwork are expected, because of the electron transfer\nthrough large 12R windows with the size ≈8˚A. In prin-\nciple, the absorptioncoefficient of the band-to-band exci-\ntation is proportional to the joint-density-of-states times\nthe transitiondipole momentsbetween groundstatesand\nexcited states. The observed gap energy of continuous\nabsorption bands, ≈0.6 eV, originates from the forma-\ntion energy of small bipolarons at supercages at low K-\nloading densities, as stated in Section IVA. Small bipo-\nlarons are optically excited to the extended states of the\nhybridized energy band of supercage network. The β-\ncagepotentialprovideswell-isolatedelectronicstates, be-cause of narrow windows. The optical excitation from 1 s\nto 1pstates is expected at 3.2 eV in Fig. 4, but the effec-\ntive potential size is expected to be slightly larger than 7\n˚A, such as 7.8 ˚A, in order to fit the observed excitation\nenergy≈2.6 eV. As discussed in Section IVB, the sur-\nrounding cations are expected to extend the confinement\npotential.\nFIG. 6. (Color online) Reflection spectra of K n/Na4K8-LSX\nat room temperature. The value of nis indicated for each\nspectrum.\nReflection spectra of K n/Na4K8-LSX (x= 4) at room\ntemperature are shown in Fig. 6. The K-loading density\nnis indicated for each spectrum. A reflection band of\nnearly metallic s-electrons of supercage clusters is seen\nbelow≈1 eV in each spectrum. The plasma edge of\nmetallic s-electrons is estimated to be ≈1 eV. With the\nincrease in n, theβ-cage cluster bands grow around ≈2.3\nand≈2.8 eV. The 2.3 eV band grows at lower values of\nn. As shown in Section IIIB, a ferrimagnetism and a\nsudden increase in the Curie constant are observed si-\nmultaneously at 6 .5< n <8.5. The 2.8 eV band of\nβ-cage clusters is assigned to the magnetic K-rich clus-\nters (small polarons) for 6 .5/lessorapproxeqln/lessorapproxeql8.5 and nonmagnetic\nK-rich clusters (small bipolarons) for 8 .5/lessorapproxeqln, as dis-\ncussed in Section IVB. The 2.3 eV band is assigned to\nnonmagnetic Na-rich clusters at β-cages.\nIn Kn/Na1.5K10.5-LSX (x= 1.5), similar reflection\nspectra are observed at room temperature, as shown in\nFig.7. Reflectionbandsof β-cageclustersareobservedat\nsimilar energies 2.2 and 2.8 eV. The 2.8 eV band appears\natn/greaterorapproxeql7.5. As shown in Section IIIB, a ferrimagnetism\nand an increase in the Curie constant are observed si-\nmultaneously at 7 .8< n/lessorapproxeql9.5. The 2.8 eV band is\nassigned to the K-rich magnetic clusters (small polarons)6\natβ-cages, as discussed in Section IVB. Reflection bands\nat 2.2, 2.3 and 2.4 eV are expected to be nonmagnetic\nNa-richβ-cage clusters with different configurations of\ncations.\nFIG. 7. (Color online) Reflection spectra of K n/Na1.5K10.5-\nLSX at room temperature. The value of nis indicated for\neach spectrum.\nB. Magnetic properties\nTemperature dependences of magnetization in\nKn/Na4K8-LSX under the low magnetic field of 10\nOe are shown in Fig. 8. The value of nis indicated\nfor each curve. The observed large magnetization\noriginates from the spontaneous magnetization, because\nof an applied magnetic field is very weak. The Curie\ntemperature increases and decreases with n. A typical\nN´ eel’s N-type ferrimagnetism with the zero minimum of\nmagnetization at the compensation temperature Tcomp\nis seen at n= 7.6, 7.8 and 7.9. A similar zero minimum\nmay be expected below 1.8 K at n= 6.7 and 7.0. A\ngradual increase in magnetization around the Curie\ntemperature is seen at n= 7.6 and 7.8 with the decrease\nin temperature, indicating that a weak inhomogeneity\nis expected to exist in the temperature of the magnetic\nphase transition. The zero minimum at Tcomp, however,\nis clearly seen.\nThe N´ eel’s N-type ferrimagnetism is explained by an\nantiferromagnetic interaction between two nonequivalent\nmagnetic sublattices A and B, one of which (A) has both\na very weak internal magnetic interaction and the satu-\nration magnetization which is larger than the magnetiza-\ntion of the other sublattice (B). The sublattice B has a\nstronger internal interaction. Below the Curie tempera-\nture, the sublattice B increases the spontaneous magne-tization. The magnetization of sublattice A follows the\nsublattice B with the opposite direction. At Tcomp, mag-\nnetizations of sublattices A and B have the same mag-\nnitude with opposite directions, and the total magneti-\nzation becomes zero. Below Tcomp, the sublattice A has\nthe magnetization larger than that of sublattices B. As\ndiscussed later in Section IVC, we introduce a model of\ntwo magnetic sublattices A and B constructed by local-\nized magnetic moments of β-cage clusters and an itin-\nerant electron ferromagnetism of supercage clusters, re-\nspectively. In Fig. 8, Tcompseems to approach the Curie\ntemperature relatively, indicating that an antiferromag-\nnetic interaction between magnetic sublattices A and B\nand/or the magnetization of sublattice A increase with n\nat the ferrimagnetic condition.\nn-dependences of the asymptotic Curie temperature\nTC, the Weiss temperature TWand the Curie constant\nin Kn/Na4K8-LSX are shown in Fig. 9. The Curie con-\nstanthasasuddenincreaseattheferrimagneticcondition\n6.5< n <8.5, as colored in blue. TWis positive and neg-\native at lower and higher values of n, respectively. The\n2.8 eV band of β-cage clusters grows at n/greaterorapproxeql6.5 in Fig. 6\nin accordance with the sudden increase in the Curie con-\nstant.\nFIG. 8. (Color online) Temperature dependences of magneti-\nzation in K n/Na4K8-LSX under the magnetic field of 10 Oe.\nThe value of nis indicated for each curve.\nThe sudden increase of the Curie constant in Fig. 9 is\nestimated to be ≈5×10−5Kemu/cm3. If we assume\nlocalized magnetic moments of β-cage clusters with spin\ns= 1/2 andg= 2, the Curie constant Cβis given by\nCβ=Nβg2µ2\nBs(s+1)\n3kB=Nβµ2\nB\nkB, (1)\nwhereNβandkBare the number density of magnetic\nclusters at β-cages and the Boltzmann constant, respec-7\nFIG. 9. (Color online) n-dependences of the asymptotic Curie\ntemperature TC, the Weiss temperature TWand the Curie\nconstant in K n/Na4K8-LSX.\ntively. The estimated value of Nβamounts to ≈15% of\nβ-cages and the saturation magnetization becomes ≈0.7\nG.\nThe background Curie constant in Fig. 9 is ndepen-\ndent, for example, ≈1.3×10−4Kemu/cm3atn≈7.5.\nThe Curie constant of an itinerant electron ferromag-\nnetism for supercage clusters, Cs, is given by\nCs=N0peff2µB2\n3kB, (2)\nwhereN0andpeffµBare the number density of su-\npercages and the effective local magnetic moment per\nsupercage, respectively. The value of peffestimated from\nthe background Curie constant is ≈1.1 which corre-\nsponds to the saturation magnetization of ≈5.3 G. In\ncase of the itinerant electron ferromagnetism, however,\nthe spontaneous magnetization at low magnetic fields is\nmuch smaller than that estimated from the Curie con-\nstant, such as ≈1/3 in the itinerant electron ferromag-\nnetism in the pressure loading of K metal into K 12-LSX\n[14]. If we assume a similar ratio, the spontaneous mag-\nnetization of supercage clusters will be ≈1.8 G at low\ntemperatures. The total magnetization will be ≈2.5 G.\nAt very high magnetic fields, the saturation of total mag-\nnetization is observed at 2.7 G as shown later in Fig. 12.\nIn order to explain the N´ eel’s N-type ferrimagnetism ob-\nserved in Fig. 8, the spontaneous magnetization of su-\npercage clusters at low temperatures will be smaller than\n≈0.7 G of the saturation magnetization at β-cage clus-\nters.\nThe temperature dependence of magnetization in\nKn/Na1.5K10.5-LSX under the magnetic field of 10 Oe\nis shown in Fig. 10. The value of nis indicated for\neach curve. The Curie temperature increases and de-\ncreases with n. The magnetization has a minimum at\nthe temperatures lower than the respective Curie tem-\nFIG. 10. (Color online) Temperature dependences of magne-\ntization in K n/Na1.5K10.5-LSX under the magnetic field of 10\nOe. The value of nis indicated for each curve.\nFIG. 11. (Color online) n-dependences of the asymptotic\nCurie temperature TC, the Weiss temperature TWand the\nCurie constant in K n/Na1.5K10.5-LSX.\nperatures, indicating that this is the N´ eel’s P-type ferri-\nmagnetism, where the magnetization of β-cage clusters\nis smaller than that of supercage clusters at any tem-\nperature. n-dependences of the asymptotic Curie tem-\nperature TC, the Weiss temperature TWand the Curie\nconstant are shown in Fig. 11. The Curie constant is\nmuch larger than that in K n/Na4K8-LSX. The Curie\nconstant has an increase at the ferrimagnetic condition\n7.8< n/lessorapproxeql9.5, as colored in blue. TWis positive and\nnegative at lower and higher values of n, respectively,\nat the ferrimagnetic condition. The 2.8 eV band of β-8\ncage clusters grows at n/greaterorapproxeql7.5 in Fig. 7. The increase\nin the Curie constant at n≈8.5 is roughly estimated to\nbe≈1×10−4Kemu/cm3which corresponds to localized\nmagnetic moments with spin 1/2 distributed at ≈30%\nofβ-cages and the saturation magnetization of ≈1.5 G.\nThebackgroundCurieconstant ≈3×10−4Kemu/cm3at\nn≈8.5correspondsto peff≈1.7. This valuecorresponds\nto the saturation magnetization of ≈8 G. As explained\nabove in K n/Na4K8-LSX, the spontaneous magnetiza-\ntion of supercage clusters will be much smaller than ≈8\nG. At very high magnetic fields, the saturation of total\nmagnetization is observed at 4.2 G, as shown later in\nFig. 12.\nFIG. 12. (Color online) The magnetization process up to\nhigh magnetic fields at 1.3 K for K n/NaxK12−x-LSX, where\nthe respective values of ( x,n) are (4, 7.7), (1.5, 8.75) and (0,\n8.9). The corresponding magnetic moment per supercage (or\nβ-cage) is indicated in the axis on the right in units of µB.\nThe magnetization process up to high magnetic fields\nat 1.3 K is shown for K n/NaxK12−x-LSX in Fig. 12,\nwhere the respective values of ( x,n) are (4, 7.7), (1.5,\n8.75) and (0, 8.9). The corresponding magnetic mo-\nment per supercage (or β-cage) is indicated in the axis\non the right in units of µB. The magnetization process in\nK7.7/Na4K8-LSX displays a weak hump around 3 .5×104\nOe, and the saturation at 2.7 G after the clear bend at\n22.7×104Oe. A hump in K 8.75/Na1.5K10.5-LSX is un-\nclear, but is expected around ≈8×104Oe. The magneti-\nzation process in K 8.9/K12-LSX displays a hump around\n16×104Oe, and the saturation at ≈6 G after the bend\nat≈32×104Oe. As discussed later in Section IVD, the\nmagnetization process of ferrimagnetism in the model of\nclassical magnetic moment has a flat magnetization up\nto the spin-flop field, and a constant slope up to the sat-\nuration field. The observed results, however, have round\nshapes at the beginning of magnetization and above thespin-flop field. This shape is explained by the increase in\nmagnetization of the itinerant electron ferromagnetism\nof the supercage clusters.\nC. Electrical properties\nAn electrical conductivity and its temperature depen-\ndence give an important information on carriersin solids,\nespecially in correlated polaron systems. The electrical\nconductivity σwith different types of carriers are given\nby\nσ=/summationdisplay\njeµjNj, (3)\nwheree,µjandNjare the elementary electric charge,\nthej-th carrier mobility, and the number density of j-th\ncarriers, respectively. There are following two limiting\nmodels in the electrical conductivity having the Arrhe-\nnius law [62]. In the band gap model with nearlytemper-\nature independent mobility, the conductivity is propor-\ntional to the number density of thermally activated free\ncarriersNj, and is expressed by the Arrhenius law. The\ngapenergyisgivenbytwotimesofthethermalactivation\nenergy. In the small polaron hopping model, the Arrhe-\nnius law can be applied to the temperature dependence\nof mobility approximately, where the thermal activation\nenergy is related to the polaron formation energy, etc. A\ndisorder and an electron correlation can have important\ncontributions to the electrical conductivity in addition to\nabove mentioned mechanisms. The electrical resistivity\nρis given by 1 /σ.\nThe temperature dependences of ρin Kn/Na4K8-LSX\nat various values of nare shown in Fig. 13. The value of\nnis indicated for each curve. The temperature of sam-\nple was decreased from 300 K. The value of ρat 300 K\ndecreases with n. With the decrease in temperature, ρ\nbasically increases, because of the decrease in the mo-\nbility of small polaron hopping. A weak anomaly is seen\naround150K. A similar anomalyand a temperature hys-\nteresis in ρhave been clearly observed around 150 K in\nKn/Na7.3K4.7-LSX [12]. In Fig. 13, ρatn= 5.1 and\n5.8 slightly increases at low temperatures, and is finite\nat the lowest temperature 2 K. This result indicates that\na finite number of free carriers (large polarons) are dis-\ntributed at low temperatures. Although ρatn= 7.1\nand 7.8 is much lower than that at n= 5.1 or 5.8 above\n≈50 K,ρatn= 7.1 and 7.8 quickly increases at very\nlow temperatures and exceeds values at n= 5.1 or 5.8.\nThe electrical conductivity σ= 1/ρin Kn/Na4K8-LSX is\nplotted for n= 7.1 and 8.2 in Fig. 14 as a function of the\nreciprocal of temperature, 1 /T. The thermal activation\nenergy depends on temperature. The activation energy\nEgis roughly estimated to be ≈1.2 and≈4 meV around\n3 and 15 K, respectively, for n= 8.2.\nThen-dependence of ρin Kn/Na4K8-LSX is plotted\nfor 2, 20 and 100 K in Fig. 15. The value of ρat 2 K\ndecreases with nup ton= 5.8, but increases extremely9\nFIG. 13. (Color online) Temperature dependences of the elec -\ntrical resistivity ρin Kn/Na4K8-LSX at various values of n,\nwhere temperatures are decreased from 300 K. The value of\nnis indicated for each curve.\nFIG. 14. (Color online) Temperature dependences of electri -\ncal conductivity 1 /ρatn= 7.1 and 8.2 in K n/Na4K8-LSX.\natn= 7.1 and 8.2 at the ferrimagnetic condition 6 .5<\nn <8.5 shown in Figs. 8 and 9. The value of ρat 2K for\nn= 8.2 is≈106times of that at 100 K. The increase is\nnot significant at n= 9.0. A similar increase in ρat low\ntemperatures has been observed in K n/K12-LSX at the\nferrimagnetic condition of n, and the value of ρat 2K for\nn= 9.0 is≈102times of that at 100 K [1, 9].\nFIG. 15. (Color online) n-dependence of electrical resistivity\nρat 2, 20 and 100 K in K n/Na4K8-LSX.\nIV. DISCUSSIONS\nA. Model of correlated polaron system\nIfs-electron wave functions of alkali metal clusters\nare well localized quantum-mechanically in zeolite cages,\nthe tight-binding approximation can be applied to them\n[30, 31]. A narrow energy band of s-electrons with a\nstrong electron correlation is expected in the supercage\nclusters in K n/K12-LSX, because of a large mutual\nCoulomb repulsion energy within supercages and the\nelectron transfer through 12R windows [14]. Further-\nmore,s-electrons have an interaction with the displace-\nment of alkali cations distributed in cages. Hence, s-\nelectrons have an electron-phonon deformation-potential\ninteraction as well as the electron correlation.\nIn order to take an overview of the electronic proper-\nties of alkali metals in zeolites, it is effective to introduce\nfollowing coarse-grainedparameters of the correlated po-\nlaron system given by the so-called Holstein-Hubbard\nHamiltonian [1, 63, 64]\nH=−/summationdisplay\ni,j,σtija†\niσajσ+U/summationdisplay\nini↑ni↓\n+/summationdisplay\ni/parenleftbiggP2\ni\n2m+1\n2mω2Q2\ni/parenrightbigg\n−λ/summationdisplay\niQi(ni↑+ni↓),(4)\nwhereaiσ(a†\niσ) is the annihilation (creation) operator\nof the electron with the spin σat thei-th site, and\nniσ=a†\niσaiσ.tijis the electron transfer energy between\nthei-th and the j-th sites. Uis the on-site Coulomb re-\npulsion energy (the Hubbard U). The localized phonons\n(Einstein phonons) with the mass mand the frequency\nωare assumed in the third term. QiandPiare the lat-\ntice distortion and the conjugated momentum at the i-th\nsite, respectively. In the last term, the on-site electron-\nphonon interaction is introduced by the assumption of10\nthe site diagonal coupling constant λ. Here, we define\nthe lattice relaxation energy Sas [63]\nS=λ2\nmω2. (5)\nIf we consider the electron transfer between the nearest\nneighbor sites for /angbracketlefti,j/angbracketrightonly, the first term of the right-\nhand side of Eq. (4) can be written as\n−t/summationdisplay\n/angbracketlefti,j/angbracketright,σa†\niσajσ, (6)\nwheret(>0) is the transfer energy of electron to the\nnearest neighbor site. The t-U-S-ncoarse-grained model\nof correlatedpolaron system is introduced to alkali-metal\nloaded zeolites, where nis the average number of elec-\ntrons provided by alkali atoms per site.\nSchematic illustration of the Holstein-Hubbard model\nis given in Fig. 16. Red arrows indicate spins of elec-\ntrons. If tis large enough, large polarons migrate as free\ncarriers. In cases of U > SandU < Sat small t, small\npolaron with the energy −S/2 and small bipolaron with\nthe energy U−2S, respectively, become stable as the\nself-trapped states. Small polarons and small bipolarons\ncontribute to the conductivity by their hopping process\nat finite temperatures.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \n−t\nU\nsmall polaron large polaron \nsmall bipolaron U − 2S− S/2 \nFIG. 16. (Color online) Schematic illustration of theHolst ein-\nHubbard model. Red arrows indicate spins of electrons. If t\nis large enough, large polarons are stabilized as free carri ers.\nIn cases of U > SandU < Sat small t, small polaron with\nthe energy −S/2 and small bipolaron with the energy U−2S,\nrespectively, are stable.\nIn zeolites, tis introduced through windows between\nadjoining cages. The energy band width 2 Bis given by\n2B= 2ht, where his the number of nearest neighbor\nsites.his 4 for supercage or β-cage in zeolite LSX. The\nenergy of the band bottom is located at −B. IfB <\nS/2, an electron relaxes into small polaron. The value\nof 2Bfor supercage network is roughly estimated to be\n≈2 eV for 1 pstates in LSX from the spectral width of\nthe supercage band in Fig. 5. tforβ-cage network is\nnegligibly small, because of the large separation by D6Rs\nas shown in Fig. 2. Therefore, clusters generated in β-\ncages relax into self-trapped states because of a finite\nS, and become small polarons with magnetic moment or\nsmall bipolarons without magnetic moment, as discussed\nin Section IVB.Twos-electrons in the same cage have a Coulomb re-\npulsion energy U. The value of Udepends on the size\nof cage, but is almost independent of the configuration\nof cations. The unscreened Ubetween two electrons in\nthe 1sstate is estimated to be ≈3 eV for supercage with\nthe inside diameter of ≈13˚A and≈6 eV forβ-cage with\nthe inside diameter of ≈7˚A [1]. A finite screening ef-\nfect reduces the value of unscreened U. A qualitative\ninterpretation has been given by the t-U-S-nmodel for\nvarious properties of alkali metals in different zeolites [1].\nAt lower loading densities, tis relatively small because\ns-electrons occupy lower quantum states of clusters, such\nas 1sstates, and the electron-phonon interaction Sdom-\ninates the system. Hence, small bipolarons are stabilized\nat lower loading densities. A gap energy ≈0.6 eV in ab-\nsorptionspectra ofdilutely K-loadedK n/NaxK12−x-LSX\ninFig.5isassignedtotheformationenergyofsmallbipo-\nlarons at 1 sstates in supercages. An effective value of t\nfor the energy band near the Fermi energy is expected to\nincrease with n, because s-electrons occupy higher quan-\ntum states of clusters, such as 1 pand 1dstates, and the\nmetallic states are realized at large ndepending on the\nkindofalkalimetals, etc. [1]. Ametallicstateisexpected\natn/greaterorapproxeql6 in K n/Na4K8-LSX as shown in Fig. 13, indi-\ncating that free carriers of large polarons are generated\nby 1pelectrons in supercage clusters. A similar metallic\ntransition has been observed in K n/K12-LSX [1, 9].\nB. Clusters at β-cages\nMagnetic moments of clusters in β-cages play a cru-\ncial role in magnetisms of K n/NaxK12−x-LSX. The value\noftbetween β-cages is negligibly small. If an electron\noccupies the 1 sempty state with the energy E1satβ-\ncage, a small polaron with the energy E1s−S/2 is gen-\nerated by the electron-phonon interaction according to\nthe Holstein-Hubbard model, as illustrated in Fig. 17. If\nthe second electron occupies the small polaron site, the\nsecond electron has the energy E1s+U−3S/2. As shown\nin Fig. 17(a), small polarons with magnetic moments are\ngenerated in β-cages at U > S, if the Fermi energy EF\nsatisfies\nE1s−S\n2< EF< E1s+U−3S\n2. (7)\nWith the increase in EFwithn, small bipolaron with the\nenergy2E1s+U−2Sin the spin-singletstate isgenerated\nby the occupation of the second electron, if EFsatisfies\nE1s+U−3S\n2< EF. (8)\nThis model means that small polarons with the mag-\nnetic moments are stabilized only at the condition given\nby Eq. (7) for EF. On the other hand, there is no choice\nforEFatU < Sin Eq. (7), and small polarons are un-\nstable at any value of EF. This is because the pairing\nof small polarons forms small bipolarons with the energy11\n2E1s+U−2Swhich is more stable than the separatepair\nof small polarons with the total energy 2 E1s−S, as illus-\ntrated in Fig. 17(b), indicating that small polarons with\nthe magnetic moments are not stabilized at any value of\nnatU < S.\n(β-cage) (β-cage)−S2U−3S2\nU−S−S2\nU−3S2U−S(U>S) (U<S)\n1sE1s E1ssmall bipolaron \nsmall polaron \nsmall polaron \nsmall bipolaron 1s(a) (b) \nFIG. 17. (Color online) Schematic illustration of electron ic\nconfigurations of clusters in β-cages at (a) U > Sand (b)\nU < S, accordingtotheHolstein-Hubbardmodel. Redarrows\nindicate spins of electrons. The value of tis negligibly small\nbetween β-cages. Small polarons and small bipolarons in β-\ncages are formed depending on the relative magnitudes of U\nandSand the Fermi energy EF. See text in detail.\nThe value of Sstrongly depends on the kind of cations\nand their arrangement such as the number and the loca-\ntions of cations. Generally, Sfor Na-rich cluster is larger\nthan that for K-rich one, because of the larger ionization\nof Na atom. The value of Sincreases with the number\nof cations which contribute to the formation of cluster.\nGenerally, cations in zeolites are located near the\naluminosilicate framework, because of the attractive\nCoulomb force between cations and negatively charged\nframework. However, cations keep the mutual distance,\nbecause of the repulsive Coulomb force among them. In\neachβ-cage of zeolite LSX, there are three cation sites,\nI, I’ and II, which are located at the center of D6R, the\njust side of D6R in β-cage and the center of 6R in su-\npercage, respectively, as illustrated in Fig. 18 [11]. There\nare 12 cation sites for β-cage (four sites of I, four sites of\nI’ and four sites of II). Because site I is shared with ad-\njoiningβ-cages, there are 10 cation sites per β-cage. By\nthe loading of guest alkali metal, the number of cation\nincrease. At the same time, the locations of cations are\nadjusted by the interaction with the s-electronsshared in\nclusters, as expressed by the electron-phonon interaction\nin the Holstein-Hubbard model.\nAccording to the structure analysis in hydrated\nNaxK12−x-LSX, Na cations occupy preferably site I [65].\nSites I and II have the full occupancy, but site I’ has a\nhalf occupancy. According to the structure simulation\nof dehydrated zeolite LSX, the simultaneous occupations\nat sites I and I’ are expected, unlikely in other zeolites\n[66, 67]. The total average number of cations is ≈10 for\noneβ-cage, and the average number becomes ≈8 perβ-\ncage because of the sharing of site I between adjoining\nβ-cages. The 8 of 12 cations are distributed around each!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nSite I Site I' \nSite I' Site II \nFIG. 18. (Color online) Schematic illustration of cation si tes\nI, I’ and II around β-cage.\nβ-cage. Other 4 cations are distributed in supercage.\nBy the loading of guest alkali metal, s-electrons are\nshared with cations, and the metallic bonding among\ncations stabilizes cation-rich clusters in β-cages. In\nNan/Na12-LSX, a full occupation of Na cations are ob-\nserved simultaneously at sites I, I’ and II for n= 9.4 and\n16.7,namely12cationsforeach β-cagecluster(10cations\nperβ-cage cluster) [11]. In the simplest model, the pos-\nsible numbers of cations for the cluster in β-cage are 10,\n11 and 12 with the increase in n, where the numbers of\ncations at site I’ are 2, 3 and 4, respectively. According\nto this model, three kinds of β-cage clusters are expected\nwith respective optical excitation energies. The optical\nexcitation energy from 1 sto 1pstates is mainly deter-\nmined by the confinement potential size of s-electrons.\nThe size is basically determined by that of β-cage, but\nthese additional cations can extend slightly the effective\nsize of the confinement potential. The origin of the dif-\nferent excitation energies of β-cage clusters around 2.5\neV in Figs. 5, 6 and 7 is assigned to the difference in the\nnumber of cations and the kind of cations.\nIn an Na-K alloy system, a stronger cohesion effect for\nNa atoms makes Na-rich clusters more stable [12]. Na\nclusters in Na 12-LSX are nonmagnetic, because of a large\nS[8]. Atn <6.5 in K n/Na4K8-LSX, Na-rich clusters are\nexpected to be stabilized at β-cages as small bipolarons\nat the condition of U < Sin Fig. 17(b). The 2.3 eV\nreflection band in Fig. 6 are assigned to such Na-rich\nsmall bipolarons. The candidate of magnetic clusters of\nsmall polarons is K-rich ones. At 6 .5< n <8.5, K-\nrich small polarons are expected to be stabilized at the\ncondition of Eq. (7) for U > Sin Fig. 17(a), and are\nobserved at 2.8 eV reflection band in Fig. 6, in addition\nto Na-rich small bipolarons at 2.3 eV. At n >8.5, K-\nrich small bipolarons are stabilized at the condition of\nEq. (8).\nAt higher K-loading densities by the pressure loading\nin Kn/Na4K8-LSX, a new ferrimagnetism has been ob-\nservedattheloadingpressureof ≈0.5GPa[7]. TheCurie\nconstant is ≈3.5×10−4Kemu/cm3which is assigned to\nthe contribution of magnetic sublattices of β-cage clus-\nters and supercageones. The spontaneousmagnetization\nis much smaller than that expected from the Curie con-12\nstant, because of the cancellation of magnetizations by\nthe antiferromagnetic interaction between two magnetic\nsublattices in ferrimagnetism. The magnetic moments of\nβ-cage clusters under the pressure loading are assigned\nto small polarons at 1 pstates.\nIn Kn/Na7.3K4.7-LSX, the increase in localized mag-\nnetic moments have been observed clearly at 8 .2< n <\n9.7 in the increase in the Curie constant, and a nearly\npure ferromagnetism has been observed at 8 .4< n <9.7\nin the insulating phase [12]. Simultaneously, a reflection\nband of β-cage clusters at 2.8 eV has been observed at\nn >8. The origin of the magnetism is assigned to the\nferromagnetic superexchange coupling between magnetic\nmoments of β-cage clusters (small polarons) through sp3\nclosed-shell clusters in supercages. In reflection spec-\ntra,β-cage clusters are observed at 2.4 eV for n/greaterorapproxeql4\n[12]. These clusters are nonmagnetic and and assigned\nto the cace of U < Sshown in Fig. 17(b), where Na-rich\nclusters are preferentially stabilized. Clusters observed\nat 2.8 eV at 8 < n/lessorapproxeql9.7 are assigned to K-rich ones\n(small polarons) with magnetic moments at β-cages for\nU > S, and they become nonmagnetic (small bipolarons)\natn/greaterorapproxeql9.7, as illustrated in Fig. 17(a).\nIn Kn/K12-LSX, pure K clusters in β-cages can be\nmagnetic (small polarons) at large n. A ferrimagnetism\nby the antiferromagnetic interaction between localized\nmoments of β-cage clustersand the itinerant electron fer-\nromagnetism of supercage clusters has been observed at\nn≈9[1,9]. Thisferrimagnetismdisappearsat n≈11by\nthe pressure loading at ≈0.3 GPa, because of the gener-\nation of nonmagnetic β-cage clusters (small bipolarons)\n[14].\nC. Supercage clusters and their interaction with\nβ-cage clusters\nIn zeolite sodalite (SOD frameworkstructure), β-cages\nare arrayed in a body centered cubic structure by the\nsharing of 6Rs with eight adjoining β-cages. An an-\ntiferromagnetism of clusters in β-cages of sodalite has\nbeen observed clearly by the antiferromagnetic interac-\ntion through 6Rs [1, 41–55]. In zeolite LSX, each β-cage\nshares 6Rs with four adjoining supercages. The antifer-\nromagnetic interaction between β-cage clusters and su-\npercageonesoccursthrough6Rs, whereoneup-spinin β-\ncagearrangesdown-spinsin fouradjoiningsupercages, as\nillustrated in Fig. 19. These four supercages with a com-\nmon adjoining β-cage are the second nearest neighbors\nwith each other. Each supercage shares 12Rs with four\nadjoining supercages, and electrons in supercage clusters\nitinerate over many supercages as large polarons. If the\nnumber densityofmagnetic β-cageclustersincreases, the\nlong range magnetic ordering of an itinerant electron fer-\nromagnetism at supercage clusters is assisted geometri-\ncally by the antiferromagnetic interaction with the mag-\nnetic moments of β-cage clusters. At the same time, the\nmagnetic moments of β-cage clusters are ordered in theferrimagnetism, although the direct interaction between\nβ-cage clusters is absent. The hybridization effect of β-\ncage clusters with itinerant electrons of supercage clus-\nters is expected to play an important role in electrical\nproperties, if many β-cages are filled with small polarons\nwith magnetic moments.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nβ-cage supercage \nsupercage -supercage \nsupercage -β-cage \nFIG. 19. (Color online) Schematic illustration of cluster n et-\nworks in zeolite LSX. Clusters at supercages have an interac -\ntion network of a diamond structure. Each clusters at β-cages\nhas an interaction with clusters at four adjoining supercag es.\nThe direct interaction between β-cage clusters is absent. See\ntext in detail.\nIn the Kondo system, localized electron spins of mag-\nnetic atoms dilutely distributed in metal have an inter-\naction with conduction electron spins, and an electri-\ncal resistivity gradually increases at very low tempera-\ntures. Because of the Coulomb repulsion between local-\nized electrons at the magnetic atom, up-spins and down-\nspins of conduction electrons near the Fermi energy con-\ntribute equivalently to the localized electronic state, and\nthe Kondo singlet state is formed at very low tempera-\ntures. In the Kondo lattice system, an array of magnetic\natoms provide a remarkable increase in resistivity at low\ntemperatures, as observed in a typical Kondo insulator\nYbB12[68,69]. Theresistivitydecreasesunderhighmag-\nnetic fields up to ≈50×104Oe in YbB 12[70].\nIn Kn/Na4K8-LSX, a remarkable increase in resistiv-\nity is observed at low temperatures in Fig. 13. This re-\nsult resembles the Kondo insulator YbB 12. A similar\nincrease has been observed in K n/K12-LSX [1, 9]. The\nactivation energy indicated in Fig. 14 is temperature de-\npendent as observed in YbB 12[68, 69]. However, there is\nan essential difference between the Kondo insulator and\nKn/NaxK12−x-LSX in magnetism. The metallic narrow\nband at supercage clusters in K n/NaxK12−x-LSX is fer-\nromagnetic at low temperatures both by the intraband\nelectron-electron interaction and by the antiferromag-\nnetic interaction with magnetic clusters at β-cages. A\nenergy gap model of the ferrimagnetism is schematically\nillustrated in Fig. 20. Electrons in magnetic clusters at\nβ-cages have an antiferromagnetic interaction with itin-\nerant electrons at supercages, and the energy gap opens\nat the Fermi energy EFatlow temperatures. Achangeof13\nresistivity, however,is not observedundermagneticfields\nup to 13 ×104Oe in K n/Na4K8-LSX within the experi-\nmental accuracy, indicating that the gap in the itinerant\nelectron ferromagnetism seems to be kept under these\nmagnetic fields. A detailed theory is needed to explain\nthese results in the future.\nantiferromagnetic \nDOS \n(β-cage)E\n(supercage)EF\nDOS E\n(supercage)EFU−STC>T TC<T\nFIG. 20. (Color online) Schematic illustration of the energ y\ngap model of electronic states at ferrimagnetism ( TC> T)\nand paramagnetism ( TC< T) in K n/NaxK12−x-LSX. Local-\nizedelectrons in β-cageshaveanantiferromagnetic interaction\nwith itinerant electrons of narrow energy band of supercage\nclusters. The gap is opened at the Fermi energy EFat the\nferrimagnetism.\nIn Figs. 9 and 11, the Weiss temperature at the fer-\nrimagnetic region is positive and negative at lower and\nhigher values of n, respectively. The Weiss temperature\nTWin the meanfield theoryoflocalizedmomentsis given\nby Eq. (A18) in Appendix A, where the intra-sublattice\nmean field coefficient of β-cage clusters, λββ, is assumed\nto be zero. The asymmetry of TWcan not be explained\nbythen-dependenceofthe Curieconstantof β-cageclus-\nters,Cβ, which is defined by Eq. (A11), because the\nnumber density of magnetic clusters in β-cages,Nβ, is\nsymmetric for ferrimagnetism according to the model il-\nlustrated in Fig. 17(a). According to Eq. (A20), a neg-\native value of the Weiss temperature is expected at the\ncondition Csλss<2Cβλsβ, where λssandλsβare the\nintra-sublattice mean field coefficient of supercage clus-\nters and the inter-sublattice mean field coefficient be-\ntween supercage clusters and β-cage ones, respectively.\nCshere is the Curie constant of supercage clusters in the\nlocalized moment model and is given by Eq. (A10). The\nmain reason of the asymmetry of TWis expected to be\nthe increase in λsβwithn. According to the model at\nU > Sillustrated in Fig. 17(a), λsβincreases with n,\nbecause the Fermi energy EFincreases with nand then\nthe hybridization between electrons of supercage clusters\nand the localized electrons at β-cage clusters increases\nwithn.D. Magnetization process of ferrimagnetism\nThe magnetization process at 1.3 K in K n/NaxK12−x-\nLSX shown in Fig. 12 displays curves rounded out. The\nmagnetization process of ferrimagnetism at T= 0 is il-\nlustrated schematically in Fig. 21. In an ordinary fer-\nrimagnetism of classical magnetic moments, a constant\nmagnetization is observed up to a spin-flop field, and\na constant increase in magnetization up to the satura-\ntion field, as indicated by black lines. In the ferrimag-\nnetism in K n/NaxK12−x-LSX, the magnetic sublattice at\nsupercages is an itinerant electron ferromagnetism, and\nthe magnetization, Ms, increases with the applied mag-\nnetic field, because of the suppression of magnetization\nby the dynamical spin fluctuation [14, 71–74]. At low\nfields, the dominant magnetization of the magnetic sub-\nlattice is oriented to the applied magnetic field. For ex-\nample, the dominant magnetization in the N´ eel’s N-type\nferrimagnetismisthemagneticsublatticeat β-cages,Mβ,\nbelow the compensation temperature. According to the\nmean field theory, the effective field from MβtoMsis op-\nposite to the external field, and the total magnetization\nis given by Mβ−Ms. With the increase in the external\nfield,Msdecreasesand the total magnetizationincreases.\nAbove the spin-flop field, the angle between MsandMβ\ndecreases and Msincreases with the external field. The\ntotal magnetization increases up to the saturation value\nMβ+Ms(max), as indicated by red curves in Fig. 21. In\nK7.7/Na4K8-LSX,Mβ−Ms≈0.3 G at low fields, and\nMβ+Ms(max)≈2.7 G in Fig. 12. If we assign Mβ≈0.7\nG in the sudden increase in the Curie constant in Fig. 9,\nMsat low fields and Ms(max) are estimated to be ≈0.4\nand≈2.0 G, respectively.\n!\"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \nM\nMβMβ\nMsMs(max)\nHspin-flop \nFIG. 21. (Color online) Schematic illustration of the magne -\ntization process of ferrimagnetism up to high magnetic field s.\nMβandMsare magnetizations of magnetic sublattices at β-\ncages and supercages, respectively. See text in detail.14\nV. SUMMARY\nWe measured electronic properties in detail for\nKn/NaxK12−x-LSX mainly for x= 4. Ferrimag-\nnetic properties are observed in K n/Na4K8-LSX and\nKn/Na1.5K10.5-LSX. At the same time, the Curie con-\nstantincreases,andareflectionbandof β-cageclustersat\n2.8eVisobservedinaccordancewiththeferrimagnetism.\nAn electrical resistivity indicates metallic value at n/greaterorapproxeql6\nin Kn/Na4K8-LSX. The ferrimagnetism is explained by\nthe antiferromagnetic interaction between the magnetic\nsublattice of localized moments at β-cage clusters and\nthat of itinerant electron ferromagnetism at supercage\nclusters. The electrical resistivity increases extraordinar-\nily at low temperatures in ferrimagnetic samples. We try\nto explain the anomaly in the electrical resistivity by the\nanalogyof the Kondo insulator, where itinerant electrons\nof supercage clusters interact with localized electrons of\nβ-cage clusters. However, itinerant electrons of the nar-\nrow energy band of supercage clusters are ferromagnetic,\ndifferently from nonmagnetic electrons of the ordinary\nenergy band in the Kondo insulator.\nACKNOWLEDGMENTS\nWe are deeply grateful to Profs. R. Arita, K. Naka-\nmura, and H. Aoki for theoretical studies and discus-\nsions. We also thank Mr. S. Tamiya (Osaka Univer-\nsity) for chemical analysis. This work was supported\nby Grant-in-Aid for Scientific Research on Priority Ar-\neas (No. JP19051009), Grant-in-Aid for Scientific Re-\nsearch (A) (No. JP24244059 and No. JP13304027) and\n(C) (No. JP26400334), Grant-in-Aid for Creative Sci-\nentific Research “New Phases of Matter in Multidisci-\nplinary Approaches” (No. JP15GS0213), Global COE\nProgram “Core Research and Engineering of Advanced\nMaterials-Interdisciplinary Education Center for Materi-\nals Science” (G10), the 21st Century COE Program “To-\nwards a new basic science: depth and synthesis” (G17),\nMEXT Japan.\nAppendix A: Ferrimagnetism\nWe calculate a ferrimagnetism by the use of the\nmean field (molecular field) theory. We assume two\nnonequivalent magnetic sublattices of localized moments\ncorresponding to supercage clusters and β-cage ones\nin Kn/NaxK12−x-LSX. The geometrical arrangement\nshown in Fig. 19 and the itinerant electron ferromag-\nnetism of supercage clusters are not considered.\nWedefinemeanfieldsforsupercageclustersand β-cage\nclusters, HmsandHmβ, respectively, as\nHms=λssMs−λsβMβ, (A1)\nHmβ=λββMβ−λsβMs, (A2)whereMsandMβare magnetizations of respective mag-\nnetic sublattices for the ferrimagnetism, and λss,λββand\nλsβtheintra-sublatticemeanfieldcoefficientofsupercage\nclusters, that of β-cage clusters and the inter-sublattice\nmean field coefficient between supercage clusters and β-\ncage ones, respectively. The minus sign of the second\nterm in the right hand side of above equations means\nan antiferromagnetic interaction between two magnetic\nsublattices.\nThe magnetizations of both sublattices under the ex-\nternal magnetic field Hat the temperature Tare given\nas\nMs=NsgsµBBJs/parenleftbigg\ngsµBJsH+Hms\nkBT/parenrightbigg\n,(A3)\nMβ=NβgβµBBJβ/parenleftbigg\ngβµBJβH+Hmβ\nkBT/parenrightbigg\n,(A4)\nwhereNsandNβare the number densities of supercage\nclustersand β-cageones,respectively, gsandgβthegval-\nues ofrespectiveclusters, and JsandJβthe total angular\nmomentum quantum numbers of respective clusters. kB\nis the Boltzmann constant. BJ(y) is the Brillouin func-\ntion, and is given for |y| ≪1 as\nBJ(y) =J+1\n3Jy. (A5)\nAt sufficiently high temperatures of paramagnetism, fol-\nlowing conditions are satisfied:\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglegsµBJsH+Hms\nkBT/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1, (A6)\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglegβµBJβH+Hmβ\nkBT/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1. (A7)\nThen, we obtain following magnetizations by using\nEqs. (A1) and (A2) as\nMs=Cs\nT(H+λssMs−λsβMβ),(A8)\nMβ=Cβ\nT(H+λββMβ−λsβMs),(A9)\nwhere the Curie constants of supercage clusters and β-\ncage ones, CsandCβ, respectively, are given as\nCs=Nsg2\nsµ2\nBJs(Js+1)\n3kB, (A10)\nCβ=Nβg2\nβµ2\nBJβ(Jβ+1)\n3kB. (A11)\nFrom Eqs. (A8) and (A9), the total magnetic suscep-\ntibilityχis given by\nχ=Ms\nH+Mβ\nH\n=T(Cs+Cβ)−CsCβ(2λsβ+λββ+λss)\nT2−T(Csλss+Cβλββ)+CsCβ/parenleftbig\nλssλββ−λsβ2/parenrightbig.(A12)15\nThe Curie temperature TCis obtained from Eq. (A12)\nby the divergence condition at the higher temperature as\nTC=Csλss+Cβλββ\n2\n+/radicalBig\n(Csλss−Cβλββ)2+4CsCβλsβ2\n2.(A13)\nAt sufficiently high temperatures, χis expected to ap-\nproache the Curie-Weiss law\nχ≈Cs+Cβ\nT−TW, (A14)\nwhereCs+CβandTWare the total Curie constant and\nthe Weiss temperature, respectively. We obtain the re-\nlation at sufficiently high temperatures from Eq. (A12)as\nCs+Cβ\nχ≈T+CsCβ(2λsβ+λββ+λss)\nCs+Cβ\n−(Csλss+Cβλββ).(A15)\nFinally, we extract TWfrom Eqs. (A14) and (A15) as\nTW=−CsCβ(2λsβ+λββ+λss)\nCs+Cβ\n+Csλss+Cβλββ(A16)\nIf we assume no intra-sublattice interaction of β-cage\nclusters as λββ= 0, we obtain TCandTWas\nTC=Csλss\n2\n1+/radicalBigg\n1+4Cβλ2\nsβ\nCsλ2ss\n,(A17)\nTW=Cs\nCs+Cβ(Csλss−2Cβλsβ).(A18)\nThe positive and negative values of TWare obtained as\nTW>0 atCsλss>2Cβλsβ, (A19)\nTW<0 atCsλss<2Cβλsβ.(A20)\n[1] T. Nakano and Y. Nozue, Adv. 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Ser. 868, 012002 (2017)."    },    {        "title": "2203.00818v1.Emergence_of_insulating_ferrimagnetism_and_perpendicular_magnetic_anisotropy_in_3d_5d_perovskite_oxide_composite_films_for_insulator_spintronic.pdf",        "content": " 1 Emergence of insulating ferr imagnetism and \nperpendicular magnetic anisotropy in 3d -5d \nperovskite oxide composite films for insulator \nspintronics  \n \nZeliang Ren1,2,3†, Bin Lao2,3†, Xuan Zheng2,3,4, Lei Liao5, Zengxing Lu2,3, Sheng Li2,3, \nYongjie Yang2,3, Bingshan Cao1,2,3, Lijie Wen2,3,6, Kenan Zhao2,3, Lifen Wang5, Xuedong  \nBai5, Xianfeng Hao6, Zhaoliang Liao7*, Zhiming Wang2,3,8* and Run -Wei Li2,3,8 \n \n1Nano Science and Technology Institute, University of Science and Technology of \nChina, Hefei 230026, Anhui, China  \n2CAS Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials \nTechnology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China  \n3Zhejiang Province Key Laboratory of Magnetic Materials and Application Technolo gy, \nNingbo Institute of Materials Technology and Engineering, Chinese Academy of \nSciences, Ningbo 315201, China  \n4New Materials Institute, University of Nottingham Ningbo China, Ningbo 315100, \nChina  \n5Beijing National Laboratory for Condensed Matter Physics,  Institute of Physics, \nChinese Academy of Sciences, Beijing  100190, China  \n6Key Laboratory of Applied Chemistry, College of Environmental and Chemical \nEngineering, Yanshan University, Qinhuangdao 066004, China  \n7National Synchrotron Radiation Laboratory, Un iversity of Science and Technology of \nChina, Hefei 230026, Anhui, China  \n8Center of Materials Science and Optoelectronics Engineering, University of Chinese \nAcademy of Sciences, Beijing 100049, China  \n   2  \nAbstract : Magnetic insulators with strong perpendicular magnetic anisotropy (PMA) \nplay a key role in exploring pure spin current phenomena and developing ultralow -\ndissipation spintronic devices , thereby it is highly desirable to develop new material \nplatforms. Her e we report epitaxial growth of La 2/3Sr1/3MnO 3 (LSMO) -SrIrO 3 (SIO) \ncomposite oxide films (LSMIO) with different crystalline orientations fabricated by \nsequential two -target ablation process using pulsed laser deposition. The LSMIO films \nexhibit high crysta lline quality with homogeneous mixture of LSMO and SIO at atomic \nlevel. Ferr imagnetic and insulating transport characteristics are observed, with the \ntemperature -dependent electric resistivity well fitted by Mott variable -range -hopping \nmodel. Moreover, the  LSMIO films show strong PMA. Through further constructing \nall perovskite oxide heterostructures of the ferr imagnetic insulator LSMIO and a strong \nspin-orbital coupled SIO layer, pronounced spin Hall magnetoresistance (SMR) and \nspin Hall -like anomalous Hall effect (SH -AHE) were observed. These results illustrate \nthe potential application of the ferr imagnetic insulator LSMIO in developing all -oxide \nultralow -dissipation spintronic  devices . \n \nKeywords: Perovskite oxide, magnetic insulator, perpendicular magneti c anisotropy, \nspin hall magnetoresistance,  spintronics  \n   3 INTRODUCTION  \nIn order to satisfy the ever -increasing demanding of information storage capacity and \nprocessing speed, reducing the energy consumption of electronic devices becomes \nmore and more important in the microelectronics industry. In contrast to conventional \nmicroelectronics relying only the charge of electrons, spintronics use the additional spin \nof electrons to encode, st ore, process and transmit data, which is very promising to \nbring new capabilities to microelectronic devices1-3. Esp ecially, spintronics based on \nferromag netic/ferrimagnetic magnetic insulators  (FMI)  has attracted tremendous \nattention for developing ultralow -dissipation devices. The local magnetic moments in \nFMI act as ideal media for pure spin angular momentum  propagat ing while the thermal \nconsumptions such as Joule heating can be efficiently avoided by suppressing the \nmoving  of electrons4-6. Additionally, magnets with perpendicular magnetic anisotropy \n(PMA), where the magnetic easy axis preferentially points toward normal to the surface, \nis essentially required to reduce the devices size while maintaining high thermal \nstability7-11. Combination of FMI and PMA further provides exciting functionalities for \nlowering current threshold and enhancing mobilit y of domain wall displacement, as \nwell as achieving long decay length during spin -wave propagation12. Therefore, the \ndevelopment o f FMI with PMA is much meaningful for the development of high -\nperformance spintronic devices.  \n \nMagnetic insulators  are frequently found in transition metal oxides with various crystal \nstructures, such as rocksalt, perovskite, spinel, and garnet. Among them , perovskite \nABO 3 oxides are particularly promising candidates to achieve high -quality all -oxides \nspintronic devices because of their versatile functionalities, including spin source \ngenerator, spin current detector and so on13, 14. However, magnetic insulators  in \nperovskites often favor antiferromagneti sm instead of ferromagnetic ordering, due to \nthe super -exchange interaction between neighboring magnetic ions according to \nGoodenough -Kanamori -Anderson rules15. Moreover, FMI with strong PMA in \nperovskite oxides are rarely reported. Manganite La 1-xSrxMnO 3 (LSMO), as an \narchetypal colossal magnetoresistance material s, have been widely studied on account \nof its rich phase diagrams where multiple electric and magnetic properties are present16, \n17. Previous studies had demonstrated that PMA can be obtained in LSMO films and \nheterostructures through engineering the lattice and orbital degrees of freedom18-20. \nRecently, strong PMA can be triggered in LSMO when interfacing with strong spin -\norbital coupled SrIrO 3 (SIO)  21, 22. However, the LSMO maintained metallic  in most \ncases, unless large  strain was imposed 23-25. Through A - or/and B -site cation \nsubstitutions in ABO 3 the spin polarization of band structure can be directly controlled ,  4 leading to modif ied exchange interactions and associated magnetic and electrical \nground st ates26. Thanks to the advances in epitaxial synthesis techniques, composite \nfilms have emerged as powerful platforms for realizing FMI and strong PMA in LSMO -\nbased perovskite oxides since deliberating control of specific atoms can be realized27. \nGiven  these reasons, emergent FMI with strong PMA can be obtaine d via engineering \ncation substitution in LSMO and SIO, which enables to explore pure spin current \nphenomena in all -oxide heterostructures.  \n \nIn this letter, we reported a facile method to fabricate LSMO -SIO (LSMIO) composite \noxide films that are mixed homog enously at atomic level. The LSMIO films are FMI \nwith PMA, whose temperature -dependent resistivity was fitted well by Mott variable -\nrange -hopping model. By further constructing heterostructures of FMI LSMIO and \nstrong spin -orbital coupled SIO layers, we ex plore d pure spin current phenomena and \nobserved pronounced spin Hall magnetoresistance (SMR) and spin Hall -like anomalous \nHall effect (SH -AHE) . These results demonstrate that FMI LSMIO is promising for \nexploring insulator spintronics in all oxides heterost ructures and developing low -\ndissipation spintronic devices.  \n \nRESULTS AND DISCUSSION  \nFigure 1(a) shows the schematics of sequential two -target deposition process of LSMIO \nfilms using pulsed laser deposition (PLD) . During the growth, the LSMO and SIO \ntargets  were rotated repeatedly and periodically. In each repeat, sub -monolayer 0.3 unit \ncell (u.c.) LSMO and 0.2 u.c. SIO were deposited, ensuring homogeneous mixture of \ntwo ceramic targets at the atomic scale. The LSMIO films have a nominal stoichiometry \nof La 0.4Sr0.6Mn 0.6Ir0.4O3. The crystalline orientation of LSMIO films can be controlled \nby changing substrate orientation. The crystalline structures were characterized by X-\nRay diffraction ( XRD) 2θ-θ scans as shown in Fig. 1(b). The peaks of LSMIO films are \nlabeled and are locate d on the left side of SrTiO 3 (STO) substrates  closely . This result \nindicates that the films are slightly compressed  (<0.1%)  and of high -quality single \ncrystal without impurity phases. To further cha racterize the interface quality and \nstructural homogeneity, scanning transmission electron microscopy (STEM ) \nmeasurements on the (001) -oriented LSMIO films, as shown in Figs. 1(c) -1(h), were \nperformed. The sharp contrast across the LSMIO and STO substrates  in Fig. 1(c), \nsupplemented by a high -angle annular dark -field (HAADF) STEM image of the \nLSMIO films in Fig. 1(d), indicates high crystalline quality of the films. Figures 1(e) -\n(h) show the corresponding energy dispersive X -ray spectroscopy ( EDS) mappings of \nLa, Sr, Mn and Ir elements. All measured elements are distributed homogeneously at \natomic level in the composite films without observable clustering regions.   5  \nTo investigate the magnetic properties of LSMIO composite films, we measured \nmagnetic hystere sis ( M-H) loops and temperature -dependent magnetization ( M-T) \ncurves. Figures 2(a) -2(c) display the M-H loops observed with the magnetic field \napplied parallel ( H⊥c) and perpendicular ( H∥c) to the film plane for differently \noriented LSMIO composite films. The coercive field is found to be 0.67 T, which is \nmuch larger than that of soft ferromagnetic LSMO films. On the other hand, we note \nthat the saturation moments are determined to be around  2.4 μ B/u.c., which are \nconsiderably smaller than that in LSMO films28. The reduction of the saturation \nmoments  can be explained by the antiferromagnetic coupling in the Mn -O-Ir bonds as \nevidenced from the first -principle calculations shown below, which leads to the \nferrimagnetic ground state in LSMIO composite films.  Figure 2(d) show the \ncorresponding magnetization M as a function of temperature. T he Curie temperature \n(TC) becomes lower for all the LSMIO films  as compared to pure LSMO bulk material. \nIntriguingly, by comparing the magnetic hysteresis loops measured und er field parallel \nand perpendicular to films respectively, it is found that all easy magnetization axes are \nperpendicular to the film plane for all LSMIO films regardless of the crystal orientation \nwhile the magnetic anisotropy energy are 1.21 ×106, 1.55 ×106 and 1.56 ×106 erg/cm3 \nrespectively , which are comparable with the PMA reported in LSMO/SIO \nheterostructures22. Therefore, it is feasible to obtain perovskite FMI with strong PMA \nin LSMIO composite films.  Considering the negligible compressive strain imposed by \nthe SrTiO 3 substrates, the observed strong PMA  is not due to magneto -elastic \nanisotropy as been observed previously in LSMO films or heterostructures when under \nlarge compressive strain24, 25. The strong PMA is due to the magnetic crystalline \nanisotropy originated from strong spin -orbital coupling and Mn -O-Ir bonding21, 22, 29 \nTo further understand the electrical properties of LSMIO composite films, we have \nperformed the temperature -dependent electric resistivity measurements. A s shown in \nFig. 2(e), the resistivity increases monotonically as the temperature decreases, \nindicating that all the LSMIO composite films are insulators . This is in sharp contrast \nto individual LSMO and SIO films which are metallic and semi -metallic in the ir bulk \ncounterparts, respectively. To illustrate the origin of the emergent FMI state, we have \nfitted the resistivity data with thermal activation, Efros –Shklovskii variable -range -\nhopping (ES -VRH) and Mott variable -range -hopping (Mott -VRH) models, as show n \nin Figs. S3 (b) -(c) respectively. Figure 2(f) depicts the linear fitting curves using Mott -\nVRH model, which agrees well to the measured resistivity. Here, the Mott -VRH model \nis expressed as the following equation:   6 𝜌(𝑇)=𝜌(0)𝑒𝑥𝑝(𝑇𝑀𝑇⁄)(1𝑑+1⁄)\n \nWhere 𝜌0 is resistance coefficient, 𝑇𝑀 is characteristic temperature and the exponent \n𝑑 depends on the VRH mechanism. For the Mott VRH conduction, the exponent d is \ndimension dependent and has a value of 𝑑=3 in a three -dimensional system30-32. \nFrom the fitted characteristic temperature 𝑇𝑀, we can infer the Mott  hopping energy \n(𝐸𝑀) as: \n𝐸𝑀=1\n4𝑘𝐵𝑇(𝑇𝑀𝑇⁄)14⁄\n \nThe fitted hopping energy for (001) -, (110) - and (111) -oriented LSMIO composite \nfilms are determined to be 72.2, 93.7 and 117.2 meV at room temperature (300 K), \nrespectively.  The Mott -VRH model describes tha t the localized electron at the Fermi \nlevel moves to another localized state in an optimum hoping distance which is \ndetermined by the tradeoff between the lowest energy differences and the shortest \nhopping distances in a disordered system33. Given that the Mn -O-Ir bonds are \ndominated in  the LSMIO composite oxide films, the Mott -VRH behavior suggests that \nthe carriers hopping through the Mn -O-Ir bonds are suppressed and form localized \nstates. This situation is in sharp contrast with metallic Mn -O-Mn bonds in LSMO films, \nwhere the electron s can itinerantly hop between neighboring Mn3+ and Mn4+ ions \nthrough the double -exchange mechanism17, 34. \n \nTo gain fur ther insight into the electronic and magnetic properties of LSMIO \nnanocomposite films, we have performed first -principles calculations based on the \ndouble perovskite model LaSrMnIrO 6 (Fig. 3(a)). From the total energy results of the \ndistinct magnetic solutions ( schematically shown in Fig . 4S and Tab. S2 in \nSupplementary Materials ), we found that t he ferrimagnetic, uncompensated \nantiferromagnetic alignment, due to the unequal magnetic moments on Mn and Ir sites, \nis the most stable state for LSMIO, affir ming  the experimental observations.  Moreover, \nthe GGA+U schemes prefer quite stable magnetic solution with magnetic moments of \nMn (~ 4.3 µB) and Ir (~ -0.9 µB) ions, accompanied by considerable induced magnetic \nmoments ( ~ -0.1 µB) for all the surrounding O2- anions as demonstrated  in Fig . 3(b).  \nUnexpected, the magnetic moment at Mn sites is slightly larger than the ideal spin -only \n4 μB for Mn3+ cations, suggesting the special magnetic coupling mechanism behind.  \nFigure 3(c) shows that t he ferrimagnetic ground state is insulating with a band gap of \n0.2 eV, which agrees well with the experimental temperature -dependent electric  7 resistivity data. The antiferromagnetic coupling and unexpected large magnetic \nmoment on Mn3+ site can be interpre ted in the framework of superexchange interactions \nthrough the virtual hopping bridged by O 2 p spin-up electrons, as illustrated  in Fig. 3(d), \nwhich is derived from the partial density of states and spin density plots. We can see \nthat the virtual hopping f rom Ir 𝑡2𝑔↑to the empty Mn 𝑒𝑔↑via the bridged O  2𝑝↑states \naccounts for the antiferromagnetic coupling and the abnormal large computed Mn3+ \nmagnetic moment as well as the exclusively negative sign of the induced O2- magnetic \nmoments.  \n \nFinally, t o explore the application of FMI LSMIO in pure spin current phenomenon, \nwe fabricated a LSMIO/SIO heterostructure and performed temperature -dependent \nmagnetoresistance measurements. Figures 4(a)-4(b) show schematic structure of Hall \nbar devices and measure ment geometry. As illustrated in Fig. 4(a), the heterostructure \nconsists of the FMI LSMIO and a strong spin -orbital coupled SIO layer. The SIO layer \ncan efficiently convert the charge current into spin current35, 36, so a pronounced SMR \ncan be expected due to the asymmetry between absorption and reflection of the spin \ncurrent at the LSMIO/SIO interface. Figure 4(c) depi cts the angle dependent \nmagnetoresistance measurements where external field rotates in the y -z plane around \nangle β. It is necessary to distinguish the difference between SMR and anisotropic \nmagnetoresistance (AMR). SMR signal is known to be described as:  \n𝜌𝑥𝑥=𝜌0−∆ρ𝑚𝑦2 \nwhile AMR signal is described as:  \n𝜌𝑥𝑥=𝜌0−∆ρ𝑚𝑥2 \nIn these two formulas, the 𝜌0 is constant resistivity offset, ∆ρ is amplitude of the \nresistivity change as a function of the magnetization orientation and m i (i = x, y) is the \nproject ions of the magnetization orientation unit vector along x and y axis in the \ncoordinate system37-39. Note that 𝜌𝑥𝑥=𝑥𝑥𝑤𝑡/𝐽𝑞𝑙 and 𝜌𝑥𝑦=𝑥𝑦𝑡/𝐽𝑞, where w is \nchannel width ( w = 10 um), the l is the length of the separated voltage contacts ( l = 50 \num) and the t is the thickness of SIO layer ( t = 5 nm). During the measurement, the \nexternal field was fixed to 4 T which is much larger than the coercive  field of LSMIO \n(0.67 T) in order to obtain collinear magnetization. Figure 4(c) shows that the resistivity \ndata exhibit the cos2β dependence. This is in consistent with the SMR signal since my \nhas cosβ dependence, while AMR keep constant in angle β depend ent measurements  8 since mx equals to 0. These measurements demonstrate that the SMR can be detected in \nall perovskite oxide heterostructures.  \nFigure 4(d) shows the Hall resistance measurements with the magnetic field applied \nperpendicular to the sample and varied from -4 T to 4 T. By subtracting a linear \ncontribution from ordinary Hall effect, a clear hysteresis loop curve is visible, which \nresembles the AHE. Figure 4(e) shows the temperature -dependent Hall resistance after \nsubtracting a linear term. The coe rcive field inferred from the Hall hysteresis loop is \nabout 0.67 T at 10 K, which is similar as that of LSMIO films measured by SQUID \nshown in Fig. 2(a). Moreover, the coercive field decreases as a function of temperature \nand becomes invisible at above 120  K. These results demonstrate that the evolution of \nAHE with temperature resembles that of the magnetization versus temperature curve as \nshown in Fig. 2(d), indicating that the observed AHE is closely related to the PMA of \nLSMIO composite films. We note th at interfacial magnetism and AHE has been \nintensively studied in LSMO/SIO superlattices and heterostructures21, 40 -46. Our \nobserved AHE is quite different from that the AHE reported in ref.  41, where a large \nAHE is originated from SIO layers due to magnetic proximity effect. Our AHE stem s \nfrom the reflection of the spin current at the interface where an out -of-plane component \n(mz) of the LSMIO magnetization rotates the spin orientation of th e spin current and \ngenerates a transverse voltage via the inverse spin Hall effect (ISHE). Such AHE is \ntermed as SH -AHE47, 48. Therefore, both SMR and SH -AHE indicate occurrence of the \nspin transfer at the interface of the conduct or and ferrimagnetic insulator. The spin \ncurrent can be injected into insulators which gives a possibility of transmitting spin \ninformation through an insulator. Such behavior would have an advantage of ultralow \npower dissipation due to absence of Joule he ating.  \n \nCONCLUSION  \nIn conclusion, we have synthesized LSMIO composite oxide films with different \ncrystalline orientations by sequential two -target deposition process. The LSMIO \ncomposite films exhibit high crystalline quality with homogeneous mixture of L SMO \nand SIO at atomic level. The electric and magnetic measurements, complemented by \nfirst-principle calculations, indicate that the LSMIO composite films are FMI with \nstrong PMA . The temperature -dependent electric resistivity is well described by Mott -\nVRH model. Furthermore, t hrough fabricating heterostructures of the FMI LSMIO and \na strong spin -orbital coupled SIO layer, we observed pronounced SMR and SH -AHE. \nThese results illustra te the potential application of LSMIO in developing all -oxide \nultralow -dissipation spintronic  devices . Given the scarcity of perovskite FMI with \nstrong PMA, our result provides a promising real material candidate to boost the \ndevelopment of insulating spin tronics devices.   9  \nEXPRIMENT AL SECTION  \nSample  preparation : LSMIO composite films were synthesized by PLD, equipped with \nKrF (λ=248 nm) excimer laser and high reflection high energy electron diffraction \n(RHEED). Prior to the film deposition, (001) -, (110) - and (111) -oriented STO \nsubstrates were etched by NH 4F-buffered hydrofluoric acid followed by annealing at \n1100 ℃ for 90 minutes. During films deposition, the temperature of substrates, oxygen \npartial pressure and laser fluence were set to 700 ℃, 0.1 mbar an d 1.5 J/cm2 at the \nfrequency of 2 Hz, respectively. The film growth was monitored by RHEED and the \nthickness was determined to be 12 nm.  \nSample c haracterizations:  The quality and structure of the films was characterized by \nXRD (Bruker), STEM and EDS (JEOL  Grand ARM 300), respectively. The magnetic \nand electric properties were measured by Quantum Design Magnetic Property \nMeasurement System (MPMS -SQUID, Quantum Design) and Physical Property \nMeasurement System (PPMS, Quantum Design). Further investigations of  spin-related \nSMR and AHE were performed on Hall bar devices composed of LSMIO and SIO using \na home -build low -temperature and high -intensity magnetic field magneto -electric \nsystem. The LSMIO/SIO heterostructures were prepared by PLD as the same condition \nwith LSMIO films. The as grown (001) -oriented LSMIO/SIO heterostructure was \npatterned into Hall bars by photolithography and Ar ion etching.  \nFirst -principle calculations: In order to describe the structure of LSMIO composite \nfilms, we constructed the canoni cal perovskite -type model within 2× 2× 2 supercell \n(La 4Sr4Mn 4Ir4O24, 40 atoms).  The spin-polarized density functional theory (DFT) \ncalculations are achieved with the plane wave basis set as implemented in the Vienna \nab initio  simulation package49, 50. For the exchange correlation functional, the \ngeneralized gradient approximation (GGA) is used according to the Perdew -Burke -\nErnzerhof scheme51, and GGA+U approaches52, 53 are also performed for including the \nstatic electron correlation, with the Hubbard U = 4 eV (2 eV) and Hund J = 0.8 eV (0.4 \neV) for Mn  3d (Ir 5d) states. A  plane -wave energy cutoff of 400 eV was employed . The \nk-point meshes over the total Brillouin zone were sampled by 4×4×4 and 6 ×6×6 grids \nconstructed according to the Monkhorst -Pack scheme54, 55  for structural relaxations and \nelectronic calculations, respectively . We optimized the lattice vectors, volumes, as well \nas internal atomic with the relaxation terminated once the total energies and residual \natomic forces converged to < 1× 10-5 eV and < |0.01| eV/Å, respectively.  \n \n◼ AUTHOR INFORMATION  \nCorresponding Author   10 zliao@ustc.edu.cn . \nzhiming.wang@nimte.ac.cn . \nAuthor contribution  \n†Z.R. and B.L. contributed equally to this work . Z.W. designed the experiments. Z.R. \nperformed sample  growth. Structure characterization, magnetic and transport \nmeasurements were performed by Z.R., B.L., X.Z., Z.L., S.L., B.C., L.W., and K.Z. \nL.L.,  L.W.  and X.B.  performed STEM measurements. Z.R. and B.L. performed data \nanalysis. X.H. performed first -principle calculations and data analysis. Z.R., B.L., X.H., \nZ.L. and Z.W. wrote the manuscript. All authors participated in discussions and \napproved the submitted manuscript.  \n \n◼ ACKNOWLEDGMENTS  \nThis work was supported by the National Key  Research and Development Program of \nChina (Nos. 2017YFA0303600, 2019YFA0307800), the National Natural Science \nFoundation of China (Nos. 12174406, U1832102, 11874367, 51931011, 51902322), \nthe Key Research Program of Frontier Sciences, Chinese Academy of Sc iences (No. \nZDBS -LY-SLH008), the Thousand Young Talents Program of China, K.C.Wong \nEducation Foundation  (GJTD -2020 -11), the 3315 Program of Ningbo, the Natural \nScience Foundation of Zhejiang province of China (No. LR20A040001), the Beijing \nNational Laborat ory for Condensed Matter Physics.  \n \n◼ REFERENCES  \n1. Bader, S. D.; Parkin, S. S. P., Spintronics. Annu. Rev. Condens. Matter Phys. 2010,  1, 71. \n2. Shiraishi, M.; Ikoma, T., Molecular spintronics. Physica E Low Dimens. Syst. Nanostruct. 2011,  43, \n1295.  \n3. 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B., Yue Wang, Generalized gradient approximation for the exchange -correlation \nhole of a many -electron system. Phys. Rev. B 1996,  54. \n55. Elsasser, C.;  Fahnle, M.;  Chan, C. T.; Ho, K. M., Density -functional energies and forces with \nGaussian -broadened fractional occupations. Phys.  Rev.,  B Condens.  Matter 1994,  49, 13975.  \n \n   15  \nFigure 1. (a) Schematic illustration of sequential two -target deposition process of LSMIO films. (b) XRD \n2θ-θ scans of LSMIO films with (001), (110) and (111) crystall ine orientations. (c) HAADF -STEM \nimages of the (001) -oriented LSMIO/STO. The high -magnification image in (b) correspond to the single \ncrystalline LSMIO films. (e) -(h) The corresponding EDS mappings of La, Mn, Ir and Sr elements, \nrespectively, confirming ca tion distribution homogeneously at atomic level.  \n \n \n \nFigure 2. (a) -(c) Magnetic hysteresis loops measured with the magnetic field H applied along in -plane \n(black) and out -of-plane (red) for LSMIO composite films with different crystalline orientations. (d) -(e) \nThe corresponding magnetization M and electric r esistivity  as a function of temperature T for different \noriented LSMIO composite films. (f) The resistivity data are plotted with lnR~1/T^[1/(d+1)] (d = 3) to \nfit with the three -dimensional Mott -VRH model.  \n \n \n 16  \nFigure 3. (a) Double perovskite structure of L aSrMnIrO 6, in which the Mn atoms (medium purple spheres) \nand the Ir atoms (medium brown ones) are surrounded by O 6 octahedron denoted by small red spheres, \nwhile the light (dark) green spheres are Sr (La) atoms. (b) Isosurfaces of the spin density (0.02 e/ Å2) for \ndouble double perovskite LaSrMnIrO 6, within the ferrimagnetic magnetic ground state. Yellow and cyan \ncolor denote spin up and spin down, respectively. (c) Total and partial density of states of Mn -3d and Ir -\n5d orbitals for LaSrMnIrO 6. Majority and minority spin are presented in the top and bottom channels, \nrespectively. (d) Schematic representation of the superexchange interaction through the Mn -O-Ir bonds. \nThe virtual electron hopping results into the antiferromagnetic coupling between Mn -O-Ir bond s. \n \nFigure 4. (a) Illustration of the SMR in LSMIO/SIO heterostructure. (b) Schematic sample structure and \nmeasurement geometry. (c) The angle dependent SMR signal. The inset shows the measurement \ngeometry with external field rotating in the y -z plane around angle β . (d) Transverse Hall resistivity ρxy \nmeasured with the magnetic field applied perpendicular to the film plane; (e) Temperature dependent \nAHE sign  \n  \n 17 Suppl emental Informat ion \n1. Structural characterization of LSMIO and LSMIO/SIO heterostructure.  \n \n \n \nFigure S1.  XRD scans of LSMIO and LSMIO/SIO heterostructure.  \n \nAs shown in Fig. S1, the peak of SIO and LSMIO can be seen clearly in LSMIO/SIO \nheterostructure while there is only one peak of LMSIO in single LSMIO layer which \nmeans there is only one single crystal of LSMO -SIO composite films.  \n \n2. Structural characterization of LSMIO composite films by STEM  \n \n 18  \nFigure S2 (a) High -resolution HAADF -STEM image of the LSMIO films (b -f) EDS \nmaps of La, Mn, Sr,  O and Ir for LSMIO films, Scale bar:2nm.  \n \nWe can see the h omogen eously mixture of atoms in LSMIO films in Figs S2(b) -(f). \n \n \n3. The fittings  of measured resistivity of LSMIO.  \n \n \n \n 19  \nFigure S 3. (a) Temperature -dependent electric resistivity of LSMIO composite films with different \ncrystalline orientations. (b) -(d) The resistivity data are plotted with lnR~1/T ^[1/(d+1) ] (d = 0,1,3) \nto fitted with the thermal activation, Efros –Shklovskii -VRH and Mott -VRH model respectively.   \n \nIn this picture, we plot the resistivity with different physical model with thermal \nactivation (d=0), Efros–Shklovskii -VRH  (d=1) and 3 -dimentional Mott -VRH (d=3). \nHowever, the nonlinearity of the curves in the Fig . S3(b)-(c) indicate that the thermal \nactivation model cannot well fit the resistivities of LSMIO while only the Mott -VRH \nfits well to the measured resistivity which can be seen in Fig . S3(d).  \n \n \n4. DFT calculation s. \n \n \nIn order to describe the structure of LSMIO composit e films, we constructe d the \ncanonical perovskite -type model within 2× 2× 2  supercell (La 4Sr4Mn 4Ir4O24, 40 atoms), \nwith aid of the experimental characterizations, which can be pictured as a framework \nconsisting of alternative corners sharing MnO 6 and IrO 6 octahedra separated by \nLa3+/Sr2+ ions, due to the quite different charge states and ionic sizes of 3 d and 5 d \ncations. Concerning the La3+/Sr2+ ions distributions, the columnar La3+/Sr2+ cation order \nis taken (Fig. 3(a)), as it is prevailing among the nine pos sible double double AA ′BB′O6 \nperovskite1. Owning to the rotation and tilting of MnO 6 and IrO 6 octahedra, the Mn -O-\nIr bond angles significantly deviate from 180 º. \n \n \n \nFigure S4.  Different A -cations configurations of LSMIO composite considered in this \nstudy. (left) columnar type (middle) rocksalt type (right) layer type.  \n 20  \n \nTable S1. Relative energies (meV/ f.u.) obtained by GGA scheme within the magnetic \nground state, i.e. ferrimag netic ordering. The results suggest that the columnar type A -\ncations pattern is preferable in LSMIO composite films.  \n \n Columnar  Rocksalt  Layer  \nEnergy (meV)  0.0 15.7 86.7 \n \n \nFigure S5.  Different magnetic configurations of LSMIO composite considered in this \nstudy. Only Mn atoms (purple sphere s) and the Ir atoms (brown ones) are shown for \nsimplify. Green (yellow) arrows indicate spin up (down) character.  \n \n \nTable S 2. The calculated total energies (eV/ 2× 2× 2  supercell) obtained by GGA and \nGGA+U (UMn = 4.0 eV , UIr = 2.0 eV ) methods. The FIM (ferrim agnetic) is the ground \nstate.  \n \n FM FIM AFM1  AFM2  \nEnergy (eV)  GGA  -302.0302  -302.0391  -301.9835  -301.9835  \nGGA+U  -286.4180  -286.9023  -286.2655  -286.5611  \n \n \nAFM1 \n AFM2 \n FIM  \n FM  21 We performed electronic calculations on four distinct magnetic alignments: \nferromagnetic (FM), ferrimagnetic (FIM) (i.e., with Mn ions being antiferromagnetic \n(AFM) coupled to six neighboring Ir ions and vice versa) and two distinct AFM ones,  \nwithin the normal GGA/GGA+ U (UMn = 4.0 eV , U Ir = 2.0 eV ) prescriptions with the \ngoal of identifying the magnetic ground state. From the total energy results of the \ndistinct magnetic solutions, we found that t he ferrimagnetic, uncompensated \nantiferromagnetic alignment, due to the un equal magnetic moments on Mn and Ir sites, \nis the most stable state for LSMIO , irrespective of whether the Hubbard U values was \ntaken into account or not, affirming  the experimental magnetic observations.  \n The conspicuous feature of electronic structure is  that most of majority channel \nof Mn 3 d orbitals are occupied, forming a broad valence band ranged from -7.0 eV to \nthe Fermi level, while the corresponding minority channel is entirely empty centered at \n4.5 eV above the Fermi level, revealing that electronic configuration is close to the\n13\ng2etg\nstate and the oxidation state is 3+. Unexpected, the magnetic moment at Mn sites \nis slightly larger than the ideal spin -only 4 μB for Mn3+ cations, suggesting the special \nmagnetic coupling m echanism behind.  For the Ir states, on one hand, the occupied \nminority Os t2g states are completely separated from the empty eg states by an energy \ninterval of ~ 4 eV, accompanied by a moderate spin splitting of ~ 2 eV. On the other \nhand, the spin up t2g states are partially filled, splitting into a small band gap under the \nHubbard U correlation effect. As a consequence, electronic configuration is close to the \nnominal\n13\ng2t state and the oxidation state is 5+. In addition, there are some  Ir eg obitals \npresent around -6 eV below the Fermi level drowned in the O 2p manifold, unfolding \nthe strong covalency between the Ir 5d and O 2p states. The covalent bonding is so \nstrong that the computed Ir magnetic moment (~ 0.9 µB) is remarkable smaller than the \nideal spin -only value (~2 µB). \nThe antiferromagnetic  coupling and unexpected large magnetic moment on Mn3+ \nsite can be interpreted in the framework of superexchange interactions through the \nvirtual hopping bridged by O 2 p spin-up electrons . We can see that in contrast to the \nlarge energy separation of more than 5 eV between the down -spin Mn3+ and Ir5+ t2g \nstates, the up -spin Mn3+ eg and Ir5+ t2g states are separated by on the small band gap of \n0.2 eV, and their hybridization via O 2p up -spin orbitals can push the occupied Mn3+\n13\ng2etg\nand Ir5+\n13\ng2tbands downwards, consequently stabilizes the antiferromagnetic  \ninteraction and then the ferrimagnetic ground state. In other words, the virtual hopping \nfrom Ir5+\n\ng2tto the empty Mn3+\n\ngevia the bridged O\np2 states accounts for the \nantiferromagnetic cou pling and the abnormal large computed Mn3+ magnetic moment \nas well as the exclusively negative sign of the induced O2- magnetic moments.  \n \n "    },    {        "title": "2005.09864v1.Frequency_mixing_in_a_ferrimagnetic_sphere_resonator.pdf",        "content": "arXiv:2005.09864v1  [quant-ph]  20 May 2020Frequency mixing in a ferrimagnetic sphere resonator\nCijy Mathai,1Sergei Masis,1Oleg Shtempluck,1Shay Hacohen-Gourgy,2and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n2Physics Department, Technion, Haifa 32000 Israel\n(Dated: May 21, 2020)\nFrequency mixing in ferrimagnetic resonators based on yttr ium and calcium vanadium iron gar-\nnets (YIG and CVBIG) is employed for studying their nonlinea r interactions. The ferrimagnetic\nKittel mode is driven by applying a pump tone at a frequency cl ose to resonance. We explore\ntwo nonlinear frequency mixing configurations. In the first o ne, mixing between a transverse pump\ntone and an added longitudinal weak signal is explored, and t he experimental results are compared\nwith the predictions of the Landau-Zener-Stuckelberg mode l. In the second one, intermodulation\nmeasurements are employed by mixing pump and signal tones bo th in the transverse direction for\nstudying a bifurcation between a stable spiral and a stable n ode attractors. Our results are appli-\ncable for developing sensitive signal receivers with high g ain for both the radio frequency and the\nmicrowave bands.\nI. INTRODUCTION\nThe physics of magnons in ferromagnetic resonators\n[1–3] has been extensively studied in the backdrop of\nBose-Einstein condensation [4], optomagnonics [5–7], and\nspintronics [8]. Owing to the high magnon life time of\nthe order of a few microseconds, such ferromagnetic in-\nsulators have become the natural choice of microwave\n(MW) resonators in synthesizers [9], narrow band filters\n[10], and parametric amplifiers [11]. Exploring the non-\nlinearity associated with such systems is gaining atten-\ntion. A variety of magnon nonlinear dynamical effects\nhave been studied in the context of auto-oscillations [12],\noptical cooling [13], frequency mixing [14, 15] and bista-\nbility [16–20]. Applications of nonlinearity for quantum\ndata processing have been explored in [21, 22]. Nonlin-\near interactions between the electromagnetic (EM) MW\nfield coherent photons and these resonators can be sig-\nnificantly enhanced with relatively low power around the\nresonance frequency of the oscillator. Studying such non-\nlinear interactions is important due to the realization of\nhybrid quantum systems for quantum memory and opti-\ncal transducer related applications [23–28].\nHere we study the nonlinear frequency mixing process\nin these ferromagnetic resonators based on two config-\nurations. In the first configuration we study frequency\nmixing of transverse and longitudinal driving tones that\nare simultaneously applied to the magnon resonator. The\nsignal tone is in the radio frequency (RF) band, and it\nis applied in the longitudinal direction, parallel to the\nexternal static magnetic field. This process can be em-\nployed for frequency conversion between the RF and the\nMW bands. Here we find that the measured response can\nbe well described using the Landau-Zener-Stuckelberg\nmodel [29, 30]. In the second configuration, Kerr non-\nlinearity that is induced by magnetic anisotropy is stud-\nied by intermodulation measurements. This is done by\nsimultaneously applying in the transverse direction an\nintense pump and a weak signal tones both having fre-\nquencies close to resonance. The observed intermodula-\ntion frequency conversion reveals a bifurcation between a\nFIG. 1: (a) A schematic of the experimental setup used for\nstudying the ferrimagnetic (FM) sphere. Longitudinal and\ntransverse driving are applied using a radio frequency ante nna\n(RFA) and a microwave antenna (MWA), respectively. (b)\nReal image of the DUT.\nstable spiral and a stable node [31]. These nonlinear ef-\nfects may find applications in signal sensing, parametric\namplification and other related applications.\nThe spherical resonators under test are made of yt-\ntrium iron garnet (YIG) and calcium vanadium bismuth\niron garnet (CVBIG) with a radius of Rs= 125µm. They\nhost magnonic excitations with relatively low damping\nand large spin densities. These spheres are anisotropic\nferrimagnetic crystals with strong Faraday rotation an-\ngles and high refractive index as compared to other iron\ngarnets. A schematic image of our device under test\n(DUT) is shown in Fig. 1. The ferrimagnetic sphere\nis held by vacuum through a ferrule. A fixed magnet is\nemployed for fully magnetizing the sphere. A loop an-\ntenna (coil) is used to apply a transverse (longitudinal)\ndriving in the MW (RF) band. All measurements are\nperformed at room temperature.2\nII. LANDAU-ZENER-STUCKELBERG\nINTERFEROMETRY\nLandau-Zener-Stuckelberg interferometry is based on a\nmixing process between transverse and longitudinal driv-\ning frequencies that are simultaneously applied to a res-\nonator [29, 30]. The polarization vector Pevolves in\ntimetaccording to the Bloch-Landau-Lifshitz equation\ndP/dt=P×Ω+Γ, whereΩ=γeBis the rotation vec-\ntor, with Bbeing the externally applied magnetic induc-\ntion and γe= 28GHzT−1being the gyromagnetic ratio,\nand the vector Γ=−Γ2Pxˆ x−Γ2Pyˆ y−Γ1(Pz−Pz,s)ˆ z\nrepresents the contribution of damping, with Γ1= 1/T1\nandΓ2= 1/T2being the longitudinal and transverse re-\nlaxation rates, respectively, and Pz,sbeing the steady\nstate polarization. Consider the case where Ω(t) =\nω1(cos(ωt)ˆ x+sin(ωt)ˆ y) +ω0ˆ z. Here ω1andωare\nboth real constants, and ω0oscillates in time accord-\ning toω0=ωc+ωbsin(ωmt), where ωc,ωbandωm\nare all real constants. Nonlinearity of the Bloch-Landau-\nLifshitz equation gives rise to frequency mixing between\nthe transverse driving at angular frequency ωand the\nlongitudinal driving at angular frequency ωm. The reso-\nnance condition of the l’th order frequency mixing pro-\ncess reads ω+lωm=ωc, where lis an integer. The\ncomplex amplitude P+(in a rotating frame) of the cor-\nresponding l’th side band is given by (see appendix D of\nRef. [32])\nP+=iω1ζ\nΓ2\n2/parenleftBig\n1+iωd\nΓ2/parenrightBig\nPz,s\n/parenleftBig\n1+ω2\nd\nΓ2\n2/parenrightBig\nΓ1\nΓ2+/parenleftBig\nω1ζ\nΓ2/parenrightBig2, (1)\nwhereζ=Jl(ωb/ωm),Jlis thel′th Bessel function of\nthe first kind, and the detuning angular frequency ωdis\ngiven by ωd=ω+lωm−ωc.\nThe schematic of the experimental setup employed to\nexplore this frequency mixing process is shown in Fig.\n2(a). The device under test (DUT 1) comprises of the\nferrimagnetic resonator coupled to both the MW loop\nantenna and the RF coil. The Kittel mode frequency is\ntuned by the static magnetic field to the value 2.305GHz .\nThe sphere is simultaneously driven by a pump with a\npower of 0 dBm that is applied to the MW loop antenna\nand an RF signal with a frequency of 0.5MHz that is ap-\nplied to the RF coil. Spectrum analyzer measurements of\nthe signal reflected from the MW loop antenna are shown\nin Fig. 2(b) as a function of the spectrum analyzer angu-\nlar frequency ωSAand the driving MW angular frequency\nωMWthat is injected into the loop antenna. The theoret-\nical prediction that is derived using Eq. ( 1) is presented\nby Fig. 2(c). The values of parameters that are used\nfor the calculation are listed in the caption of Fig. 2.\nThe comparison between the measured [see Fig. 2(b)]\nand calculated [see Fig. 2(c)] response yields a good\nagreement.\nFIG. 2: Landau-Zener-Stuckelberg interferometry. Experi -\nmental (b) and theoretical (c) spectral response obtained b y\nthe frequency mixing of transverse and longitudinal drivin g\nsignals applied simultaneously to the magnon resonator. Th e\ntheoretical color-coded plot (c) is derived using Eq. (1), u s-\ning the following parameters’ values ωm/(2π) = 0.5MHz ,\nω1/(2π) = 0.5MHz ,ωb/(2π) = 0.5MHz ,Γ1/(2π) = 1.0MHz\nandΓ2/(2π) = 2.0MHz .\nIII. ANISOTROPY-INDUCED KERR\nNONLINEARITY\nThe experimental setup used for intermodulation mea-\nsurements is shown in Fig. 3(a). Here the device un-\nder test (DUT 2) is the same as that shown in Fig. 1,\nwhere the RF antenna (RFA) is removed from the setup.\nThe nonlinearity gives rise to bistability, which in turn\nyields a hysteretic resonance curve, which is obtained via\nthe forward and backward sweeping directions [see Fig.\n3(b)]. The measured response becomes bistable when\nthe input pump power Ppis of the order of mW. The\nsubsequent idler tones generated due to the nonlinear\nfrequency mixing of pump and signal tones in the ferri-\nmagnetic resonator are shown in Fig. 3(c).\nThe technique of Bosonization can be applied to model\nthe nonlinearity in ferrimagnetic sphere resonators [33].\nIn this approach, the Hamiltonian HMis expressed in the\nform/planckover2pi1−1HM=ωcNM+KMN2\nM+QMN4\nM+···, where\nωc=µ0γeHis the angular frequency of the Kittel mode\n[7, 34],µ0= 4π×10−7NA−2is the permeability of free\nspace,His the externally applied uniform magnetic field\n(which is assumed to be parallel to the ˆ zaxis),NMis a\nnumber operator, KMis the so-called Kerr frequency, and\nQMis the coefficient of quartic nonlinearity. When non-\nlinearity is taken into account to lowest nonvanishing or-\nder only, i.e. when the quartic and all higher order terms\nare disregarded, the response can be described using the\nDuffing-Kerr model. This model predicts that the re-\nsponse of the system to an externally applied monochro-3\nmatic driving can become bistable.\nFIG. 3: Intermodulation. (a) An intense pump and a rela-\ntively weak signal are simultaneously injected into the MW\nloop antenna, and the reflected signal is measured using a\nspectrum analyzer. (b) Experimentally obtained hystereti c\nresonance curve (with no signal) showing bistability corre -\nsponding to the forward and backward microwave frequency\nsweep directions. (c) Spectrum analyzer measurement of the\nreflected signal intensity as a function of the detuning fre-\nquency with respect to the pump frequency. The idler peaks\nare generated as a result of the nonlinear pump-signal mixin g.\nIn general, the number of magnons /angbracketleftNM/angbracketrightin a reso-\nnantly driven sphere having total linear damping rate\nγcwith pump power Ppis given for the case of critical\ncoupling by /angbracketleftNM/angbracketright ≃Pp/(/planckover2pi1ωcγc). On the other hand,\nthe expected number of magnons /angbracketleftNM/angbracketrightat the onset of\nDuffing-Kerr bistability is ≃γc/|KM|[see Eq. (42) of\nRef. [35] and note that, for simplicity, cubic nonlinear\ndamping is disregarded]. Thus, from the measured val-\nues of the linear damping rate γc/(2π)≃1MHz and\nPp≃1mW , at the bistability onset point one obtains\nKM/(2π)≃ −2×10−9Hz(the minus signs indicates\nthat the Kerr nonlinearity gives rise to softening). Note,\nhowever, that the above estimate, which is based on the\nDuffing-Kerr model, is valid provided that the quartic\nand all higher order terms can be disregarded near (and\nbelow) the bistability onset. For the quartic term this\ncondition can be expressed as |QM| ≪ |KM|3/γ2\nc.\nThe values of KMandQMare estimated below for\nthe case where nonlinearity originates from magnetic\nanisotropy. The Stoner–Wohlfarth energy EMis ex-\npressed as a function of the magnetization vector M=\nMˆ uM, and the first-order Kc1and second-order Kc2\nanisotropy constants as [36]\nEM\nVs=−µ0M·H+Kc1sin2φ+Kc2sin4φ , (2)\nwhereVs= 4πR3\ns/3is the volume of the sphere hav-\ning radius Rs, andφis the angle between ˆ uMand theunit vector ˆ uAparallel to the easy axis. It is assumed\nthat the sphere is fully magnetized, i.e. |M| ≃Ms,\nwhereMsis the saturation magnetization. In terms\nof the dimensionless angular momentum vector Σ=\n−2MVs/(/planckover2pi1γe)≡(Σx,Σy,Σz)Eq. ( 2) is rewritten as\nEM−/planckover2pi1ωK1(1+Kc2/Kc1) =HM, where\n/planckover2pi1−1HM=ωcΣz\n2+/parenleftbigg\n1+2Kc2\nKc1/parenrightbiggKM(Σ·ˆ uA)2\n4\n+Kc2\nKc1K2\nM(Σ·ˆ uA)4\n16ωK1,\n(3)\nωK1=/planckover2pi1−1VsKc1andKM=/planckover2pi1γ2\neKc1//parenleftbig\nVsM2\ns/parenrightbig\nis the Kerr\nfrequency [17].\nIn the Holstein-Primakoff transformation [37], the\noperators Σ±= Σx±iΣyandΣzare expressed as\nΣ+=B†(Ns−N′\nM)1/2,Σ−= (Ns−N′\nM)1/2Band\nΣz=−Ns+ 2N′\nM, where Nsis the total number\nof spins, and where N′\nM=B†Bis a number opera-\ntor. If the operator Bsatisfies the Bosonic commu-\ntation relation/bracketleftbig\nB,B†/bracketrightbig\n= 1then the following holds\n[Σz,Σ+] = 2Σ +,[Σz,Σ−] =−2Σ−and[Σ+,Σ−] =\nΣz. The approximation (Ns−N′\nM)1/2≃N1/2\nsleads to\nΣ·ˆ uA=N1/2\ns/parenleftbig\nB†uA++BuA−/parenrightbig\n+2NMuAz, whereuA±=\n[(ˆ uA·ˆ x)∓i(ˆ uA·ˆ y)]/2,uAz=ˆ uA·ˆ z, and the magnon\nnumber operator NMis defined by NM=N′\nM−Ns/2.\nThis approximation is valid near the bistability onset pro-\nvided that γc/(|KM|Ns)≪1. For YIG, the spin density\nisρs= 4.2×1021cm−3, thus for a sphere of radius Rs=\n125µmthe number of spins is Ns=Vsρs= 3.4×1016,\nhence for the current experiment γc/(|KM|Ns)≃10−1.\nThis estimate suggests that inaccuracy originating from\nthis approximation may be significant for the current ex-\nperiment near and above the bistability threshold.\nSecond-order anisotropy gives rise to a quartic non-\nlinear term in the Hamiltonian ( 3) with a coefficient\nQM≃(Kc2/Kc1)/parenleftbig\nK2\nM/ωK1/parenrightbig\n(the exact value depends\non the angle φbetween the magnetization vector and\nthe easy axis). Near or below the bistability onset the\nquartic term can be safely disregarded provided that\n(Kc2/Kc1)/parenleftbig\nγ2\nc/(ωK1|KM|)/parenrightbig\n≪1. When this condition\nis satisfied the Hamiltonian ( 3) for the case where ˆ uAis\nparallel to ˆ z(i.e.uAz= 1anduA+=uA−= 0) approxi-\nmately becomes\n/planckover2pi1−1HM=ωcNM+KMN2\nM. (4)\nThe term proportional to KMrepresents the anisotropy-\ninduced Kerr nonlinearity.\nFor YIG Ms= 140kA /m,Kc1=−610J/m3at297K\n(room temperature), hence for a sphere of radius Rs=\n125µmthe expected value of the Kerr coefficient is given\nbyKM/(2π) =−2.0×10−9Hz. This value well agrees\nwith the above estimation of KM/(2π)based on the mea-\nsured input power at the bistability onset. For YIG\nKc2/Kc1= 4.8×10−2(Kc2/Kc1= 4.3×10−2) at a tem-\nperature of T= 4.2K(T= 294K ) [7]. Based on these4\nvalues one finds that for the sphere resonators used in the\ncurrent experiment (Kc2/Kc1)/parenleftbig\nγ2\nc/(ωK1|KM|)/parenrightbig\n≃10−6,\nhence the second-order anisotropy term (proportional to\nKc2) in Eq. ( 3) can be safely disregarded in the vicinity\nof the bistability onset.\nIV. STABLE SPIRAL AND STABLE NODE\nTo explore the regime of weak nonlinear response, con-\nsider a resonator being driven by a monochromatic pump\ntone having amplitude bcand angular frequency ωp. The\ntime evolution in a frame rotating at the pump driving\nfrequency is assumed to have the form\ndCc\ndt+Θc=Fc, (5)\nwhere the operator Ccis related to the resonator’s anni-\nhilation operator AcbyCc=Aceiωpt, the term Θc=\nΘc/parenleftbig\nCc,C†\nc/parenrightbig\n, which is expressed as a function of both\nCcandC†\nc, is assumed to be time independent, and Fc\nis a noise term having a vanishing expectation value.\nThe complex number Bcrepresents a fixed point, for\nwhichΘc(Bc,B∗\nc) = 0 . By expressing the solution as\nCc=Bc+ccand considering the operator ccas small,\none obtains a linearized equation of motion from Eq. ( 5)\ngiven by\ndcc\ndt+W1cc+W2c†\nc=Fc, (6)\nwhereW1=∂Θc/∂CcandW2=∂Θc/∂C†\nc(both deriva-\ntives are evaluated at the fixed point Cc=Bc).\nThe stability properties of the fixed point depend on\nthe eigenvalues λc1andλc2of the2×2matrixW, whose\nelements are given by W11=W∗\n22=W1andW12=\nW∗\n21=W2[see Eq. ( 6)]. In terms of the trace TW=\nW1+W∗\n1and the determinant DW=|W1|2− |W2|2of\nthe matrix W, the eigenvalues are given by λc1=TW/2+\nυWandλc2=TW/2−υW, where the coefficient υW\nis given by υW=/radicalBig\n(TW/2)2−DW. Note that in the\nlinear regime, i.e. when W2= 0, the eigenvalues become\nλc1=W1andλc2=W∗\n1. For the general case, when both\nλc1andλc2have a positive real part, the fixed point is\nlocally stable. Two types of stable fixed points can be\nidentified. For the so-called stable spiral, the coefficient\nυWis pure imaginary [i.e. (TW/2)2−DW<0], and\nconsequently λc2=λ∗\nc1, whereas both λc1andλc2are\npure real for the so-called stable node, for which υWis\npure real. A bifurcation between a stable spiral and a\nstable node occurs when υWvanishes.\nFurther insight can be gained by geometrically an-\nalyzing the dynamics near an attractor. To that\nend the operators ccandFcare treated as complex\nnumbers. The equation of motion ( 6) for the com-\nplex variable cccan be rewritten as d¯ξ/dt+W′¯ξ=\n¯f, where ¯ξ=/parenleftbig\nReal/parenleftbig\ncceiφ/parenrightbig\n,Imag/parenleftbig\ncceiφ/parenrightbig/parenrightbigTand¯f=/parenleftbig\nReal/parenleftbig\nFceiφ/parenrightbig\n,Imag/parenleftbig\nFceiφ/parenrightbig/parenrightbigTare both two-dimensional\nreal vectors, and where the rotation angle φis real.\nTransformation into the so-called system of principle axes\nis obtained when the angle φis taken to be given by\ne2iφ=W1W∗\n2/|W1W2|. For this case the 2×2real ma-\ntrixW′becomes\nW′=/parenleftbigg\ncosθ1−sinθ1\nsinθ1cosθ1/parenrightbigg/parenleftbigg\nW+0\n0W−/parenrightbigg\n,(7)\nwhereθ1= arg(W1)and where W±=|W1|±|W2|. Thus,\nmultiplication by the matrix W′can be interpreted for\nthis case as a squeezing with coefficients W±followed by\na rotation by the angle θ1.\nThe flow near an attractor is governed by the eigen-\nvectors of the 2×2real matrix W′. For the case where\nυWis pure real the angle αWbetween these eigenvectors\nis found to be given by sinαW=υW/|W2|. Thus at the\nbifurcation between a stable spiral and a stable node, i.e.\nwhenυW= 0, the two eigenvectors of W′become parallel\nto one another. In the opposite limit, when υW=|W2|,\ni.e. when W1becomes real, and consequently the ma-\ntrixWbecomes Hermitian, the two eigenvectors become\northogonal to one another (i.e. αW=π/2).\nThe bifurcation between a stable spiral and a stable\nnode can be observed by measuring the intermodulation\nconversion gain GIof the resonator. This is done by\ninjecting another input tone (in addition to the pump\ntone), which is commonly referred to as the signal, at\nangular frequency ωp+ω. The intermodulation gain is\ndefined by GI(ω) =|gI(ω)|2, wheregI(ω)is the ratio be-\ntween the output tone at angular frequency ωp−ω, which\nis commonly referred to as the idler, and the input signal\nat angular frequency ωp+ω. In terms of the eigenvalues\nλc1andλc2the gain GIis given by [35]\nGI=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2γc1W2\n(λc1−iω)(λc2−iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (8)\nwhereγc1is the coupling coefficient (in units of\nrate) between the feedline that is used to de-\nliver the input and output signals and the res-\nonator. For the case of a stable spiral, i.e.\nwhenλc2=λ∗\nc1, one has |(λc1−iω)(λc2−iω)|2=/bracketleftBig\nλ′2+(λ′′−ω)2/bracketrightBig/bracketleftBig\nλ′2+(λ′′+ω)2/bracketrightBig\n, where λ′= Reλc1\nandλ′′= Imλc1(i.e.λc1=λ′+iλ′′), whereas for the case\nof a stable node, i.e. when both λc1andλc2are pure real,\none has|(λc1−iω)(λc2−iω)|2=/parenleftbig\nλ2\nc1+ω2/parenrightbig/parenleftbig\nλ2\nc2+ω2/parenrightbig\n.\nFor the case of a resonator having Kerr nonlin-\nearity and cubic nonlinear damping Θcis given by\nΘc= [i∆c+γc+(iKc+γc3)Nc]Cc+i√2γc1eiφc1bc,\nwhere∆c=ωc−ωpis the driving detuning, the total\nrate of linear damping is γc=γc1+γc2, the rate γc1\ncharacterizes the coupling coefficient between the feed-\nline and the resonator, γc2is the rate of internal lin-\near damping, γc3is the rate of internal cubic damping,\nKcis the Kerr coefficient, Nc=A†\ncAcis the resonator\nnumber operator, and φc1is a phase coefficient charac-5\nFIG. 4: Stability map of a driven Duffing-Kerr resonator.\nThe BOP is the point/parenleftbig\n∆c/(∆c)BOP,bc/(bc)BOP/parenrightbig\n= (−1,1).\nIn both 'C'(dark blue) and 'R'(light blue) regions, there is\na single locally stable attractor, whereas there are two in t he\nregions 'CC'(light green), 'CR'(orange) and 'RR'(red). The\nletter 'C'is used to label a stable spiral, whereas the letter\n'R'labels a stable node.\nterizing the coupling between the feedline and the res-\nonator [35]. The rates W1andW2are given by W1=\ni∆c+γc+2(iKc+γc3)|Bc|2andW2= (iKc+γc3)B2\nc.\nThe condition Θc(Bc,B∗\nc) = 0 can be expressed as a cu-\nbic polynomial equation for the number of magnons Ec=\n|Bc|2given by/bracketleftBig\n(∆c+KcEc)2+(γc+γc3Ec)2/bracketrightBig\nEc=\n2γc1|bc|2. The eigenvalues can be expressed in terms\nofEcasλc1,2=TW/2±υW, where TW/2 =γc+\n2γc3EcandυW=/radicalbig\n(∆−−∆c)(∆++∆c), where∆±=/parenleftbigg/radicalBig\n1+(γc3/Kc)2±2/parenrightbigg\nKcEc. The stability map of the\nsystem is shown in Fig. 4. Both driving detuning ∆cand\ndriving amplitude bcare normalized with the correspond-\ning values at the bistability onset point (BOP) (∆c)BOP\nand(bc)BOP[see Eqs. (46) and (47) of Ref. [35]]. Inside\nthe regions 'C'and'R'of mono-stability ( 'CC','CR'\nand'RR'of bistability) the resonator has one (two) lo-\ncally stable attractors. A stable spiral (node), for which\nλc2=λ∗\nc1(bothλc1andλc2are pure real), is labeled by\n'C'('R').\nIn the bistable region, the cubic polynomial equation\nhas 3 real solutions for Ec. The corresponding values\nof the complex amplitude Bcare labeled as C1,C2and\nC3. In the flow map shown in Fig. 5, which is obtained\nby numerically integrating the equation of motion ( 5)\nfor the noiseless case Fc= 0, the point C1is a stable\nnode, the point C2is a saddle point and the point C3\nis a stable spiral. The red and blue lines represent flow\ntoward the stable node attractor at C1and the stable\nspiral attractor at C3, respectively. The green line is the\nseperatrix, namely the boundary between the basins of-8-6-4-202468-10-8-6-4-2\nC3C1\nC2\nRe(C)Im(C)\n(a)\n4.74.754.84.854.94.9555.05-2.7-2.6-2.5-2.4-2.3-2.2-2.1-2\nC1\nC2\nRe(C)Im(C)\n(b)\nFIG. 5: Flow map of a Duffing oscillator in the region of\nbistability. The point C1is a stable node, the point C2is a\nsaddle point, and the point C3is a stable spiral. A closer view\nof the region near C1andC2is shown in (b).\nattraction of the attractors at C1andC3. A closer view\nof the region near C1andC2is shown in Fig. 5(b).\nThe intermodulation conversion gain GIinduced by\nthe Kerr nonlinearity is measured with the ferrimagnetic\nresonator DUT 2[see Fig. 3(a)], and the results are com-\npared with the theoretical prediction given by Eq. ( 8).\nIn these measurements the pump frequency ωpis tuned\nclose to the resonance frequency ωc. The measured gain\nGIis shown in the color-coded plots in Fig. 6(for three\ndifferent values of the pump frequency ωp) as a function\nof the detuning between the signal and pump frequencies\nω/(2π)and the pump power Pp.\nThe overlaid black dotted lines in Fig. 6indicate the\ncalculated values of the imaginary part of the eigenvalues\nλ′′= Imλc1and−λ′′= Imλc2. The calculation is based\non the above-discussed Duffing-Kerr model. At the point6\nFIG. 6: Intermodulation gain GIas a function of detuning\nbetween the signal and pump frequencies ω/(2π)and pump\npowerPp(in dBm units). The pump frequency ωp/(2π)is\n(a)3.8674GHz (b)3.8704GHz and (c)3.8734GHz . The sig-\nnal power is −15dBm. Note that GIis measured with ω >0\nonly, and the plots are generated by mirror reflection of the\ndata around the point ω= 0. Note also that for clarity the\nregion near the pump frequency, i.e. close to ω= 0, has been\nremoved from the plot. The width of this region, in which\nthe intense pump peak is observed, depends on the resolu-\ntion bandwidth setting of the spectrum analyzer. The black\ndotted lines indicate the calculated values of the imaginar y\npart of the eigenvalues λ′′= Imλc1and−λ′′= Imλc2. The\npump amplitude bcand pump detuning ∆cused for the cal-\nculation of the eigenvalues are determined from the measure d\nvalue of Pp= 0.4dBm for the pump power and the value of\n∆c/(2π) = 1.3MHz for the pump detuning at the BOP.\nwhereλ′′vanishes, a bifurcation from stable spiral to sta-\nble node occurs. As can be seen from comparing panels(a), (b) and (c) of Fig. 6, the pump power Ppat which\nthis bifurcation occurs depends on the pump frequency\nωp. This bifurcation represents the transition between\nthe regions 'CC'and'CR'in the stability map shown in\nFig.4. A bifurcation from the bistable to the monos-\ntable regions occurs at a higher value of the pump power\nPp. This bifurcation gives rise to the sudden change in\nthe measured response shown in Fig. 6. In the stability\nmap shown in Fig. 4, this bifurcation corresponds to the\ntransition between the regions 'CR'and'C'.\nV. CONCLUSION\nWe present two nonlinear effects that can be used for\nsignal sensing and amplification. The first one is based\non the so-called Landau-Zener-Stuckelberg process [29]\nof frequency mixing between transverse and longitudi-\nnal driving tones that are simultaneously applied to the\nmagnon resonator. This process can be employed for fre-\nquency conversion between the RF and the MW bands.\nThe second nonlinear effect, which originates from mag-\nnetization anisotropy, can be exploited for developing in-\ntermodulation receivers in the MW band. Measurements\nof the intermodulation response near the onset of the\nDuffing-Kerr bistability reveal a bifurcation between a\nstable spiral attractor and a stable node attractor. Above\nthis bifurcation, i.e. where the attractor becomes a stable\nnode, the technique of noise squeezing can be employed\nin order to enhance the signal to noise ratio [35].\nVI. ACKNOWLEDGMENTS\nWe thank Amir Capua for helpful discussions. 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Banerjee, N. Teichert, K. Siewierska, Z. Gercsi, G. Atcheson, P. Stamenov,  \nK. Rode, J. M. D. Coey and J. Besbas*  \n1 CRANN, AMBER and School of Physics, Trinity College, Dublin 2, Ireland  \nEnergy -efficient control of magnetization without the help of a magnetic field is a key \ngoal of spintronics1,2. Purely heat -induced single -pulse all -optical toggle switching has \nbeen demonstrated, but so far only in Gd ba sed amorphous ferrimagnet films3–6. In this \nwork, we demonstrate toggle switching in  the half-metallic compensated ferrimagnetic \nHeusler alloys Mn 2Ru xGa, which have  two crystallographically -inequivalent Mn \nsublattices7. Moreover, we observe the switching at room temperature in samples that are \nimmune to external magnetic fields in excess of 1 T, provided they exhibit compensation \nabove room temperature. Observation s of the effect in compensated ferrimagnets without \nGd challenges our understanding of all -optical switching.  The dynamic behavior \nindicates that Mn 2Ru xGa switches in 2 ps or less.  Our findings widen the basis for fast \noptical switching of magnetization and break new ground for engineered materials that \ncan be u sed for nonvolatile ultrafast switches using ultrashort pulses of light.  \nKeywords: Single -pulse toggle switching; Half -metallic ferrimag nets; compensation \ntemperature;  Mn 2Ru xGa; picosecond spin dynamics;  \n*besbasj@tcd.ie  \n  \n  2 Driven by the demands for high speed, low cost and high -density magnetic recording, \nresearch in spintronics has always sought insight into new classes of magnetic materials and \ndevices that show efficient and reproducible magnetization switching. In this respect, interest \nin the magneti c properties of antiferromagnetically coupled sub -lattice systems has gained \nmomentum in the last decade. The total or partial cancellation of the moments makes these \nsystems insensitive to stray magnetic fields, and the interaction between the sublattice moments \nintroduces phenomena that are absent in conventional ferromagnets, opening new opportunities \nfor magnetic recording and information processing2,8–10.  \nAn efficient way of controlling magnetism is to use ultrashort laser pulses1,11. XMCD \ninvestigations in 2011 by Radu et al.3 of the dynamics of the Gd and Fe atomic moments in a \nthin layer of amorphous ferrimagnetic Gd 25Fe65.6Co9.4 after a 50 fs laser pulse , revealed a \ntransient parallel alignment of the moments that was the precursor of swi tching.  They were  \nfollowed  by the discovery of single -pulse all -optical toggle switching of the magnetization in \nthe same material by Ostler et al.12. A general basis for fast all -optical switching in multi -\nsublattice magnets was then proposed by Mentink et al.13. Amorphous Gd x(Fe,Co) 100-x with x \n≈ 25 is a metallic ferrimagnet with localized 4 f-shell magnetic moments on the Gd sublattice \nand delocalized 3 d-band moments on the Fe -Co sublattice. Upon excitation by a femtosecond \nlaser pulse,  the Fe -Co undergoes sub -picosecond demagnetization leading to practically \ncomplete loss of the ordered 3 d shell magnetization, an effect that had been originally observed \nin ferromagnetic nickel14. Concomitantly, the Gd atoms  experience a slower loss of m agnetic \nalignment , with partial transfer of angular momentum from the  Gd f shell to the FeCo d shell3, \nentail ing a transient parallel alignment of the moments  of the demagnetizing Gd and the re-\nmagnetiz ing FeCo  that ultimately leads to magnetization toggle  switching on a picosecond \ntimescale3,12. As the suggested mechanism for single pulse all optical switching  (SP-AOS ) \nrelies on an ultrafast interplay between two inequivalent spin sublattic es, one with a slower \nresponse to the laser (the Gd 4 f electrons ) and the other with a faster one (the Fe 3 d electrons ), \nsubsequent researches on SP -AOS concentrated on rare -earth based ferrimagnets15. In these  \nswitching  measurements, it is useful to distinguish the  very short timescale on whi ch the future \ndirection of t he n et magnetization is decided, and the longer timescale necessary for the \nequilibrium magnetization to be established  and respond to an external magnetic field.  In \npractice helicity -independent SP -AOS has only been demonstrated in ferrimagnetic 3 Gdx(Fe,Co )100-x thin films3, Gdx(Fe,Co) 100-x spin valves5 and in synthetic Gd/Co ferrimagnets4; \nIt has not been seen in other rare -earth based ferrimagnet s such as amorphous Tb 27Co7316,17 \nwhere the 4 f electrons experience strong spin -orbit coupling. Its thermal ori gin is established \nby the independence of the effect on the polarization and helicity of the light3,12, and the \nequivalent effect produced by pulses of hot electrons18. A related phenomenon has been \nreported in ferrimagnetic TbFeCo, however, under specific  structural condition s19 and in \nferromagnetic Pt/Co/Pt structures, when the laser spot size matches that of the ferromagnetic \ndomains20. A different type of single -pulse, non -thermal, non -toggle switching has been \nreported with linearly -polarized light in insulating Co -doped yttrium iron garnet21. \n In this work, we present a new, rare -earth -free, ferrimagnet that exhibits SP -AOS where \nthe two sublattices should not according to the prevalent thermodynamical models13 have \ndrastically different response time to laser excitation. We report all -optical toggle switching in \nthe ferrimagnetic Heusler alloys Mn 2RuxGa (MRG)7 where both magnetic sublattices are \ncomposed of manganese, and establish MRG as a versatile alternative to Gd x(FeCo) 1-x for SP -\nAOS applications.  In MRG, the Mn atoms occupy two inequivalent sub -lattices at Wyckoff \npositions 4 a and 4 c in the cubic 𝐹4´3𝑚 structure (See supplementary Fig. S1 a), with \nantiferromagnetic intersublattice coupling7. At low temperature the magnetization of the \nMn(4 c) sublattice is dominant, but as temperature increases the magnetization of the Mn(4 c) \nfalls faster than that of the Mn(4 a) sublattice, leading to a compensation temperature Tcomp \nwhere the two are equal and opposite as the  coercivity tends to diverge when t he net \nmagnetization crosses zero22. The value of Tcomp can be varied by changing the Ru \nconcentration  x, so it is possible to make MRG peculiarly insensitive to external magnetic fields \nby decreasing its magnetisation23. The electronic structures of the t wo sublattices are different. \nMRG has a spin gap ~1 eV close to the Fermi energy, which led to its identification as the first \nexample of a half -metallic  ferrimagnet7; the Mn(4 c) electrons have a high spin -polarized \ndensity of states whereas that of the Mn (4a) electrons is much lower. The unusual electronic \nstructure accounts for an anomalous Hall effect (AHE) that is greater than those seen in \ncommon ferromagnets23 and a strong magneto -optical Kerr effect (MOKE), even when the net \nmagnetization vanishes at  Tcomp23,24, because both AHE and MOKE probe mainly the spin -4 polarized conduction band associated with Mn in the 4c position. Domains can be directly \nimaged in the Kerr microscope, regardless of the net magnetization 25. \nIn our experiments, we investigated  SP-AOS in 19 MRG thin films having different Ru \ncontents with Tcomp above or below room temperature (RT). The films are deposited on MgO \n(100) substrates, which leads to a slight tetragonal distortion of the cubic XA -type structure \n(from space group 216, 𝐹4´3𝑚 to space group 119, I 4´m2), which is responsible for the \nperpendicular magnetic anisotropy of the MRG films. Optical pulses of 800 nm waveleng th \nand about 200 fs duration  were generated by a mode -locked Ti -sapphire laser seeding a 1 kHz \namplifier. Fi gure 1 displays the result s of irradiating a Mn 2Ru1.0Ga film by a single 200 fs pulse \nwith a Gaussian intensity profile, as observed by ex situ  Kerr microscopy. Here the light or \ndark contrast indicates an orientation of the Mn(4 c) sublattice  into or out of the plane. For either \ninitial magnetization direction, a single laser pulse of sufficient intensity will switch the \nmagnetization d irection in the irradiated area  (The elliptical shape of the switched domain is \ncaused by astigmatism of the focusing lens). Pulses  where the average energy density is sub -\nthreshold  leave the magnetization unchanged , except just at the centre,  where the intensity \nexceeds threshold (Fig 1a) . The whole irradiated spot is switched at 1.5 µJ (Fig 1b) , but at  3 µJ \na multidomain pattern appears in the center of the irradiated zone  (Fig 1c) , where the \ntemperature of the film has transiently exceeded the Curie temperature of the sample (~500 K)7 \nleading to re -magnetization in sub -micron domains close in size to the resolu tion of the Kerr \nmicroscope. They are much smaller than t he ~ 100 µm domains normally observed at room \ntemperature during the reversal process after saturating the magnetization25. It is established \nthat such temperatures can be  reached in equilibrium betw een the  lattice and spin system in the \nvery first picoseconds following optical excitation in transition metal compounds14, after which \nthe system re -magnetizes randomly in the stray field during the cool down. The multidomain \npattern is directly surrounde d by a ring -shaped switched domain, which shows that SP -AOS \ninvolves  a significant transient demagnetization. The variation of the size of the switched \ndomain area with increasing pulse energy has been employed to calculate the threshold fluence \nfor switch ing (See Supplementary Information VI). Interestingly, we never observed SP -AOS \nin any MRG film having Tcomp below RT (see Supplementary Information V). We verified  that \nthe observed sequence of switching originates solely from laser induced heating, by re peating 5 the experiment with circularly polarized laser pulses of opposite helicities as well as different \ndirections of linear polarization with respect to the MRG crystallographic directions (not \nshown) . The SP -AOS occurred identically in all cases, which  eliminates the possibili ty of any \ncontribution from magnetic circular dichroism26 or from transient spin -orbit torques generated \nby the electric field. On further increasing the laser power to 5 μJ, the center of the irradiated \nspot on the film is ablated.  \nFigure 2 depicts the results of the irradiation with 1 to 5 successive laser pulses on \nMn 2Ru1.0Ga. The panels show different regions that were subjected to the given numbers of \nshots. Consistently, the irradiation by a series of laser pulses leads  to a toggling of the direction \nof the magnetization, which was investigated for up to 12 consecutive pulses.  \nMRG possesses a low net magnetization and a high anisotropy field. Therefore, the \ncoercive field of the films usually exceeds 0.2 T and can reach  values as high as 10 T23 if the \ntemperature is very close to Tcomp. It is interesting to see whether a highly -coercive sample can \nbe switched by light at RT. Figure 3 shows the toggling of magnetization following a sequence \nof pulses in a film of Mn 2Ru0.75Ga with coercivity exceeding 1 T. That sample could not be \nsaturated in our electromagnet and it was therefore measured in its virgin state, which is \ncharacterized by a distribution of magnetic domains with a predominance of magnetization \ndirected toward the substrate. Toggling of each of the individual domains by the light pulse is \nobserved even though the sample is insensitive to an external magnetic field of 1 T. The \nthreshold fluence for this sample was approximately one third of that of Mn 2Ru1.0Ga. Th e SP-\nAOS observed in samples with compensation temperature  close to RT is particularly important \nfor three reasons: 1) the threshold fluence required for switching is small, potentially enabling \nenergy -efficient applications in the future27; 2) the coerciv ity of MRG diverges close to Tcomp, \nwhich makes the magnetic state impervious to external magnetic fields; 3) switching of micron -\nsized domains is possible, which are much smaller than the laser spot size.  \nNext we turn to the dynamics of the excitation and   reversal processes. The magnetism \nin MRG originates from the 3 d moments of Mn(4 a) and Mn(4 c) sublattices which are \nantiferromagnetically coupled. The  femtosecond  laser pulse is expected to disrupt the inter -site \nexchange (< 0.1 eV) and rapidly destroy th e magnetic order; while the intra -atomic, on -site \nexchange that depends on stronger Coulomb interactions (3 – 5 eV) should not be completely 6 destroyed. The aftermath of the pulse therefore involves re -establishment of magnetic order \nfrom  the atomic moments , which in a ferrimagnet c ould include effects of angular momentum  \ntransfer between sites. To investigate this possibility, we have studied the magnetization \ndynamics using time -resolved polar MOKE (TR -MOKE) in the two -color collinear pump \nprobe geometry. In this part of the study, we  compare  two samples, Mn 2Ru1.0Ga and \nMn 2Ru0.65Ga. They  have Tcomp of 390 K and 165 K respectively and their coercive fields at \nroom temperature  are similar (~ 460 mT), as shown in the optically measured hysteresis loops \nin Fig.  4a. The loops have opposite signs, as expected , because the Mn(4 c) sublattice, which \ngives the dominant contribution to the MOKE signal, aligns parallel to the applied field below \nTcomp and antiparallel above. Intense laser pulses of wavelength 800 nm wer e used as the pump \nbeam to excite the magnetization dynamics, and the Mn(4c) sublattice magnetization was \nsubsequently probed in a stroboscopic manner using weaker 400 nm pulses. A field of 500 mT \nwas applied perpendicular to the films to ensure an identic al initial state before each pump \npulse. Figure 4b shows the TR-MOKE signal for different pump fluences for Mn 2Ru0.65Ga, \nwhich does not switch because Tcomp is below RT. Following the laser excitation, the transient \nMOKE signal shows a step -like change, ca used by the ultrafast destruction of the magnetic \norder  as the electron temperature shoots up.  Subsequently, the magnetization regains its initia l \nstate in tens or hundreds of ps, depending on laser fluence. This behavior is typical of \nferromagnetic metals . It should be noted here that even though the MOKE response indicates \nfull demagneti zation of the Mn(4 c) electrons, this does not mean we have transiently exceeded \nthe Curie temperature in the lattice. At a fluence of 1 5.1 mJ cm-2, a trace of magnetic order \nmight still persist in the magneto -optically silent Mn (4a) sublattice, to ensure that when the \nsystem re -magneti zes, it does so uniformly. At greater  fluence s we observe a multi -domain \nstate after laser irradiation, indicating t hat the system has  been thermally demagnetized  when \nthe electrons re -establish thermal equilibrium with a lattice that is now above Tc. \nOn changing to Mn 2Ru1.0Ga, we f ind a strong dependence of the TR -MOKE signal on \nlaser fluence, which is quite different below and above the threshold fluence Fth for SP -AOS \n(See Fig. 4c 1 and 4c 2 respectively). Below threshold , the behavior is like that of Mn 2Ru0.65Ga \nat similar fluence;  the recovery takes about 10 ps, and  an increase in the fast demagnetization \nsignal is observed with increasing fluence (Fig 4 d). Upon crossing the fluence threshold \nindicated by the yellow  bar, the following new features appear in the signal: i) an increase in 7 the pump fluence now leads to a decrease  in the demagne tization amplitude at zero delay (Fig. \n4c3), contrary to the previous case and to Gdx(FeCo )1-x28. The behavior of the demagnetization \namplitude with increas ing pump fluence is seen in Fig. 4d , where it fall s away from full \ndemagnetization above Fth; ii) As the system relaxes, the signal undergoes a rapid partial \nrecovery within 2 ps (See Fig. 4c 3), after which it reverses, at the point marked by the arrow. \nThis anomaly , which  has not been reported in Gdx(FeCo )1-x, appears in the sample \ncompensating above room temperature and only at fluences where toggle switching is observed. \nThere, at 2 ps, the film must have switched, in the short -time sense mentioned in the \nintroduction , because at longer times the Mn(4 c) sublattice continues to re -magnetize in a  \ndirection opposite to the one it had originally, before being turned back after 50 ps by the weak \napplied field and gradually recovering over the course of several hundred picoseconds. \nExtrapolating the initial slope to negative saturation gives a switchi ng tim e, in the longer sense, \nof about 200 ps.  The slope from 2 – 50 ps is comparable , but opposite in sign to that of the first \nsample (Fig 2b) and the timescale for recovery at high fluence is similar.  \nAnother sample with Tcomp = 250 K, excited at 210 K  or 230 K was found to behave  \nsimilarly29, and the time for thermalization of the electrons and the lattice is deduced there from \na 4-temperature  model to be 2 ps. Furthermore, Bonfiglio et al.29  have shown that magnetic \norder is already beginning to be re-established in MRG within 1 ps, permitting efficient \nexchange scattering and transfer of angular momentum from one sublattice to the other, even \nat extremely short timescales. We speculate that for MRG most of the demagneti zation is due \nto ex change of a ngular momentum between the sublattices, but a quite different wavelength of \nthe probe pulse would be needed to reveal the behavior of the 4 a sublattice and see whether \nthe transient parallel alignment of the two subattice moments  that is seen in XMCD  in \nGdx(FeCo )1-x3 is also present in MRG. The question of where the local memory of the \nmagnetization is stored just after the  electrons are demagnetized by the  laser pulse is germane \nto the explanation of the behavior shown in Fig 1 both below and above the sw itching threshold \nFth. If not in a substantially slower demagnetization rate for the silent 4 a sublattice, then the \nlocal angular momentum might be transient ly parked in optical phonon modes, or the nuclear \nspins.  8  In summary, we have demonstrated single -pulse all -optical thermal switching in less \nthan 2 ps in films of the half -metallic compensated Heusler ferrimagn et Mn 2RuxGa, where both \nmagnetic sublattices are composed of manganese atoms, occupying different crystallographic \nsites. These results extend the scope of the phenomenon beyond a limited range of amorphous \nGdx(Fe,Co) 100-x alloys with x ≈ 25, where the mag netic sublattices are defined chemically . A \ncomparison of the two systems is provided in Supplemental Information VIII. The Heusler \nalloys are a huge family, with an established body of knowledge about their magnetic and \nelectronic properties that will all ow us to advance our understanding of the SP -AOS \nphenomenon and design materials that can be the basis of future nonvolatile opto -magnetic \nswitches. Beyond the newly -demonstrated quality of MRG as an opto -magnetic material, its \nlarge intrinsic spin -orbit t orque, which relies on the absence of inversion symmetry of the \nMn(4 c) sublattice  opens prospects for new multifunctionality30,31. Therefore, MRG and its \nchemically tailored successors offer the prospects of both new insights into condensed matter \non a fem tosecond timescale and new technological prospects taking advantage of ultra -fast \ncontrol of the magnetic state without any reliance on a magnetic field. \n9 Methods:  MRG films with different Ru content were grown on MgO(001) substrates at 350◦C \nby DC magnetron sputtering in a Shamrock system with a base pressure of 2×10−8 Torr. They \nwere co -sputtered from Ru and stoichiometric Mn 2Ga targets. The Ru concentration was \ncontrolled by varying the Mn 2Ga target plasma power while fixing the Ru power. The samp les \nwere then capped with a protective layer of 2 nm of Al 2O3. \nFemtosecond laser pulses were generated by Ti -sapphire laser seeding a 1 kHz \namplifier with a Q -switched cavity. Their central wavelength was 800 nm and the pulse \nduration was about 200 fs. The  amplifier can be operated in continuous mode where a train of \npulses is generated at a repetition rate of 1 kHz or in single pulse mode where the emission of \none single pulse can be externally triggered. In some cases, 400 nm laser pulses were obtained \nby second harmonic generation in a β-BaB 2O4 crystal.  \n Prior to laser irradiation, the films were saturated at room temperature in the 1 T \nperpendicular magnetic field of our Evico Kerr microscope. Different areas of the films were \nthen irradiated with severa l linearly polarized laser pulses of different powers, followed by ex -\nsitu imaging of the final results in the Kerr microscope. For the imaging, a polarized beam is \nfocused onto the sample using a microscope objective. A rotation in the polarization due to  \nKerr effect occurs in the reflected beam, which then passes through an analyzer, before \nreaching the camera. In order to increase the contrast, we keep the axis of the analyzer few \ndegrees away from cross position in both directions and acquire two images . Subsequently, the \nimages are subtracted to extract the final image.  \n For the dynamic measurements, the laser beam with wavelength 800 nm was split into \na pump beam and a frequency -doubled probe beam at 400 nm. The intensity of the probe was  \n10 kept low. Bot h pump and probe were linearly polarized and collinear. The spot sizes were \nmeasured to be about 150 and 70 µm respectively. The dynamical magneto -optic Kerr rotation \nwas measured using a balanced detection scheme and acquired using a lock -in amplifier and  a \nmechanical chopper at 500 Hz in the pump beam. The pump/probe delay was varied using a \nmechanical delay line.  \n   \n11 Figure 1| Single -pulse all -optical switching (SP -AOS) in Mn 2Ru 1.0Ga. A film uniformly -\nmagnetized  out of the plane (top) or into the plane (bottom) is irradiated by a single 800 nm \npulse focused onto a spot 150µm x 100µm. The pulse energy in a is 0.9 µJ, which only exceeds \nthe switching threshold in a small region at the centre  where the fluence is hi ghest . The 1.5µJ \npulse in b switches the whole irradiated area, whereas the 3µJ pulse in c heats the film at t he \ncentre of the spot above the  Curie temperature, produci ng a fine multidomain pattern. The most \nintense pulses (5 µJ) lead to ablation of the film.  \nFigure 2| Toggling of the magnetization in Mn 2Ru 1.0Ga. Magnetization pattern as a \nfunction of the number of applied pulses . Pulse energy was 1.2  µJ. \nFigure 3| Toggling of magnetization in a high -coercivity Mn 2Ru 0.75Ga film.  Repeated \ntoggling of the m icron -scale domain pattern of a virgin -state sample is observed with \nrepeated pulses. There was a net imbalance of domains pointing in and out of the plane.  \nFigure 4| Time Resolved Magnetization Dynamics in Mn 2Ru xGa. a, Hysteresis loops \nmeasured by MOKE  in Mn 2Ru0.65Ga and Mn 2Ru1.0Ga, which have compensation temperature \nbelow and above RT respectively. b, Transient Kerr signals of Mn 2Ru0.65Ga for different pump \nfluences. The variation of the Kerr signal is normalized to the total Kerr rotation at RT. c1, \nTransient Kerr signal of Mn 2Ru1.0Ga for fluences below the switching threshold ; c2, similar \ndata including fluences above threshold ; c3, A zoom of c2 in a shorter time window; the anomaly \nmarked by the arrow is discussed in the text. d, Variation of the demagnetization amplitude at \nzero delay for different pump fluences. The yellow shaded region indicates the threshold \nfluence for switching.  The solid line is a guide to eye.  \n  \n12 References  \n1. Kimel, A. V. & Li, M. Writing magnetic memory wit h ultrashort light pulses. Nat. \nRev. Mater.  4, 189 –200 (2019).  \n2. Lalieu, M. L. M., Lavrijsen, R. & Koopmans, B. Integrating all -optical switching with \nspintronics. Nat. Commun.  10, 110 (2019).  \n3. Radu, I. et al.  Transient ferromagnetic -like state mediatin g ultrafast reversal of \nantiferromagnetically coupled spins. Nature  472, 205 –209 (2011).  \n4. Lalieu, M. L. M., Peeters, M. J. G., Haenen, S. R. R., Lavrijsen, R. & Koopmans, B. \nDeterministic all -optical switching of synthetic ferrimagnets using single femto second laser \npulses. Phys. Rev. B  96, 220411(R) (2017).  \n5. Iihama, S. et al.  Single -Shot Multi -Level All -Optical Magnetization Switching \nMediated by Spin Transport. Adv. Mater.  30, 1804004 (2018).  \n6. Gorchon, J. et al.  Single shot ultrafast all optical magnetization switching of \nferromagnetic Co/Pt multilayers. Appl. Phys. Lett.  111, 042401 (2017).  \n7. Kurt, H. et al.  Cubic Mn 2Ga Thin Films: Crossing the Spin Gap with Ruthenium. \nPhys. Rev. 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Wienholdt, S., Hinzke, D., Carva, K., Oppeneer, P. M. & Nowak, U. Orbital -resolved \nspin model for thermal magnetization switching in rare -earth -based f errimagnets. Phys. Rev. \nB 88, 020406(R) (2013).  \n16. El Hadri, M. S. et al.  Two types of all -optical magnetization switching mechanisms \nusing femtosecond laser pulses. Phys. Rev. B  94, 064412 (2016).   \n13 17. A. R. Khorsand et al.  Element -Specific Probing of Ult rafast Spin Dynamics in \nMultisublattice Magnets with Visible Light. Phys. Rev. Lett.  110, 107205 (2013).  \n18. Xu, Y. et al.  Ultrafast Magnetization Manipulation Using Single Femtosecond Light \nand Hot -Electron Pulses. Adv. Mater.  29, 1703474 (2017).  \n19. Liu, T.-M. et al.  Nanoscale Confinement of All -Optical Magnetic Switching in \nTbFeCo - Competition with Nanoscale Heterogeneity. Nano Lett.  15, 6862 –6868.  \n20. Vomir , M., Albrecht, M. & Bigot, J. -Y. Single shot all optical switching of intrinsic \nmicron size magnetic domains of Pt/Co/Pt ferromagnetic stack. Appl. Phys. Lett.  111, 242404 \n(2017).  \n21. Stupakiewicz, A., Szerenos, K., Afanasiev, D., Kirilyuk, A. & Kimel, A.  V. Ultrafast \nnonthermal photo -magnetic recording in a transparent medium. Nature  542, 71 (2017).  \n22. Betto, D. et al.  Site-specific magnetism of half -metallic Mn 2Rux thin films determined \nby x-ray absorption spectroscopy. Phys. Rev. B  91, 094410 (2015).  \n23. Thiyagarajah, N. et al.  Giant spontaneous Hall effect in zero -moment Mn 2RuxGa. \nAppl. Phys. Lett.  106, 122402 (2015).  \n24. Fleisher, K. et al.  Magneto -optic Kerr effect in a spin -polarized zero moment \nferrimagnet. Phys. Rev. B  98, 134445 (2018).  \n25. Siew ierska, K. E., Teichert, N., Schäffer, R. & Coey, J. M. D. Imaging Domains in a \nZero -Moment Half Metal. IEEE Trans. Magn.  55, 2600104 (2019).  \n26. Khorsand, A. R. et al.  Role of Magnetic Circular Dichroism in All -Optical Magnetic \nRecording. Phys. Rev. Lett.  108, 127205 (2012).  \n27. Barker, J. et al.  Two-magnon bound state causes ultrafast thermally induced \nmagnetisation switching. Sci. Rep.  3, (2013).  \n28. Stanciu, C. D. et al.  Subpicosecond Magnetization Reversal across Ferrimagnetic \nCompensation Points. Phys . Rev. Lett.  99, 217204 (2007).  \n29. Bonfiglio, G. et al.  Sub-picosecond exchange -relaxation in the compensated \nferrimagnet Mn 2RuxGa. arXiv  2003.01420 , (2020).  \n30. Lenne, S. et al.  Giant spin -orbit torque in a single ferrimagnetic metal layer. arXiv  \n1903.0 4432 (2019).  \n31. Finley, J., Lee, C. -H., Huang, P. Y. & Liu, L. Spin -Orbit Torque Switching in a \nNearly Compensated Heusler Ferrimagnet. Adv. Mater.  31, 1805361 (2019).  \n \n  \n14  \n \nAuthor contributions:  \nC. B., J. B . and J. M. D. C. designed the project. Experimental work was done by C. B., N. T.  \nand J. B. Growth and characterization of the samples were carried out by G. A. and K. S.  \nNumerical simulations were performed by Z. G. and J. B. ;  C. B., J. B., P. S., J. M. D. C and \nK. R. interpreted the data. Al l authors discussed the results. C. B., J. B. , K. R.  and J. M. D. C. \nwrote the paper.  \n \nAcknowledgements:  \nThis project has received funding from Science Foundation Ireland through contracts \n16/IA/4534 ZEMS and 12/RC/2278 AMBER and from the European Union’s  FET-Open \nresearch programme under grant agreement No 737038. C.B. is grateful to Irish Research \nCouncil for her post -doctoral fellowship. N. T. would like to acknowledge funding from the \nEuropean Union’s Horizon 2020 research and innovation programme unde r the Marie \nSkłodowska -Curie EDGE grant agreement No 713567.  \n \nThe authors declare no competing financial interest.  \n \n \n \n  \n15  \n \n \nFigure 1|  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n16  \n \nFigure 2|  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n17  \n \n \nFigure 3|  \n \n \n \n \n \n \n \n \n \n \n \n \n50 μmMnet \n18  \n \nFigure 4|  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n4 8 12 160.250.500.751.00\nFluence (mJ.cm-2)Demag Ampl. \n0 10 20 30-0.8-0.40.0DMz/ Mz\nDelay (ps)\n0102030405060-0.8-0.40.0DMz/ Mz\n 4.3 mJ.cm -2\n 6.4 mJ.cm -2\n 7.5 mJ.cm -2\nDelay (ps)\n050200 400 600-0.9-0.6-0.30.0\n  7.5 mJ.cm -2\n 15.1 mJ.cm -2\n 17.2 mJ.cm- 2DMz/ Mz\nDelay (ps)b\n-1.2 -0.6 0.0 0.6 1.2-0.80.00.81.6 Mn2Ru0.65Ga\n Mn2Ru1.0GaKerr Signal (Norm.)\nm0 H (T)a\nd c3c1c2 a\n050200 400 6000.00.40.8DMz/ Mz\nDelay (ps) 7.5 mJ.cm-2\n 15.1 mJ.cm-2\n 17.2 mJ.cm-2 \n19 Supplementary Information  \nSingle pulse all -optical toggle switching of magnetization without Gd: The example of \nMn 2Ru xGa \nC. Banerjee, N. Teichert, K. Siewierska, Z. Gercsi, G. Atcheson, P. Stamenov,  \nK. Rode, J. M. D. Coey and J. Besbas*  \n1 CRANN, AMBER and School of Physics, Trinity College, Dublin 2, Ireland  \n*besbasj@tcd.ie  \n \nI. STRUCTURAL AND MAGNETIC CHARACTERIZATION OF Mn 2Ru1.0Ga \n \n \n \n \n \n \n \n \n \nFigure S1: (a) Diagram of the inverted Heusler  (XA) crystal unit cell of  typical MRG  film. \nCharacterisation of Mn 2Ru1.0Ga (MRG): ( b) X ray reflectivity pattern of the MRG thin film. \nFitting gives a thickness of 42. 8 nm and a density of 8.2 g . cm-3, (c) X ray diffraction pattern \nof MRG thin film on MgO  (001) substrate. ( d) Reciprocal space map of MgO (113) peak and \nMRG (204) peak with lattice parameters calculated with respect to th e MRG unit cell .  \n  \n1 2 3 4100102104106Intensity\n2(o)\n30 40 50 60 70100102104106\nMRG (004)MgO (002)Intensity\n2 (o)MRG (002)a b\nc\n d \n20 MRG  crystallises in an inverted  Heusler (XA) structure of space group F -43m with two \ncrystallographically inequivalent magnetic Mn atoms at Wyckoff positions 4 a and 4 c and Ga \nand Ru atoms occupy the 4 b and 4 d positions, respectively, as shown  in Fig. S1 (a ). Note the \nMn(4c) sublattice is non -centrosymmetric.  Due to the biaxial non -volume c onserving strain of \nthe MgO substrate the unit cell  of MRG is tetragonally distorted and hence the space group is \nreduced to I -42m. X-ray data on the Mn 2Ru1.0Ga film  are shown in Fig. S1 (b), (c) and (d ). The \nX-ray reflectivity ( XRR) pattern shown in Fig. S1 (b ) has been fitted using X'Pert Reflectivity \nsoftware and the film thickness was calculated to be 42. 8 nm. The X -ray diffraction (XRD) \npattern in Fig. S1  (c) exhibits (002) and (004) reflections from the MRG, together with peaks \nfrom the MgO substrate.  The c -parameter calculated from  the (004) reflection is 604.7 pm.  A \nreciprocal space map (RSM) of the MRG film in Fig. S1 (d ) confirms the c -parameter obtained \nfrom XRD, and shows a distribution of a -parameters around the central value of 595.8 pm, \nwhich  corresponds to that of MgO. This demonstrates how substrate strain induces a ~ 1% \ntetragonal elongation of the MRG unit cell since the c/a ratio is 1.01, giving rise to the \nperpendicular magnetic anisotropy found in all MRG films .  \n \n \n \n \n \n \n \n \n \n \n  \n21 II. DEPENDENCE  OF MAGNETIZATION ON APPLIED MAGNETIC FIELD FOR \nMn 2Ru0.65Ga AND Mn 2Ru1.0Ga \n \n \n \n \n \n \nFigure S2 : Magnetization versus field applied in plane and out of plane at 300 K (a) for the \nMn 2Ru0.65Ga film and (b) for the Mn 2Ru1.0Ga film. \nThe dependence of magnetization on the field applied out of plane and in plane at 300 K relative \nto the surface of the thin film  was measured for Mn 2Ru0.65Ga and Mn 2Ru1.0Ga films  (See Fig. \nS2), which have Tcomp  below and above  RT respectively.  From the curves, t he saturation \nmagnetization s are 40 and 65 kA . m-1, respectively.  Interestingly, we observe a soft component \nin the out-of-plane, as well as the in-plane  magnetization , which  originates  partly  from the non -\ncollinearity of the two  exchange -coupled antiferromagnetically aligned  Mn magnetic \nsublattices.   \n \n \n \n \n \n \n \n  \n-4 -2 0 2 4-40040Magnetisation (kA m-1)\nm0H (T) Out of Plane\n In Plane\n-4 -2 0 2 4-80-4004080Magnetisation (kA m-1)\nm0H (T) Out of Plane\n In Planea b \n22 III. MEASUREMENT OF OPTICAL AND TRANSPORT MAGNETOMETRY OF \nMn 2Ru1.0Ga \n \n \n \n \n \n \nFigure S3 : Measured hysteresis loops of Mn 2Ru1.0Ga via (a) Kerr microscopy and (b) \nanomalous Hall effect.  \nWe have  measured the magnetic hysteresis loop s using polar Kerr effect ( = 630 nm) and \nanomalous Hall effect in the perpendicular Mn 2Ru1.0Ga fi lm, which are compared in Fig. S3 . \nIn contrast to the two-step switching observed in the S QUID  loop (See Fig. S 2 (b)), here the \nsquare hysteresis  loops  exhibit straightforward single -step switching , with an average \nswitching field of 460 mT in both cases. Whereas the SQUID probes the net magnetic moment, \nthe difference of the nearly -equal sublattice contributions, the optical and electrical transport, \non the other hand, relies on the distribution of spin-polarized electrons at  or near  the Fermi \nenergy, which in MRG reflects the 4c sublattice magnetization. Consequently,  the hysteresis \nshown in Fig. S3  as well as the  domain images presented here reflects the local magnetization \nstate of the 4c sublattice. It is therefore possible to explore the local magnetization state even \nat compensation.  \n \n \n \n \n  \n-0.8 -0.4 0.0 0.4 0.8-0.8-0.40.00.40.8Rxy ()\nm0 H (T)\n-0.8 -0.4 0.0 0.4 0.80255075\nm0 H (T)Kerr signal (a.u.)a b \n23 IV. DETERMINATION OF Tcomp FOR Mn 2Ru1.0Ga \n \n \n \n \n \n \n \nFigure S4: Optically  measured hysteresis loops for Mn 2Ru1.0Ga at different temperatures (a)  \nand the corresponding variation of coercive field (b).  \n \nFigure S4  (a) presents  the hysteresis loops measured by polar Kerr effect in Mn 2Ru1.0Ga at \ndifferent temperatures. On approaching Tcomp, the net moment falls, which increases the \nanisotropy field and  coercivity , until they  diverge at Tcomp. In addition, as the optical \nmeasurement s probes the Mn (4c) sublattice only, a change in the sign  of the hysteresis loop is \nseen upon crossing Tcomp. The variation of the coercive field as a  function of temperature is \nshown in Fig. S4b , from which Tcomp  for this  sample is estimated to be ~ 390 K. \n \n \n \n \n \n \n  \n300 350 400 4500.00.20.40.6Coercivity (T)\nTemperature (K)\n-0.8 -0.4 0.0 0.4 0.8Kerr Signal (a.u.)\nField (T) 292 K\n 303 K\n 313 K\n 424 K\n 431 K\n 454 K\n 523 Ka b \n24 V. RESULT OF PULSE ENERGY DEPENDENT MEASUREMENT IN  MRG \nSAMPLE WHERE AOS WAS NOT OBSERVED  \n \n \n \n \n \nFigure S5 : Typical Kerr microscope images of MRG after single laser pulse of various pulse \nenergies  were irradiated on different positions of the surface. The results show a multidomain \npattern has formed.  \n \nIn the main text, single -pulse all-optical switching ( SP-AOS) is presented for MRG samples \nhaving Tcomp above room temperature. No toggling was not observed for the MRG samples \nhaving Tcomp below room temperature. Figure S 5 shows typical Kerr micrographs after the \npulse , for different pulse energies.  The images reveal the presence of multidomain state with \nan onset at ~ 1 µJ.  This is different to the Mn 2Ru1.0Ga results, where a ring of switched domain \nwas observed around t he thermally demagnetized region.  \n \n \n \n \n \n \n \n  \n1.5 µJ 2.0 µJ 2.5 µJ\n3.0 µJ 3.5 µJ\n 4.0 µJ\n1.0 µJ\n50 μm \n25 VI. DETERMINATION OF THE SPOT SIZE AND THRSHOLD FLUENCE FOR \nSWITCHING  \n \n \n \n \n \n \n \nFigure S 6: Switched domain size as a function of pulse energy for Mn 2Ru1.0Ga. The red solid \nline is a fit to Eqn. 1.  \nAs mentioned in  the main text,  we have employ ed the growth of the switched domain size  with \nincreasing pulse energy for Mn 2Ru1.0Ga to calculate the  laser  spot size as well as the threshold \nfluence for switching using the Liu methodS1. This method exploits the fact that at the edge of \nthe magnetic contrast, the excitation fluence is equal to the threshold fluence. By assuming a \nGaussian pulse -shape , the switched area can be determined as,  \n 𝐴𝑆=𝐴0𝑙𝑛(𝐸\n𝐸𝑡ℎ)                                                                                                            ………(1)  \n where AS is the switched area corresponding to pulse energy E, A0 is the laser spot area and Eth \nis the threshold pulse energy. Figure S 6 presents the variation of the switched area as a function \nof pulse energy, from which the spot area and the threshold pulse energy are extracted by fitting \nwith Eqn. 1 to be 18600 µm2 and 1.4  µJ respectively. The threshold fluence is  then calculated \nby di viding the threshold pulse  energy with the laser -spot area  ,which yields ~7.5 mJ/cm2.  \n \n[S1] Liu , J. M. Simple technique for measurements of  pulsed Gaussian -beam  spot sizes. Opt.  \nLett. 7, 196 (1982 ).   \n2.0 2.4 2.8 3.2 3.6 4.08121620Spot Size(103 mm2)\nPulse Energy (10-6 J) \n26 VII. SP-AOS MEASUREMENT ON Mn 2Ru0.92Ga USING LASER PULSES OF \nWAVELENGTH 400 NM  \n \n \n \n \nFigure S7 : Single pulse all optical switching measurement performed on Mn 2Ru0.92Ga using \nlaser pulses of wavelength 400 nm and pulse energy 0.9 µJ. The number of laser pulses the \nregion is exposed to is labelled in each imag e.    \n \nIn order to substantiate the thermal origin of SP -AOS in MRG,  we examined the response of \nits magnetization to the laser pulses of different wavelength and polarization. In Fig. S 7 the \nresponse  of Mn 2Ru0.92Ga is shown  for laser light of wavelength 400 nm and up to five laser \npulses. The results are in line with the observation with 800  nm (See Fig. 2  in main text ). \nEssentially, a decrease in the excitation wavelength decreases the laser spot size, thereby \nincreasing th e thermal gradient across it. This directly affects the magnetization profile of the \nirradiated region and the switching is observed in a relatively narrow ring around the thermally \ndemagnetized area, as  compared to the 800 nm case.  \n \n \n \n \n \n \n  \n1 shot 2 shots 3 shots 4 shots 5 shots\n50 μm\n \n27 VIII. COMPARISON OF TEMPERATURE DEPENDENCE OF MAGNETIZATION \nIN MRG AND Gd FeCo \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S8: Representation of the sublatttice  magnetizations with respect to temperature  \ncalculated in the mean field approach for Mn 2Ru1.0Ga and Gd(FeCo)3. \n \n \n \n \n \n \n \n \n \n \n  \nMn 2Ru 1.0Ga \n Gd(FeCo) 3 \n  \n28 TABLE S1: Outcome of single -pulse excitation  in various  MRG samples. In some cases, where \nsimilar behaviour was found for films with similar composition, only one entry is shown. \nValues of Tcomp shown in bl ack are experimentally measured , while the others (shown in blue) \nare interpolated.   \nSample  Tcomp  (K) Coercive Field  (mT)  Switching Observed?  \nMn 2Ru0.5Ga 75  150  No \nMn 2Ru0.55Ga 80  170  No \nMn 2Ru0.60Ga 130 260  No \nMn 2Ru0.62Ga 145 350  No \nMn 2Ru0.63Ga 160 370  No \nMn 2Ru0.65Ga 165 440  No \nMn 2Ru0.7Ga 245 740  No \nMn 2Ru0.9Ga 310 > 1000  Yes \nMn 2Ru0.92Ga 315 > 1000  Yes \nMn 2Ru0.93Ga 320 > 1000  Yes \nMn 2Ru0.94Ga 325 > 1000  Yes \nMn 2Ru0.95Ga 375 600  Yes \nMn 2Ru1.0Ga 390 480  Yes \n  \n29 TABLE S2: MAGNETIC PROPERTIES OF Mn 2RuxGa AND Gd(FeCo )3 \nMagnetic properties of Mn 2RuxGa and  Gd(FeCo )3 (i.e. GdFeCo) . Tc and Tcomp are respectiv ely \nthe Curie and compensation temperatures. Mi is the magnetization of the sublattice i and Mnet \nis the net magnetization of the system. τi are the characteristi c demagnetization times for \nsublattices i = 4a, 4c  in MRG and i = Gd, FeCo in Gd(FeCo) 3. τe-l is the characteristic time \nassociated to the energy transfer between the hot electronic system and the lattice.  \n Mn 2Ru xGa Gd(FeCo) 3   \nStructure  Cubic Heusler XA  Amorphous  \nTc (K) 5001 (Mn 2RuGa)   \n5006 (Gd 22Fe9.8Co68.2) \n \nTcomp (K) 3901,2 (Mn 2Ru1.0Ga)  \n2505 (Gd 25Fe65.6Co9.4) \n \nM4a/FeCo  (kA . m-1) 7901 (Mn 2Ru1.0Ga) \n5503 (Mn 2Ru0.61Ga) 6407 (Gd 25Fe65Co10) \nM4c/Gd (kA . m-1) 8601 (Mn 2Ru1.0Ga) \n5903 (Mn 2Ru0.61Ga) 10007 (Gd 25Fe65Co10) \nMnet (kA . m-1) 702 (0 K) (Mn 2Ru1.0Ga) \n403 (0 K)(Mn 2Ru0.61Ga) 3607 (0 K)  (Gd 25Fe65Co10) \nτ4a/Gd (ps) 8.04 (Mn 2Ru0.7Ga)  \n0.435 (Gd 25Fe65.6Co9.4) \n \nτ4c/ FeCo (ps) 0.5 (Mn 2Ru1.0Ga),  \n0.5/8.04 (Mn 2Ru0.7Ga) 0.15 (Gd 25Fe65.6Co9.4) \n4.0/1506 (Gd 22Fe9.8Co68.2) \n \n         τe-l (ps) 2.04 (Mn 2Ru0.7Ga) 1.55 (Gd 25Fe65.6Co9.4) \n5.06 (Gd 22Fe9.8Co68.2) \n1 From static MOKE vs Temperature measurements.  \n2 From SQUID/XRD measurements.  \n3 Fowley et al.  Phys. Rev. B  98, 220406(R) (2018).  \n4 Bonfiglio et al.  ArXiv, 2003.01420 (2020).  \n5 Radu et al.  Nature 472, 205 -209 (2011).  \n6 Mekonnen et al.  Phys. Rev. B 87, 180406(R) (2013).  \n7 Estimated for atomic spins µGd =7.6 µB and µFe/Co =1.6 µB. \n "    },    {        "title": "2308.16806v1.Tunable_magnetic_domains_in_ferrimagnetic_MnSb__2_Te__4_.pdf",        "content": "Tunable magnetic domains in ferrimagnetic MnSb 2Te4\nTatiana A. Webb,1Afrin N. Tamanna,2Xiaxin Ding,2Jikai Xu,1Lia\nKrusin-Elbaum,2,∗Cory R. Dean,1,†Dmitri N. Basov,1,‡and Abhay N. Pasupathy1, 3,§\n1Department of Physics, Columbia University, New York, NY 10027, USA\n2Department of Physics, The City College of New York, New York, NY 10027, USA\n3Condensed Matter Physics and Materials Science Division,\nBrookhaven National Laboratory, Upton, New York 11973, USA\nHighly tunable properties make Mn(Bi,Sb) 2Te4a rich playground for exploring the interplay\nbetween band topology and magnetism: On one end, MnBi 2Te4is an antiferromagnetic topological\ninsulator, while the magnetic structure of MnSb 2Te4(MST) can be tuned between antiferromagnetic\nand ferrimagnetic. Motivated to control electronic properties through real-space magnetic textures,\nwe use magnetic force microscopy (MFM) to image the domains of ferrimagnetic MST. We find\nthat magnetic field tunes between stripe and bubble domain morphologies, raising the possibility of\ntopological spin textures. Moreover, we combine in situ transport with domain manipulation and\nimaging to both write MST device properties and directly measure the scaling of the Hall response\nwith domain area. This work demonstrates measurement of the local anomalous Hall response using\nMFM, and opens the door to reconfigurable domain-based devices in the M(B,S)T family.\nThe recent discovery of MnBi 2Te4(MBT) [1–5] was\na breakthrough to realize the quantum anomalous Hall\neffect in a stoichiometric crystal, avoiding the need for\ndisorder-inducing magnetic dopants [6–9]. In addition,\ncrystals are exfoliatable down to few layer thicknesses\nenabling integration into Van der Waals heterostruc-\ntures with well developed fabrication techniques [1, 10–\n12]. This discovery was rapidly followed by work ex-\ntending MBT into a family of materials with highly\ntunable properties via crystallographic and chemical\nparadigms [4, 5, 13–21]. Substituting Sb for Bi changes\nthe doping from n-type to p-type [16, 22]. But surpris-\ningly, within MST, the magnetic order can also be tuned\n(via the concentration of magnetic defects) from A-type\nantiferromagnetic seen in MBT, where planes of Mn mo-\nments are aligned ferromagnetically (antiferromagneti-\ncally) within the plane (between planes), to ferrimag-\nnetic with net out of plane magnetization [19, 21, 23–25].\nThe ability to tune the effective inter-plane coupling from\nantiferromagnetic to ferromagnetic strongly suggests the\npresence of magnetic frustration in M(B,S)T, raising the\npossibility of stabilizing other interesting magnetic or-\nders [26, 27].\nThe ability to tune magnetic order in the M(B,S)T\nfamily opens more conventional applications of magnetic\nmaterials, where intense efforts have gone into develop-\ning materials structures with interdependent magnetic\nand electronic properties for control of charge and spin\ntransport (e.g. magnetic data storage and spintron-\nics). Growing evidence suggests that the low energy\nbands of MST are sensitive to the details of magnetic\norder [3, 19, 24, 28–30], but we do not yet have a de-\ntailed understanding of the correlation between electronic\n∗krusin@sci.ccny.cuny.edu\n†cd2478@columbia.edu\n‡db3056@columbia.edu\n§apn2108@columbia.eduproperties and real-space magnetic textures in MST. So\nfar, the use of magnetism to control electronic prop-\nerties in magnetic topological materials has been ex-\nplored primarily in terms of topological phase transi-\ntions (e.g. [10, 12, 28, 31]) and manipulating chiral edge\nmodes of the quantum anomalous Hall effect [11, 32, 33].\nEdge mode manipulation has been demonstrated via\nmagnetic domains in Cr-doped (Bi,Sb) 2Te3[32, 33] and\nvia layer-dependent magnetization in antiferromagnetic\nMBT [11], but to the best of our knowledge, the abil-\nity to write arbitrary-shaped domains in ferrimagnetic\nM(B,S)T compounds has yet to be investigated. In this\nwork, we therefore set out to investigate what magnetic\ntextures can be realized in MST, and the prospects for\nmanipulating the local magnetization to create config-\nurable devices. Specifically, we use the domain imag-\ning and writing capabilities of magnetic force microscopy\n(MFM) combined with in situ transport to directly mea-\nsure the device response to local changes in magnetiza-\ntion.\nOur interdependent transport and magnetic measure-\nments were performed on an exfoliated flake of ferrimag-\nnetic MST (Figure 1a) with average thickness 84 nm and\n±10 nm variations (Supporting Information SII). Four\ngold contacts were used to measure longitudinal Rxxand\nHall Rxyresistance. Immediately after device fabrica-\ntion, Rxxshowed a peak at 27 K on cooling, consistent\nwith typical Curie temperatures seen in MST. At 2K,\nthe hysteretic loop in Rxyand peaks in Rxxas a func-\ntion of magnetic field showed a coercive field near 10 mT\n(Supporting Information SIV). Refer to Supporting In-\nformation SI for additional details of sample fabrication\nand characterization.\nTo characterize the magnetic domains in MST, we per-\nformed MFM in a cryogenic atomic force microscope\n(AFM) with variable magnetic field Bextnormal to the\nsample surface. Figure 1c-i shows a MFM image of the\nzero-field cooled (ZFC) sample at 5 K. Because the co-\nercive field of MST is so low, we quenched the supercon-arXiv:2308.16806v1  [cond-mat.str-el]  31 Aug 20232\n12 3\n4\n10 μm(a)\n(e)\n(b)\n50\n25\n025 50\nB (mT)1\n01Rxy( )\nbefore MFM\nafter MFMiiiiiiiv\nvivΔf\n1.5 Hz\n-20 mT (v)\n(iv) 50 mT\n(vi) -30 mT\n(c) ZFC\n(ii) 12 mT\n (i)\n2 μmBext = 0 mT\n(iii) 24 mT\nxyxy(d)\n0 mT -20 mT FT\nkxky\n0 20\n|k| ( m1)\n0.00.51.01.52.0Normalized FT amplitude\n0mT ZFC\n-20.0mT~0mT FC\nFIG. 1. Evolution of stripe domains under Bext:(a)Optical micrograph of the MST device showing the MST flake with\n4 contacts for transport measurements in a Van der Pauw geometry. The arrows show the scan axes for the MFM images. (b)\nMagnetic field Bextdependence of the Hall resistance Rxymeasured at 5 K starting from a zero-field cool. Measurements were\nrecorded before (open symbols) and after (filled symbols) MFM imaging. The light orange lines are guides to the eye showing\nthe order of data acquisition. (c)Constant height MFM measurements of the magnetic domains at the center of the MST\ndevice recorded at 5 K after zero-field cooling. Measurements were interspersed with RxyandRxxmeasurements shown in (b)\nand in the Supporting Information SIV. The arrows indicate the order of data acquisition. The color scale on all images is\n1.5 Hz, but the zero values have been offset. The tip was lifted 300 nm above the SiO 2surface. (d)Amplitude of the Fourier\ntransforms of (c-i) and (c-v) after mean value subtraction. (e)Angular averaged amplitude of the Fourier transformed MFM\ndata.\nducting magnet prior to cooling the sample to ensure a\ntrue ZFC with no trapped flux. The MFM image is a\nmeasurement of ∆ f, the resonance frequency shift of the\nAFM cantilever due to the interaction of the sample’s\nstray fields with the cantilever’s magnetic tip, so we ex-\npect images to primarily detect the domain structure of\nthe ferromagnetically aligned components of MST’s fer-\nrimagnetic ordering [34, 35]. Correspondingly, the ZFC\nimage shows disordered maze-like stripe domains (Fig-\nure 1) consistent with ferromagnetic ordering in the out\nof plane direction, similar to domain images from Ge et\nal [25]. Applying magnetic field Bextnormal to the sam-\nple surface polarizes the sample (c-i to iv), increasing the\narea of the domains aligned with the field until at 50 mT,\nonly a single domain remains, giving a uniform MFM sig-\nnal. In situ transport measurements show an associated\nincrease in Rxyfrom 0.04 to 1.42 Ω. Over this range of\nBext, the contribution to Rxyfrom the linear Hall effect\nis negligible, so the change in Rxyis primarily due to the\nanomalous Hall Effect (AHE) [36] (Supporting Informa-\ntion SVII).\nReversing the magnetic field (Figure 1c-iv to vi), we\nobserve the reformation of stripe domains at -20 mT\nasRxydrops and changes sign, indicating the reversal\nof the magnetization. These field-reversed domains are\nsignificantly less disordered than the ZFC domains. To\nquantify the difference, we examine the Fourier Trans-\nforms (FT) of the ZFC and -20 mT images, shown in\nFigure 1d. Both exhibit a ring shape, or a peak in the\nangular-averaged FT (Figure 1e), indicating the domainshave a characteristic length scale, as expected from the\nenergetics of domain formation [37–39]. The peak occurs\nat wavevector |k|6.3µm−1with standard deviation σ\n4.8µm−1for the ZFC domains and |k|= 5.7 µm−1with\nσ= 2.1 µm−1for the -20 mT domains. The broader peak\nassociated with the ZFC domains indicates that during\ncooling the domains form features with a wider range of\nlength scales compared to during magnetization reversal\nat low temperature.\nTo further explore how an external field can tune the\ndomain morphology, we cooled the MST device from 35 K\nbelow Tcunder |Bext|up to 15 mT (Figure 2). With\n|Bext|larger than 10 mT, a single domain forms across\nthe entire MST flake. However, when we nominally zero\nthe magnet’s current such that a small Bext≈0 exists\nonly from trapped flux, we see circular features in the\nMFM, indicating the formation of bubble rather than\nstripe domains. MST is thus remarkably sensitive to\nsmall magnetic fields. At intermediate |Bext|(10 mT),\nthe domains formed are not uniform in size and shape,\nand it is not clear if they are intrinsically bubbles or\nstripes. We now focus on Bext≈0, where bubbles are\nclearly observed.\nThe bubble domains are highly disordered (Figure 2e).\nThe nearly isotropic Fourier transform (f) shows no\nevidence for lattice organization, and the distribution\nof wavevectors centered at |k|= 11.3 µm−1with\nσ= 7µm−1is extremely broad. Correspondingly, the\ncircular MFM features range in size from below 100 nm\nto above 300 nm, as shown by the distribution of full3\n150 300450\nd (nm )1502503504502Bz/z2FWHM (a.u.)\n5 µm(a) Bext ~ 0 -10 mT -12 mT -15 mT (d) (c) (b)\nSiO2MSTΔf\nkxky (f)\n(e)\n2 µm\n(g) (h) (i)\n(j)(k)(l)\nx (nm )250\n0 2501\n01f(Hz)\n0 100 200 300 400\nMFM FWHM (nm )0102030Count\ndz\nt\n332 nmFWHM\n97 nm197 nmxy\nFIG. 2. Field cooled domain structures. (a-d) Constant height MFM images of the magnetic domains in the MST device\nunder field cooling with the indicated Bext. The tip was lifted 300 nm above the SiO 2surface. The range and offset of the\ncolor scale has been chosen for each image independently. Color scale range: 3.7 Hz (a), 1.5 Hz (b), 1.4 Hz (c), 1.5 Hz (d).\nTemperature: 5 K (a), 10 K (b-d). (a)Bext∼0 indicates that a small unknown residual flux from the superconducting magnet\nwas present. (e)Smaller scale constant height MFM image after cooling under Bext∼0 showing clear bubble shaped domains.\nColor scale range: 4.4 Hz. Temperature: 5 K. The images in (a) and (e) are from separate cooling cycles with the same Bext.\n(f)Amplitude of the Fourier transform of (e) after mean value subtraction. (g-i) Zooms of 3 regions in (e) showing a large,\nmedium, and small size bubble. (j)Profiles through the large, medium, and small size bubbles from (g-i). (k)Histogram of the\nfull-width-at-half-max (FWHM) of the bubbles imaged via MFM, determined from horizontal and vertical profiles through all\nresolvable bubbles in (e). The MFM FWHM is not directly interpretable as the bubble domain size. (l)FWHM of ∂2Bz/∂z2\nfor the stray magnetic field generated by a bubble domain with diameter d, using sample thickness t= 81.2 nm and z= 300 nm\nmeasured from the bottom of the material.\nwidth at half maxima (FWHM, Figure 2k). The size of\nthe features seen in MFM cannot directly be interpreted\nas the size of the domains in the MST. Approximating\nthe tip as a point dipole with small oscillation amplitude,\nthe MFM image ∆ fis proportional to ∂2Bz/∂z2arising\nfrom the sample’s stray field [34, 35]. To help interpret\nthe MFM features, we model the stray field for a single\ncylindrical bubble domain at a representative height z.\nAs the domain diameter decreases below z, the spatial\npeak in ∂2Bz/∂z2decreases in intensity ( Supporting In-\nformation SV) and the FWHM saturates at a lower limit\nnear 150 nm (Figure 2l). The FWHM does not decrease\nlinearly in the domain diameter for small bubbles. Re-\nturning to the MFM data, we therefore expect small bub-\nbles may not be detectable due to weak intensity, and for\nslightly larger bubbles, the apparent size in MFM may\nsaturate at a lower limit larger than the domain diame-\nter. The MFM data, however shows bubbles with FWHM\nbelow the expected 150 nm cuttoff, likely because the\nwidth of the low intensity bubbles can be dominated by\nthe positions of the neighboring bubbles. Under repeat\ncooling, we find that some but not all bubbles form in\nthe same location (Supporting Information SIX), whichalong with their disordered organization could indicate\nsignificant pinning, either due to crystal inhomogeneity,\nor extrinsic factors such as strain. The observation of\nbubble domains under field cooling, but not when sweep-\ningBextat low temperature suggests that the bubble and\nstripe morphologies are separated by a significant energy\nbarrier, likely associated with nucleating a domain wall.\nThe domain wall structure (i.e. Bloch or N´ eel) can pro-\nduce topologically non-trivial chiral spin textures on bub-\nble domains [40], and topological bubble and skyrmion\nphases have been reported in other (Bi,Sb) 2Te3-based\nmaterials [41, 42]. While our detection of bubble do-\nmains opens the possibility of stabilizing topological spin\ntextures in M(B,S)T, our MFM measurements do not al-\nlow us to draw a conclusion about the topology of the\nbubbles. We observed no evidence of a topological Hall\neffect (THE) – a deflection of carriers due to the real\nspace Berry curvature of topological spin textures – in\ntheRxyhysteresis loops when sweeping Bextto flip the\nsample magnetization at low temperature. However, un-\nlike many skyrmion materials that display a THE as the\nskyrmion phase is formed over a finite range of Bat con-\nstant temperature, the MFM data in Figure 1 does not4\n0 2 4 6 8\nTip distance ( m)\n0.02.5V13(nV)\n(b)\nIV13\n12 3\n4\n(d)\n(c)\n(a)\nΔf V13\n12 nV1.13 Hz\nxy\n2 μm\nFIG. 3. Impact of a single domain on transport (a)\nMFM image showing a domain flipping during the scan, in-\ndicated by the black arrow. Same as Figure 1(c-vi). (b)V13\nmeasured simultaneously with (a), using a 500 nA amplitude\nAC source current. The black arrow indicates the location of\nthe domain flip, identical to the arrow in (a). Inset: Schematic\nof the V13measurement. (c)Zoom of (a) on the area show-\ning the domain flip. The fast scan direction is vertical. The\ndomain abruptly disappears from one vertical scan line to the\nnext. (d)V13averaged vertically along the fast scan direc-\ntion to show the jump in value that occurred as the domain\nflipped. The value from the first scan line has been subtracted\nto show the change ∆ V13.\nshow the bubble morphology when sweeping Bextat low\ntemperature. Further work is therefore required to de-\ntermine if the bubble domains formed on field cooling are\ntopological or trivial.\nWe have seen how the domain morphology can be con-\ntrolled with Bext; now we investigate the possibility of\nusing the stray field from the magnetic MFM tip, Btip,\nto locally manipulate the domains in MST. To reduce\nthe influence of Btipon the sample, the domain imag-\ning discussed so far was done with the tip lifted high\n(roughly 200-230 nm) above the MST surface. However,\nwhen Bextis near the coercive field, small changes in\nthe magnetic field can have a large influence on the sam-\nple magnetization, and even 200 nm from the tip, Btip\ncould be on the order of 10 mT [43], comparable to the\ncoercive field. Correspondingly, small changes in Rxy\nandRxxduring MFM imaging (Figure 1b, Supporting\nInformation SIV) demonstrate that the tip mildly influ-\nenced the sample magnetization. Moreover, tip-induced\ndomain flips are seen in some images as a domain that\nabruptly disappears partway through imaging. To quan-\ntify the tip’s influence, we applied an AC current between\ncontacts 2 and 4, and measured the induced transverse\nvoltage V13across contacts 1 and 3 during MFM imag-\ning. The MFM image in Figure 3a shows a domain flip,\nand the simultaneously acquired V13image (b,d) shows\nan abrupt change by more than 2 nV at the same loca-\ntion, demonstrating a measurable impact of the domain\nflip on the device transport. Interestingly the tip-induceddomain flips are not always in the sense of aligning the\ndomain with the tip, suggesting that the spatial gradient\nor time-dependence of Btipmay be equally important or\nmore important for overcoming energy barriers compared\nto the local Zeeman energy term.\nWe can harness the tip’s influence to controllably write\ndomains by bringing the tip close to the MST surface, in-\ncreasing Btip. For this purpose, we first used Bextto pre-\npare the sample with magnetization anti-aligned to the\ntip (Figure 4c). After zeroing Bext, we then brought the\ntip into amplitude-controlled feedback on the MST sur-\nface ( Btipon the order of 50 mT [43]), and moved the tip\nacross the surface to write a domain aligned with the tip.\nIn Figure 4, we show both linear (e) and square (h) areas\nwritten with the MFM tip, demonstrating that both nar-\nrow 1D-like and 2D domains can be written. During the\nwrite process, the square area formed a mixed domain\nstate, suggesting that because the mixed domain state is\nenergetically favored at Bext= 0, there is a maximum\nsingle-domain area of roughly several µm2(Supporting\nInformation SXI) that can be written. Decreasing tem-\nperature to increase the importance of the domain wall\nnucleation energy may increase that area.\nBy inverting the magnetization of a small area locally\nwith our MFM tip, we can directly probe that area’s\nimpact on the AHE. During domain writing we there-\nfore recorded V13as a proxy for Rxy(Supporting Infor-\nmation SVIII). While writing the line domain, V13de-\ncreased linearly (Figure 4d), matching the area-scaling\nthat one would expect for AHE contributions [36] that\nscale linearly with the sample average magnetization. V13\nrecorded while writing the square domain is also consis-\ntent with area scaling. Here, V13forms two 2D images\nfor forward (Figure 4f) and backward scans–to write the\ndomain, the tip moved up and down along each scan line\nbefore advancing one pixel at a time left to right. Typi-\ncally, V13has a finite slope on the forward pass (the blue\nhistogram in i is peaked at 0.6 nV /µm), confirming that\nthe tip is writing a magnetization, but not on the back-\nward pass (orange histogram, peaked at zero). Consid-\nering that typically each scan line advances the domain\nwall by one pixel width (53 nm), we can quantify the local\nAHE: 11 nV/( µm)2. This value is quantitatively consis-\ntent with both: (1) the linear V13seen when writing the\nline domain, and (2) the ratio of the change in V13from\nbefore to after the write step to the domain area imaged\nvia MFM (Supporting Information SXI). Moreover, the\nentire evolution of V13during the square write can be un-\nderstood in detail as a linear decrease (gray dashed line\nin g) from writing the red domain plus deviations from\nforming the inner blue domain, in abrupt steps initially\nbut then more smoothly near the end of the write.\nWe have therefore demonstrated a direct measurement\nof the scaling of the anomalous Hall effect with domain\narea. The technique can also in principle measure devi-\nations from this area scaling to probe local properties of\ninhomogenous devices (spatially varying magnetization\nor Berry curvature). Within homogenous materials, the5\n5 μm\n2 μm\nΔf(c) (a)\n(b)\n(d)\n(e)0 5 10\nTip distance ( m )\n50\n0V13(nV)\n(i)\n(j)\n10\n5\n0 5 10\ndV13/ds(nV / m)\n0102030Count\nbwd\nfwd\n5 μm\n2 μm(f)\n(g)\n(h)\nV13\n626 nV\n0 2 4 6 8\nTip distance ( m )\n0.5\n0.0V13( V)\nfwd\nbwd\nimagedomain\t\nwriteI1\n2 34V13x\ny\n0.5\n0.0 0.5\nDistance ( m)\n0.60.81.01.21.41.61.8f (Hz)\nFIG. 4. Writing magnetic domains. (a) Schematic of the device and the V13measurement used as an approximation of\nthe Hall response during domain writing. (b)Schematic of the magnetic tip over the MST sample during domain writing, with\nthe tip close to the surface, and during domain imaging, with the tip lifted high above the surface. (c)MFM showing the MST\ndevice has no domains, and has magnetization anti-aligned with the tip after ramping Bextto -50 mT and then 0 mT. Constant\nheight imaging was done with the tip lifted 300 nm above the SiO 2surface. The overlays show the approximate locations of:\nthe electrical contacts (orange lines), the tip trajectory for writing the line domain (orange arrow), and the MFM image of the\nline domain (gray box). Color scale range: 1.1 Hz. (d)V13measured while writing the line domain. 500 nA amplitude AC\nsource current. (e)Constant height MFM image after writing the line domain, with the tip lifted approximately 200 nm above\nthe MST surface. The red line indicates the location of the cut shown in (j). Color scale range: 1.9 Hz. (f)V13recorded as a\nfunction of the tip position while attempting to write a square area. See main text. 500 nA amplitude AC source current. (g)\nV13from (f) averaged vertically along the fast scan direction. The value of the first point was subtracted to show the change\n∆V13. The gray dashed line is the expected linear trend if the tip were fully polarizing the sample. (h)Constant height MFM\nimage after writing the square area. Tip lifted 300 nm above the SiO 2surface. The white lines indicate the locations of the cuts\nshown in (j). Color scale range: 1.6 Hz. (i)Histograms of the slopes of V13during each forward and backward vertical scan\nline while writing the square area (f). The vertical lines mark the slope while writing the line domain (red), and the expected\nper-pixel slope (blue) calculated from the overall change in V13and the domain area in (h). (j)Line cuts through the MFM\nimages of the line domain (red) and of the square area (gray). All cuts have been offset for comparison, and the cuts from the\nsquare domain have been inverted. Temperature: 10 K.\narea-scaling contributions and deviations represent bulk\nand boundary contributions, meaning that this technique\ncan be used to probe topological effects such as dissipa-\ntionless chiral edge conduction in a Chern insulator or\nthe topological Hall effect from chiral spin textures at\ndomain walls.\nThis work opens the door to making programmable\nmagnetic devices within ferrimagnetic compounds in the\nM(B,S)T family. M(B,S)T could be a platform for\nwritable chiral currents (e.g. [11, 32, 33]) either in a mag-\nnetic Weyl semimetal or Chern insulating state (mul-\ntiple potential band topologies have been predicted in\nMST [3, 16, 19, 22, 29, 30]). The ability to tune mag-\nnetic domains in a compound that retains magnetic or-\nder when exfoliated to the few layer limit [1, 16, 31, 44]\nraises the possibility of using M(B,S)T to introduce pro-\ngrammable magnetic landscapes (e.g. supperlattices orboundaries made of magnetic gradients) on length scales\nof 100s of nanometers to micrometers into van der Waals\nheterostructures. Generically, the tip writing process\nallows us to locally move between different metastable\nmagnetic configurations that are separated by energetic\nbarriers. So beyond writing individual domains of uni-\nform magnetization explicitly, the tip influence could be\ncombined with external fields and temperature to stabi-\nlize and write areas of non-uniform spin textures (just as\nthe mixed domain state formed in our square area was\nnot uniform) in order to create functional devices based\non boundaries between magnetic phases.6\nSUPPORTING INFORMATION\nAdditional experimental details, characterization of\nthe sample topography, images of stripe domains, elec-\ntrical transport measurements, analysis and modelling of\ndomain length scales, analysis of the repeatability of bub-\nble domain locations, and analysis of the scaling of the\nAHE with domain area (PDF)\nACKNOWLEDGMENTS\nWe thank Zachariah Addison and Nishchhal Verma\nfor helpful discussions. This work was supported by theAir Force Office of Scientific Research via grant FA9550-\n21-1-0378 (T.A.W., A.N.P.) and by NSF grants DMR-\n2210186 (D.N.B) and HRD-2112550 (L.K.-E.). Research\non topological properties of moir´ e superlattices is sup-\nported as part of Programmable Quantum Materials, an\nEnergy Frontier Research Center funded by the U.S. De-\npartment of Energy (DOE), Office of Science, Basic En-\nergy Sciences (BES), under award DE-SC0019443. Sam-\nple synthesis is supported by the the NSF MRSEC pro-\ngram through Columbia University in the Center for\nPrecision-Assembled Quantum Materials under award\nnumber DMR-2011738.\n[1] Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H.\nChen, and Y. Zhang, Quantum anomalous Hall effect in\nintrinsic magnetic topological insulator MnBi 2Te4, Sci-\nence367, 895 (2020).\n[2] M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Es-\ntyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B. Wolter,\nA. V. Koroleva, A. M. Shikin, M. Blanco-Rey, M. Hoff-\nmann, I. P. Rusinov, A. Y. Vyazovskaya, S. V. Ere-\nmeev, Y. M. Koroteev, V. M. Kuznetsov, F. Freyse,\nJ. S´ anchez-Barriga, I. R. Amiraslanov, M. B. Babanly,\nN. 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Pasupathy1, 3\n1Department of Physics, Columbia University, New York, NY 10027, USA\n2Department of Physics, The City College of New York, New York, NY 10027, USA\n3Condensed Matter Physics and Materials Science Division,\nBrookhaven National Laboratory, Upton, New York 11973, USA\nCONTENTS\nSI. Methods 2\nA. Crystal growth and structural characterization. 2\nB. Magnetic force microscopy 2\nC. Electrical transport measurements 2\nD. Modeling bubble domains 2\nSII. Sample topography 4\nSIII. Evolution of stripe domains under Bext 5\nSIV. Transport measurements 6\nSV. Modeling bubble domains 7\nSVI. Domain length scales 8\nSVII. Area of stripe domains 9\nSVIII. V13as a proxy for Rxy 10\nSIX. Repeatability of bubble locations 11\nSX. Bubble and stripe domains in bulk MST 12\nSXI. Area scaling of the Anomalous Hall Effect 14\nReferences 14arXiv:2308.16806v1  [cond-mat.str-el]  31 Aug 20232\nSI. METHODS\nA. Crystal growth and structural characterization.\nCrystals of nominally MnSb 2Te4were grown out of a Sb–Te flux [1, 2]. Mixtures of Mn (Alfa Aesar, 99.99%) and\nSb pieces (Alfa Aesar, 99.999%), and Te shot (Alfa Aesar, 99.9999%) in the molar ratio of 1:10:16 (MnTe:Sb2Te3 =\n1:5) were placed in a 2 ml alumina growth crucible and heated to 900 C and held for 12 h. After slowly cooling across\na∼10 degree window below 600 C in two weeks, the excess flux was removed by centrifugation above the melting\ntemperature of Sb 2Te3(≥620 C). Crystals produced by this flux method were typically a few mm on a side and often\ngrew in thick, block-like forms with thicknesses up to 2 mm but were easily delaminated.\nEDX: Energy Dispersive X-ray (EDX) microanalysis was performed in the Zeiss Supra 55, a field emission SEM with\na maximum resolution of 1 nm. The ferrimagnetic (FM) MST stoichiometry was determined as Mn:Sb:Te=1.3 : 2.9\n: 5.8. XRD: X-ray diffraction of crystals was performed in a Panalytical diffractometer using Cu Ka ( λ= 1.5405 ˚A)\nline from Philips high intensity ceramic sealed tube (3 kW) X-ray source with a Soller slit (0.04 rad) incident and\ndiffracted beam optics. The determined c-axis parameter 40.898 ˚Awas consistent with space group, R-3m as reported\nin literature. Magnetization: D. c. magnetization measurements were performed using Quantum Design SQUID\nMagnetometer in up to 5.5 T fields. In all magnetic measurements the samples were supported in gelcaps without\nany substrates. From the fits to Curie-Weiss law, the magnetic moment in as-grown crystals was determined as\nµeff= 5.36µB/mole.\nB. Magnetic force microscopy\nMagnetic force microscopy (MFM) measurements were performed in an Attocube cantilever-based cryogenic atomic\nforce microscope (attoAFM I with attoLIQUID 2000 cryostat), where the microscope sits in a helium exchange gas\nat low temperature. Nanosensors PPP-MFMR probes with hard magnetic coating, resonant frequencies near 75 kHz,\nand force constants near 2.8 N /m were used for all measurements. All low temperature measurements on the MST\ndevice were were performed using a single AFM probe with cantilever resonance at 78 kHz. MFM images record the\nresonant frequency shift of the probe cantilever (∆ f). Constant height MFM images were taken with the tip at a fixed\nheight above the plane of the SiO 2surface. Constant lift MFM images were taken by passing twice over each line.\nThe first pass, in amplitude-controlled feedback, recorded the surface topography. On the second pass, the tip was\nlifted by a constant offset from the topographic pass to record ∆ f. In this way, topographic and MFM images were\nrecorded over the same field of view in a line-by-line interleaved fashion. During MFM imaging, the tip oscillation\namplitude was typically 35 nm to 55 nm. MFM data shown is raw data unless otherwise noted.\nC. Electrical transport measurements\nEx situ transport measurements were performed in a 14 Tesla Quantum Design Physical property measurement\nsystem (PPMS) in 1 Torr (at low temperature) of He gas. Crystals were mechanically exfoliated onto 285 nm SiO 2/Si\nwafers. Electrical contacts in the van der Pauw (vdP) configuration were photo-lithographically patterned and a\nsputtered Au metallurgy was used. Conformal Au coating amply covered side surfaces in order to make good contacts\nto top and bottom surfaces. The vdP DC measurements were carried out using a custom-configured electronic system\nin which four measurement configurations are switched by a Keithley scanner, with the current direction reversal\nemployed for each measurement to minimize thermal emf.\nIn situ electrical tranport measurements were done using a Signal Recovery model 7265 lock-in amplifier to source\na 17.777 Hz AC voltage, and measure the response voltage at the same frequency. A 10 MΩ resistor was used to\nconvert the voltage source into a current source. According to the vdP technique [3], Rxxwas calculated from four\nmeasurements, V23,14,V12,43,V41,32, and V34,21, and Rxyfrom two measurements, V31,24andV24,13, where Vab,cd is\nthe voltage measured from c to d while applying the source current from a to b, and the contacts are numbered in\nFigure 1a of the main text. For brevity, we refer to V31,24andV24,13asV24andV13, respectively.\nD. Modeling bubble domains\nCalculations of the stray magnetic field arising from a bubble domain were done using Magpylib [4]. The sample\nmodel consisted of a rectangular slab of uniform magnetization with a cylinder of the same thickness located at the3\ncenter. The cylinder had twice the magnetization with the opposite sign, such that the net magnetization inside the\ncylinder was equal to that of the rectangular slab, but opposite in direction. The lateral size of the rectangular slab\nwas taken to be 100 mm in both directions, large enough to avoid edge effects in the vicinity of the bubble domain.\nAfter calculating the stray magnetic field above the sample surface, numerical differentiation was used to calculate\nthe stray field gradients.4\nSII. SAMPLE TOPOGRAPHY\n0 50 100\nZ (nm)0246Count\n70 75 80 85 90\nZ (nm)012Count\n17 nm\n2 μm\n129 nm(a) (b) (c)\n(d)\n2 μm\nFIG. S1. (a, b) AFM topographies (tapping mode) of the full MST device at ambient (a) conditions, and of the center\nof the device, with the 4 contacts at the corners of the image, at 35 K (b). Background subtraction: Line-by-line constant\nsubtraction based on the median of the difference between consecutive lines, followed by plane subtraction to level the image.\n(c)Histogram of (a), showing 4 peaks associated with the SiO 2surface, the gold contacts on SiO 2, the MST surface, and the\ngold contacts on MST. (d)Zoom of (c) showing just the peak from the MST surface. For the MST surface only, the mean\nheight is 83 nm with standard deviation 4 nm. Considering only the center of the MST device, where most MFM imaging was\nperformed, the mean height is 81 nm with 4 nm standard deviation.5\nSIII. EVOLUTION OF STRIPE DOMAINS UNDER Bext\n(n) -30 mT \n(k) 50 mT\n(l) -10 mT\n(m) -20 mT\n(c) 8 mT\n(e) 14 mT\n(d) 12 mT\n(b) 4 mT\n(a) 0 mT (ZFC)\n(f) 16 mT\n (g) 18 mT\nΔf1.5 Hz\n(h) 20 mT\n (i) 24 mT2 μm\n(j) 32 mT\nFIG. S2. (a-n) Constant height MFM images of the magnetic domains at the center of the MST device recorded at 5 K after\nzero-field cooling. Measurements were interspersed with RxyandRxxmeasurements shown in Figure 1b and in Figure S3d.\nThe arrows indicate the order of data acquisition. The color scale on all images is 1.5 Hz, but the zero values have been offset.\nThe tip was lifted 300 nm above the SiO 2surface.6\nSIV. TRANSPORT MEASUREMENTS\n50\n25\n0 25 50\nB (mT)1.5\n1.0\n0.5\n0.00.51.01.5Rxy( )\n50\n25\n0 25 50\nB (mT)125.2125.4125.6125.8126.0126.2Rxx( )\n50\n25\n0 25 50\nB (mT)125.0125.2125.4125.6125.8Rxx( )\n8\n6\n4\n2\n0\nB (T)3\n2\n1\n01Rxy( )\n8\n6\n4\n2\n0\nB (T)123.0123.5124.0124.5125.0125.5126.0Rxx( )\n0 50 100 150 200 250 300\nT (K)140160180200Rxx( )\nex-situ PPMS\nMFM in-situ(a)\n(f) (e) (d)(c) (b)\nbefore MFM\nafter MFM\nFIG. S3. (a)Temperature dependence of Rxx, measured ex situ just after sample fabrication (gray line). The orange symbols\narein situ measurements in the the MFM using AC source current with amplitude between 100 nA and 500 nA. (b-f) Magnetic\nfield dependence of RxxandRxyat low temperature measured ex situ just after sample fabrication at 2 K (gray line), and in\nsituin the MFM at 5 K (orange symbols). (e) and (f) are the same data as (b) and (c), but displayed over a smaller field range\nin order to make the hysteresis visible. (d)AC source current amplitude: 500 nA.7\nSV. MODELING BUBBLE DOMAINS\n100 150 200 250 300 350 400 450\n2Bz/z2FWHM (nm)\n012342Bz/z2(a.u.)\n100 200 300 400\nMFM bubble FWHM (nm)1.0\n0.5\n0.00.51.0MFM bubble intensity (Hz)\n051015202530\n(a)(b)\n(c)\n400\n200\n0 200 400\nx(nm)1\n01f(Hz)\nFWHM=97nm\n197nm\n332nm(d) FWHM=197nm\n200\n0 200\nx (nm)1.5\n1.0\n0.5\n0.02Bz/z2(a.u.)\n(e) FWHM=332nm\n200\n0 200\nx (nm)1.0\n0.5\n0.02Bz/z2(a.u.)\n(f)\nFIG. S4. (a) Constant height MFM image repeated from Figure 2e, with the bubble locations and perimeters used in\nFigure 2k overlaid. The points show the bubble locations determined as the local minima of the image. The ellipses have\nmajor and minor axes terminated by the 4 half-minima positions in the x- and y-directions. The size of the major and minor\naxes are the FWHM in the x- and y-directions. Points without ellipses denote bubbles at the edge of the image for which the\nFWHM could not be reliably determined, and these bubbles were therefore ignored in the data analysis. While most bubbles\nare well-described by the ellipses, there are several where the representation is inaccurate due to the influence of neighboring\nbubbles on the x- and y-direction linecuts. (b)Histogram of the intensities and FWHM of the bubbles in (a). The vertical\nand horizontal FWHM were included independently, such that each bubble is counted twice. (c)Calculated ∂2Bz/∂z2peak\nintensity as a function of FWHM for a ferromagnetic cylindrical bubble domain. Note that the FWHM in ∂2Bz/∂z2on the x\naxis is related to the domain diameter by Figure 2l. The height zabove the ferromagnet, and the thickness hof the ferromagnet\nare related by z+h= 300nm for comparison to the MFM data. The thicknesses are h= 81.2nm (black), h= 76.7nm,85.6nm\n(orange), h= 71.2nm,91.4nm (blue), which correspond to the mean, 15.9, 84.1, 0.5, and 99.5 percentiles of the MST’s height\ndistribution in the approximate location of (a). (d)Linecuts through 3 bubbles in the MFM image repeated from Figure 2j.\n(e, f) Linecuts through the peak in ∂2Bz/∂z2for the same handzvalues as in (c), and for two of the FWHM values shown\nin (d). The model is a good description of the mid-size bubbles (e), but the model shows a distinct flat top for large bubbles\n(d), which differs from the MFM data. Discrepancies between the model and data could arise from: (1) The size and shape\nof the tip’s magnetic coating, (2) The presence of neighboring bubbles, (3) The finite amplitude of the tip oscillation, (4) The\nfinite size of domain walls.8\nSVI. DOMAIN LENGTH SCALES\n0 10 20 30 40 50\nk (μm-1)0250500750MFM FT (a.u.)\n0200400MFM FT (a.u.)\n050100MFM FT (a.u.)\n(a)\n(b)\n(c)\nFIG. S5. Angular averaged amplitude of the Fourier transforms of MFM-imaged stripe domains during magnetization reversal\natBext=−20mT (a) and after zero-field cooling (b), as well as MFM-imaged bubble domains after field-cooling (c). The\noriginal MFM images are Figures 1c-v, 1c-i, and 2e of the main text. The blue lines are fits to a Gaussian peak with linear\nbackground giving the peak locations and standard deviations listed in the main text.9\nSVII. AREA OF STRIPE DOMAINS\n(a) (b)\n40\n20\n0 20 40\nB (mT)1\n01Rxy( )\n0.00.20.40.60.81.0\nMFM fraction\n50\n0 50\nThreshold offset (mHz)0.00.20.40.60.81.0MFM fraction\n(c) 0 mT (ZFC) (d) 4 mT (e) 8 mT (f) 12 mT (g) 14 mT\n(h) 16 mT (i) 18 mT (j) 20 mT (k) 24 mT (l) 32 mT\n(m) -20 mT (n) -30 mT2 μm\nFIG. S6. (a)Comparison of Rxy(orange) to the fraction of the MFM images covered by red domains (colored points) for\nthe data shown in Figure 1 and Figure S2. The area fractions were estimated by applying a threshold to the MFM data\nafter subtracting a 2nd degree polynomial background and applying Gaussian smoothing ( σ= 1 pixel). The threshold is the\nmidpoint between the 0.1 and 99.9 percentiles of the image. The error bars show the range of values obtained by offsetting the\nthreshold from -80 mHz to 80 mHz. Other sources of error are not represented. For the 50 mT and -10 mT images where no\ndomain contrast is visible, the area fractions were manually chosen to be 1.0. (b)MFM area fractions calculated for a range of\nthreshold values. Each color represents a single image, matching the points in (a). (c-n) Binarized MFM images obtained by\napplying the threshold to calculate the points in a. The 50 mT and -10 mT images are not included because the images show\nno domain contrast.10\nSVIII. V13AS A PROXY FOR Rxy\n32.5\n32.0\n31.5\n31.0\n30.5\n30.0\nV42( V)\n30.030.531.031.532.032.5V13( V)\nFIG. S7. V13andV42measurements for two 5 K data sets (blue: stripe domains under changing Bext; orange: bubble\ndomains under nominal Bext= 0 before and after domain manipulation with the MFM tip on 2 cools). Rxyis calculated\nasRxy= (V13+V42)/2I, where Iis the source current amplitude, so changes that appear symmetrically in V13andV42are\nattributable to changes in Rxy. The approximately linear with slope near 1 relationship between V13andV42confirms that V13\nis a reasonable proxy for monitoring changes in Rxyduring domain manipulation. AC source current with 500 nA amplitude.11\nSIX. REPEATABILITY OF BUBBLE LOCATIONS\n(a)\n(c)\n2 μm\n(e)\n(d)(b)\nFIG. S8. To check the repeatability of the bubble domain locations, we cooled twice from 35 K to 5 K under nominal\nBext= 0, without changing Bextin between cools, such that the residual field from the superconducting magnet should have\nbeen identical for both cools. (a)Large scale constant height MFM image of the resulting bubble domains (cool #1). The\nsmearing on the right side is from piezo drift at the beginning of the image. Color scale range: 4.4 Hz. (b)The bubble locations\ndetermined as the locations of all local minima of the MFM image within the area of the flake, overlaid on the MFM image.\n(c-d) Same as a and b, but for cool #2. The MFM image is repeated from Figure 2a of the main text. Color scale range:\n3.7 Hz. (e)Bubble locations from the two cools overlaid on one another. The images have been shifted and scaled slightly\nin order to line up the edges of the flake from the two images (lines). Some bubble locations are identical between the cools,\nothers do not match. For MFM imaging, the tip was lifted 300 nm above the SiO 2surface. Temperature: 5 K.12\nSX. BUBBLE AND STRIPE DOMAINS IN BULK MST\n(a) T = 35 KB ~ 0 T\n(b) 25 K (c) 15 K (d) 5 K\n1 μm 2 μm 5 μm 1 μm\n12 nm\n(e) B = 0.2 T(f) 0.05 T (g) 0.02 T (h) 0.0 T\n2 μmT = 5 K\n43 nm 37 nm 160 nm 577 nm54 nm 31 nm 36 nmΔf28.4 Hz\nIn this section, we show temperature-dependent and Bext-dependent MFM imaging of a bulk MST crystal. We\ncleaved the MST crystal in air before loading into the MFM. We note several limitations of the MFM images shown\nhere: (1) The constant lift MFM imaging technique means that MFM imaging is done after topographic scanning, so\nit is possible that the magnetic domains have been influenced by the tip; (2) The appearance of magnetic features in\nthe topographic scans means that the lift height during the MFM imaging is not accurate with respect to the sample\nsurface. Nonetheless the data shown confirms the presence of bubble domains (under small Bextcooling), and stripe\ndomains (under Bextsweep at low temperature) in bulk MST.13\nFIG. S9. (a-d) Constant lift MFM images and their simultaneously recorded topographic images at temperatures cooling\nfrom 35 K to 5 K. No magnetic contrast is visible at 35 K, above Tc. Below Tc, bubble domains are observed. The topographic\nimages show a combination of topographic features and magnetic features due to the strong tip-sample interaction. (e-h)\nConstant lift MFM images and their simultaneously recorded topographic images at 5 K, starting at 0.2 T, where no domains\nare seen, and lowering to 0 T. At 0.02 T and 0 T, stripe domains are observed. Again, when domains are present, magnetic\nfeatures appear in the nominally topographic images due to the strong tip-sample interaction. All MFM images are shown\nwith the same color scale range 28.4 Hz, but the zero value of the images have been offset. Topographic images were levelled\nby subtracting a planar background.14\nSXI. AREA SCALING OF THE ANOMALOUS HALL EFFECT\nTABLE I. Multiple measurements of the V13response to domain writing. stipis the (linear 1D) distance written by the tip.\nAis the area of domains aligned with the tip.\nExperiment dV13/dstipdV13/dA Measured as\n(nV/µm) (nV/ µm2)\nLine domain 8.4 Slope of V13during write\nSquare area 8.4 Slope of V13during the first forward scan line.\nSquare area 0.6 11 dV13/dstip: Peak of V13slope histogram for forward scan\nlines. dV13/dA=dV13/dstip/∆x, where ∆ xis the pixel\nwidth, 53 nm\nSquare area 0.0 Peak of V13slope histogram for backward scan lines\nSquare area 12 Overall change in V13during write, divided by the area\nof red domains in Figure 4h.\nThis section provides additional details about the domain writing experiments shown in Figure 4 of the main\ntext. We discuss: (1) additional details about the Hall response observed when writing the square area, as shown in\nFigure 4f-h of the main text, and (2) the consistency between multiple measurements of the Hall response to domain\narea.\nWe attempted to write a 8 µm square. The post-write imaging (Figure 4h) shows that the square was not uniformly\nmagnetized, but instead formed a mixed domain state, with an inner blue domain inside the red domain. We examine\nV13recorded during the write step to understand how this pattern formed dynamically (f, g). During the write, the\nfast and slow scan directions are vertical and horizontal, respectively, meaning that forward (f) and backward (not\nshown) V13images were recorded as the tip moved up and down along each pixel before advancing one pixel at a\ntime from left to right. The V13evolution has both smooth changes as well as abrupt jumps, which can be seen more\nclearly in (g) after averaging along the vertical fast scan direction.\nTo quantify the write process, we extract dV13/dstipas the slope of each vertical scan line for both forward and\nbackward passes (Figure 4i), where stipis the linear (1D) distance written by the tip. The forward slopes are peaked\naround 0.6 nV/ µm, while the backward slopes are peaked around 0 nV/ µm, matching the expectation that the tip\nshould typically flip the local magnetization on the forward pass and have a random influence on the backward pass.\nHowever, the forward pass of the first scan line has a much larger dV13/dstip= 8.4nV/µm, which instead matches the\nslope seen during the line writing experiment (Figure 4d). The order of magnitude difference between the slope of the\nfirst and subsequent scan lines can be explained by the width being written by the tip: the first line creates a domain\nof a finite width on the order of hundreds of nm (Figure 4j), while subsequent scan lines advance the domain wall by\none pixel width (53 nm in this case). The consistency across measurements of the V13slope when taking into account\nthe width being written strongly supports that V13changes linearly with the area of the domains aligned with the tip.\nWe quantify this response, dV13/dA, in two ways: (1) 11 nV/( µm)2, by dividing the peak slope of the forward scan\nlines by the pixel width, and (2) 12 ±1 nV/( µm)2, by taking the ratio of the overall change in V13during the write to\nthe area of red domains imaged in Figure 4h. The values of the Hall response to domain area are compiled in Table I.\nWith the relationship between V13and the sample magnetization firmly established, we return to the question of\nhow the mixed domain state of Figure 4h formed. If each scan line is fully polarized by the tip, then V13should\nchange according to the linear trend shown in gray in Figure 4g. As previously discussed, the first scan line shows\na sharp drop in V13associated with writing a finite width domain. Then during the next 4 µm,V13changes largely\nas expected, implying that the written area is nearly fully polarized, with the exception of a few upward jumps,\nsuggesting abrupt formation of areas anti-aligned with the tip (inner blue domains of h). During the later part of\nthe scan frame, the V13slope shows a larger deviation from the expectation. We therefore suggest that the initial\nformation of the inner blue domains within the write area occurs primarily in abrupt steps. But later, the domains\ngrow more smoothly. There may therefore be a maximum size, on the order of a few µm2(based on the location of\nthe first abrupt deviation) that can be uniformly polarized with our procedure at 10 K.\n[1] H. Deng, L. Zhao, K. Park, J. Yan, K. Sobczak, A. Lakra, E. Buzi, and L. Krusin-Elbaum, Topological surface currents\naccessed through reversible hydrogenation of the three-dimensional bulk, Nat. Commun. 13, 2308 (2022).\n[2] J.-Q. Yan, S. Okamoto, M. A. McGuire, A. F. May, R. J. McQueeney, and B. C. Sales, Evolution of structural, magnetic,\nand transport properties in MnBi 2−xSbxTe4, Phys. Rev. B 100, 104409 (2019).15\n[3] L. J. van der Pauw, A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Res. Repts.\n13, 1 (1958).\n[4] M. Ortner and L. G. Coliado Bandeira, Magpylib: A free python package for magnetic field computation, SoftwareX\n10.1016/j.softx.2020.100466 (2020)."    },    {        "title": "1612.06300v1.Completely_compensated_ferrimagnetism_and_sublattice_spin_crossing_in_the_half_metallic_Heusler_compound_Mn1_5FeV0_5Al.pdf",        "content": "arXiv:1612.06300v1  [cond-mat.str-el]  19 Dec 2016Completely compensated ferrimagnetism and sublattice spi n crossing in the\nhalf-metallic Heusler compound Mn 1.5FeV0.5Al.\nRolf Stinshoff,1Ajaya K. Nayak,1,2Gerhard H. Fecher,1Benjamin Balke,3\nSiham Ouardi,1Yurii Skourski,4Tetsuya Nakamura,5and Claudia Felser1\n1Max Planck Institute for Chemical Physics of Solids, 01187 D resden, Germany\n2Max Planck Institute of Microstructure Physics, 06120 Hall e, Germany\n3Institut f¨ ur Anorganische und Analytische Chemie,\nJohannes Gutenberg - Universit¨ at, 55099 Mainz, Germany\n4Dresden High Magnetic Field Laboratory (HLD), 01328 Dresde n, Germany\n5Japan Synchrotron Radiation Research Institute, SPring-8 , Hyogo 679-5198, Japan\n(Dated: March 22, 2018)\nThe Slater–Pauling rule states that L21Heusler compounds with 24 valence electrons do never\nexhibit a total spin magnetic moment. In case of strongly loc alized magnetic moments at one of the\natoms (here Mn) they will exhibit a fully compensated half-m etallic ferrimagnetic state instead, in\nparticular, when symmetry does not allow for antiferromagn etic order. With aid of magnetic and\nanomalous Hall effect measurements it is experimentally dem onstrated that Mn 1.5V0.5FeAl follows\nsuch a scenario. The ferrimagnetic state is tuned by the comp osition. A small residual magneti-\nzation, that arises due to a slight mismatch of the magnetic m oments in the different sublattices\nresults in a pronounced change of the temperature dependenc e of the ferrimagnet. A compensation\npoint is confirmed by observation of magnetic reversal and si gn change of the anomalous Hall ef-\nfect. Theoretical models are presented that correlate the e lectronic structure and the compensation\nmechanisms of the different half-metallic ferrimagnetic st ates in the Mn-V-Fe-Al Heusler system.\nPACS numbers: 75.50.Gg, 75.50.Cc, 75.30.Gw, 72.15.Jf\nKeywords: Compensated ferrimagnets, half-metallic ferri magnets, Heusler compounds\nHalf-metallic ferromagnets are promising candidates\nfor application in spintronics because they exhibit 100%\nspin polarization. They are metallic in one spin direc-\ntion and semiconducting in the other [1, 2]. However,\nferromagnets produce a large dipole field that hinders\nthe device performance. For example, the dipolar mag-\nneticanisotropybecomesverylargeforin-planemagnetic\nsystems leading to large switching fields. For this reason,\nthere is a great interest on zero magnetic moment spin-\ntronics, as such systems do not produce dipole fields and\nare extremely stable against external magnetic fields [3–\n6]. The concept of half-metallic antiferromagnetism was\nintroduced by van Leuken and de Groot [7]. It turns out,\nhowever, that symmetry does not allow half-metallic an-\ntiferromagnets and the materials are half-metallic com-\npensated ferrimagnets [8]. Recently, Hu [9] presented a\ntheoretical work on possible half-metallic antiferromag-\nnets for spintronic applications. However, the identical\nelectronic structure of both spin directions makes most\nof the conventional antiferromagnets unable to carry a\nspin polarized current.\nHeusler materials are well known for their tunable\nmagnetic structure due to the presence of one or more\nmagnetic sublattices. Depending on the constituting ele-\nments or crystal structure, ferromagnetic, ferrimagnetic,\nantiferromagnetic, or canted spin structures may be re-\nalised [10–14]. In particular, the Heusler compounds\nwithL21orC1bstructure are well known for their half-\nmetallic behaviour[1]. These materialsfollowthe Slater–\nPauling rule [15, 16] related to the half-metallicity [17–19]. According to this rule the spin magnetic moment\n(m) in cubic Heusler compounds with L21structure is\ndefined by m=Nv−24, where Nvis the accumulated\nnumber of valence electrons. As a direct consequence,\nHeusler compounds with Nv= 24 never exhibit a macro-\nscopic magnetic moment.\nIncertaincases,however,the DO3orL21Heuslercom-\npounds with 24 valence electrons are able to exhibit a\nfully compensated half-metallic behaviour [8]. In that\nconcept, the Slater–Pauling rule is combined with the\nK¨ ubler rule [20]. The latter states that Mn on the octa-\nhedrally coordinated position (4 b) in Heusler compounds\ntends to a high, localised magnetic moment. This mo-\nment has to be completely compensated by the magnetic\nmoments of the remaining atoms to satisfy the Slater–\nPauling rule.\nAlthough there are several theoretical predictions,\nmost of the suggested materials either do not exist or\nappear only in a different crystal structure [7]. Recently\nit was demonstrated that a compensated ferrimagnetic\nstate may be realized in the tetragonal Mn-Pt-Ga sys-\ntem [21]. However, it is known that Heusler materials\nwith tetragonal distortion do not show half-metallicity.\nKurtet al[22, 23] have shown that a compensated mag-\nnetic state with considerable spin polarization may be\nachieved in a cubic thin film of Mn 2RuxGa with compo-\nsition falling between C1bandL21Heusler compounds.\nDespite several attempts by different research groups\nthere is no experimental evidence of a compensated mag-\nnetic structure in the classical 24 valence electron based2\ncubicHeuslercompounds. Inthepresentworkitisshown\nby experiments and calculations that the Heusler com-\npound Mn 1.5V0.5FeAl with L21structure exhibits a com-\npletely compensated magnetic state. Further, the pres-\nence of a temperature and composition dependent sub-\nlattice spin compensation is demonstrated in the investi-\ngated system, while keeping the half-metallicity.\nPolycrystalline ingots of Mn 1.5V0.5FeAl were prepared\nby arc melting. The composition and structure of the\nsamples was determined by energy dispersive X-ray spec-\ntroscopy (EDX) and X-ray powder diffraction (XRD).\nLow field magnetic measurements were carried out by\nmeans of a vibrating sample magnetometer (MPMS 3,\nQuantum Design). Pulsed, high magnetic field exper-\niments were performed at the Dresden High Magnetic\nField Laboratory. The transport measurements were\ncarried out utilising a physical property measurement\nsystem (PPMS, Quantum Design). X-ray magnetic cir-\ncular dichroism (XMCD) investigations were performed\nat beamline BL25SU of SPring-8. The electronic struc-\nture was calculated in the local spin density approxima-\ntion. The selfconsistent electronic structure calculations\nwere carried out using the spin polarized fully relativistic\nKorringa–Kohn–Rostockermethod (SPRKKR) provided\nby Ebert et al[24, 25].\n/s53/s48/s53\n/s69/s110/s101/s114/s103/s121 /s32/s32/s32 /s69\n/s70/s32/s91/s101/s86/s93\n/s40/s98/s41\n/s32\n/s70 /s32/s45/s52/s51 /s109/s40/s97/s41\n/s40/s99/s41\n/s40/s100/s41/s65/s108\n/s77 /s110/s73\n/s77 /s110/s73/s73\n/s70/s101\n/s86\n/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s32 /s110 /s40/s69 /s41/s32/s91/s101/s86/s45/s49 \n/s93\n/s32/s32\n/s70/s32/s109/s45/s51/s109\nFIG. 1. (Color online) Crystalline and electronic structur e of\nMn1.5V0.5FeAl\n(a) The L21type cubic Heusler structure with space group\nF m3m(225). (b) The Xtype inverse cubic Heusler struc-\nture with space group F-43m(216). Different atoms are rep-\nresented by balls with different colours, shown in between th e\ntwo structures. The spin resolved density of states is shown\nforF m3min (c) and for F43min (d).\nThe XRD analysis indicates that Mn 1.5V0.5FeAl crys-tallizes in a cubic Heusler structure with a lattice param-\neter ofa= 5.83˚A. The two most energetically favoured\ncrystal structures are shown in Figure 1(a) and (b) to-\ngether with their magnetic order. In the regular L21\ncubic Heusler structure with space group F m3m(225),\nthe Al atoms occupy the 4 aposition, the 4 bposition\nis equally occupied by V and Mn atoms and a statis-\ntical distribution of the Mn and Fe atoms at 8 cis ex-\npected. In a less probable situation, an ordering of the\nMn and Fe atoms can split the 8 cposition to 4 cand\n4d, as shown in Figure 1(b). In order to determine the\ndensity of states (DOS) of Mn 1.5V0.5FeAl, the calcula-\ntions were performed using SPRKKR with coherent po-\ntential approximation (CPA) to account for the random\noccupation of the sites and for chemical disorder. Com-\nparing the total energies at the same lattice parameter,\none finds that the energy of the L21structure with space\ngroupFm3m(Figure1(a)) is 0.5meVlowercomparedto\ntheXstructure with space group F43m(Figure 1(b)).\nThe electronic structure reveals clearly the half-metallic\ncharacterof Mn 1.5V0.5FeAl for both structure types with\nchemical disorder. The gap in the minority DOS is de-\nfined by the states of the Mn atoms located on the 8 c\nand the Fe atoms located on the 8 cor 4cpositions. This\ncoincides with previous calculation for various Heusler\ncompounds, as in most cases, the gap is dominated by\nthe states arising from the atom on the 8 csite [19].\nTABLE I. Site specific magnetic moments in Mn 1.5FeV0.5Al.\nThe calculations were carried out by means of SPRKKR -\nCPA using the L21structure ( F m3m, 225) or the X-type\nstructure ( F43m, 216). All magnetic moments are given in\nµB. The total moments are given per primitive cell. The site\nspecific spin msand orbital mlmagnetic moments are given\nper atom. Note the rounding, the induced moment at Al is\n<0.006µB.\n225 216\nAtom Site msmlSite msml\nMn (8 c) 1.40 0.03 (4 d) 1.38 0.03\nFe (8 c) 0.28 0.02 (4 c) 0.35 0.03\nMn (4 b) -2.79 -0.01 (4 b) -2.86 -0.01\nV (4 b) -0.55 0.01 (4 b) -0.60 0.01\nAl (4 a) -0.01 -0.00 (4 a) -0.01 -0.00\nms,l\ntot 0.003 0.046 0.000 0.058\nmtot 0.05 0.06\nThe calculated magnetic moments of Mn 1.5FeV0.5Al\nare listed in Table I. A summation of the site specific\nmagnetic moments yields a zero total moment, as ex-\npected for a completely compensated ferrimagnet. The\nsignsofthecalculatedmagneticmomentsweresupported\nby XMCD measurements. XMCD spectra were calcu-\nlated for the L21structure with Mn on 8 cmixed with Fe3\nand on 4 bmixed with V. The two different Mn atoms\ncause a zero-crossing with pronounced maximum and\nminimum at the L3edge, that is also clearly revealed\nin the measured spectra. Although the site specific mo-\nment of the Mn atoms could not be determined due to\noverlap of the lines from the Mn atoms at two different\nsites, the total sum moment and moments obtained for\nFe and V from the XMCD measurements well matched\nwith the theoretical values.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s48/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s40/s97/s41/s77 /s40/s84 /s41/s32/s91\n/s66/s93\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s91/s75/s93/s32/s32/s48/s46/s49/s32/s84\n/s32/s32\n/s40/s98/s41\n/s32/s48/s46/s53/s32/s84\n/s32/s48/s46/s49/s32/s84/s32/s32/s77 /s40/s84 /s41/s32/s47/s32 /s77 /s40/s48/s41\n/s40/s99/s41\n/s109\n/s73 /s73 /s32/s61/s32 /s50/s42 /s109\n/s73 /s40/s100/s41\n/s109\n/s73 /s73 /s32/s60/s32 /s50/s42 /s109\n/s73 \n/s82/s101/s108/s97/s116/s105/s118/s101/s32/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s47/s32 /s84\n/s67\n/s32/s32\n/s32/s77\n/s73 /s73 \n/s32/s124 /s77\n/s116/s111/s116/s124 \n/s32/s77\n/s73 \nFIG. 2. (Color online) Magnetisation of Mn 1.5FeV0.5Al.\n(a) shows the temperature dependence of the magnetiza-\ntionM(T), measured for a completely compensated sample.\n(b) shows the temperature dependence of the magnetization\nmeasured for an overcompensated sample in different fields.\nThe theoretical behaviour of a two sublattice ferrimagnet i n\nthe molecular field approximation is shown in (c) and (d):\n(c) shows the magnetization for a completely compensated\nferrimagnet with |mII|=|2mI|, (d) shows the case with\n|mII|<|2mI|.miare the average magnetic moments of\nthe atoms on the ithsublattice at T= 0.\nFigure 2a shows the temperature dependence of the\nmagnetisation M(T) for a completely compensated sam-\nple. As expected, the magnetization vanishes at 0 K as is\ntypical for a completely compensated ferrimagnet. The\nmagnetisation stays close to Zero up to about 50 K. The\nCurie temperature appears at about 335 K.\nM(T) curves measured for a slightly overcompensated\nsample in different induction fields are shown in Fig-\nure 2b. From the M(T) curves measured in an induction\nfield of 0.1 T a Curie temperature ( TC) of about 308 K is\ndetermined. By decreasingthe temperature the magneti-\nzation first completely reduces to zero at 127 K and then\nincreasesagainbyloweringthe temperaturebelow127K.\nThis type of magnetic behaviour indicates the presence\nof a compensation point of the ferrimagnetic order. Thecompletely compensated behaviour is very sensitive to\nthe composition of the sample as will be shown next.\nThe temperature dependence of the total and the sub-\nlatticemagneticmomentsweresimulatedusingamolecu-\nlarfieldmodel foratwosublatticeferrimagnet. Inpartic-\nular the equations introduced by Stearn [26] for binary\ncompounds (Fe 3Al, Fe 3Si) with Heusler type structure\nwere used. This model may come close to the L21struc-\nture with space group 225. It is assumed that the mag-\nnetic moment mIof the atoms in sublattice I is smaller\n(half) but that twiceasmanyatomsareoccupyinglattice\nI. That is, lattice I describes the 8 csite, whereas lattice\nII corresponds to the 4 bsites with higher magnetic mo-\nmentsmIIbut only half as many atoms ( nI/nII= 2)\ncompared to lattice I. The completely compensated fer-\nrimagnet appears when nImI=nIImII. The exchange\nintegrals Jijshould be largest for interactions between\nthe atomsin sites I and II. Further the exchangeintegrals\nbetween atoms of type I should be much smaller com-\npared to the atoms of type II, with the latter being close\ntothose betweentype I andII atoms. In particularit was\nassumed that JI/JI−II= 1/2andJII/JI−II= 2/3. The\nresults are shown in Figures 2c and 2d that compare the\ncompletelycompensatedcasewithaslightlyovercompen-\nsated case. Figure 2c describes a ferrimagnet where the\ncompensation point appears at T= 0 and the magneti-\nzation stays nearly Zero up to about T/TC≈1/5. For\nmII<|2mI|, a compensation point appears (Figure 2d).\nThe latter is classified as a N´ eel N-type ferrimagnet [27].\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s49/s54/s45/s48/s46/s49/s50/s45/s48/s46/s48/s56/s45/s48/s46/s48/s52/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s77/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32/s32/s32 /s77 /s40/s72 /s41/s32/s91\n/s66/s93\n/s73/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s32/s50/s54/s51/s32/s75\n/s32/s32/s32/s32/s32/s50/s32/s75\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49\n/s40/s97/s41/s40/s98/s41/s48/s72\n/s99/s32/s91/s84/s93\n/s84 /s32/s91/s75/s93\nFIG. 3. (Color online) Field dependent magnetisation of\nMn1.5FeV0.5Al.\nShown is the magnetisation M(H) at 2 K and 263 K. Inset\n(a)shows thelowtemperaturebehaviouronan enlargedscale .\nInset(b)shows thetemperature variation ofthe coercive fie ld.\nFigure 3 shows the field dependence of the magneti-\nsation for two different temperatures, at the maximum\nof the magnetisation (263 K) and close to Zero (2 K).\nAt high temperatures the material appears very soft but\nhas a small remanence and coercive field at low tem-4\nperature. The inset (b) shows that the coercive field is\nconstant below 50 K where the magnetisation vanishes\n(compare Figure 2a). Above this critical temperature,\nthe magnetisation softens with increasing temperature.\nThe appearance of a coercive field is a typical effect at\nthe compensation point [28]. In certain cases it is as-\nsumed to diverge at the compensation point, whereas it\nclearly saturates below the critical temperature in the\ncompletely compensated half-metallic ferrimagnet.\nSo far the occurrence and some magnetic properties\nof a completely compensated half-metallic ferrimagnet is\ndemonstrated. The compensation phenomenon is better\nstudied, however,in the slightlyovercompensatedsample\nwith a compensation point at a finite temperature. The\nremaining part is thus devoted to this case.\nTheM(H) loops measured at different temperatures\ndemonstrate the compensation phenomenon (Figure 4a)\nin the overcompensated sample. A nearly linear hystere-\nsis loop with almost zero spontaneous magnetization is\nfound in the vicinity of the compensation temperature\n(127 K). The M(H) loops measured for temperatures\nbelow and above the compensation point exhibit a soft\nmagnetic behaviour. The most important point is that\nbothM(T) andM(H) measurements hint on a satura-\ntion magnetization that is less than 0.1 µBaway from the\ncompensation point. This suggests that the sample vir-\ntually exhibits a nearly compensated magnetic state over\nthe full temperature range.\nIt isseenfromFigure2b thatthe minimum atthecom-\npensation point shifts slightly with increasing induction\nfield. For a deeper understanding of this effect we have\nmeasured Zero field cooled (ZFC) and field cooled (FC)\nM(T) curvesin avery small field of 2 mT (Figure 4b). In\nthiscasetheFCcurve,whichshowsapositivemagnetiza-\ntion at higher temperature, crosses the temperature axis\nat 127 K to give a negative magnetization at low tem-\nperatures. The ZFC curve follows an exactly opposite\nbehaviour to that of the FC curve. The zero-crossing of\nthe magnetization clearly indicates a sublattice magnetic\ncompensation at 127 K. Similar magnetic reversal at the\ncompensation point has been observed in systems with\nspin-orbital compensation [29]. Pulsed magnetic field\nmeasurements at 1.5 K and at the compensation point\n(127 K) show a linear magnetic response with fields up\nto 55 T without any spin-flop transition. This clearly in-\ndicates a strong exchange coupling between the different\nmagnetic sublattices in Mn 1.5V0.5FeAl.\nThe magnetic measurements shown in Figures 2(a)\nand (b) only give an indication of a sublattice spin cross-\ning at the compensation point. Anomalous Hall effect\n(AHE) measurements have been performed at different\ntemperatures, to allow for a direct observation of the\nspin crossing across the compensation point (see Fig-\nure 4). The AHE measured at 50 K and 100 K shows\na negative sign, i.e, negative (positive) value in positive\n(negative) field. At the compensation point the AHE be-/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s97/s41\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s55/s32/s75\n/s32/s49/s53/s48/s32/s75/s77 /s32/s91\n/s66/s93\n/s32/s32\n/s40/s98/s41\n/s32/s32\n/s32/s90/s70/s67\n/s32/s70/s67/s32/s32/s40\n/s48/s72 /s32/s61/s32/s50/s32/s109/s84/s41\n/s32/s32\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s32 /s84 /s32/s91/s75/s93/s32/s45/s48/s46/s49/s32/s84\n/s32/s43/s48/s46/s49/s32/s84\n/s32/s49/s48/s48/s32/s75/s32/s49/s53/s48/s32/s75\n/s32/s49/s50/s55/s32/s75/s120/s121/s32/s91 /s99/s109/s93\n/s73/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s32\n/s32/s40/s99/s41 /s40/s100/s41\nFIG. 4. (Color online) Magnetic and transport properties of\novercompensated Mn 1.5FeV0.5Al.\n(a) Isothermal magnetization loops, M(H), at different tem-\nperatures. (b) ZFC and FC M(T) curves measured in a small\nfield of 2 mT. (c) Field dependence of the Hall effect mea-\nsured at different temperatures. (d) Temperature dependenc e\nof Hall resistivity measured in ±0.1 T.\ncomes virtually zero. Above the compensation point for\nT= 150 K and 200 K, a positive anomalous Hall effect\nis observed. The change in sign of the AHE can be seen\nin the temperature dependence of the AHE measured in\na field of ±0.1 T (Figure 4d). The AHE changes from\na negative (positive) maximum around 100 K to a pos-\nitive (negative) maximum at 200 K when measured in\na field of 0.1 T (-0.1 T). The two curves cross the zero\nline at about 130 K. Since the AHE is an intrinsic prop-\nerty of ferro- and ferrimagnets, a small uncompensated\nmoment below and abovethe compensation point will re-\nsult in a non-vanishing AHE. The most important point\nis that the AHE changes its sign, which clearly indicates\nthe change of the sublattice magnetic structure across\nthe compensation point. The AHE is an intrinsic man-\nifestation of a Berry curvature, that changes due to the\nchange of the sublattice magnetic moment from spin-up\nto spin-down, resulting in a change of the sign across the\ncompensation point.\nIn conclusion, the existence of a completely compen-\nsated ferrimagnetic state in the half-metallic L21cubic\nHeusler compound Mn 1.5V0.5FeAl has been experimen-\ntally demonstrated. Although there have been several\ntheoretical works regarding realization of a fully com-\npensated magnetic state in the L21cubic Heusler com-\npounds with 24 valence electrons, no successful exper-\nimental attempt has been made until now. This work\nalso establishes the existence of a temperature depen-\ndent sublattice spin crossing in half-metallic ferrimag-\nnets. The compensation temperature can be varied by\nan intentional variation of the stoichiometry. Recently, it\nhasbeendemonstratedthatantiferromagnetsmaybeuti-5\nlized as a principal component in spintronic devices, es-\npecially in tunnel magnetoresistance based devices. The\npresent half-metallic compensated ferrimagnet adds the\nadvantage of nearly 100% spin polarization, which is ex-\ntremely important for spintronics.\nThe authors thank N. Demitri for assistance during\nthe XRD experiment at ELETTRA. This work is funded\nby the Deutsche Forschungs Gemeinschaft (project 1.3-\nA in research unit 1464 ASPIMATT ) and by the ERC\nAdvanced Grant (291472) Idea Heusler . The experi-\nments at the High Magnetic Field Laboratory Dresden\n(HLD) were supported by Euro-MagNET II under the\nEuropean Union contract 228043. X-ray diffraction mea-\nsurements were performed at beamline XRD1 of the\nELETTRA Synchrotron (Trieste, Italy) under proposal\nnumber 20145509. Synchrotron based HAXPES and\nXMCD measurements were performed at BL47XU and\nBL25SU of SPring-8 with approval of JASRI, Proposal\nNos. 2008A0017 and 2008A1606, respectively.\n[1] R. A. de Groot, F. M. Mueller, P. G. van Engen, and K.\nH. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).\n[2] M. I. Katsnelson, V. Yu. Irkhin, L. Chioncel, A. I. Licht-\nenstein, and R. A. de Groot, Rev. Mod. Phys. 80, 315\n(2008).\n[3] Y. Soh, and R. K. Kummamuru, Phil. Trans. R. Soc. A\n369, 3646 (2011).\n[4] B. G. Park, J.Wunderlich, X. Marti, V. Holy, Y.\nKurosaki, M. Yamada, H. Yamamoto, A. Nishide, J.\nHayakawa, H. Takahashi, A. B. Shick, and T. Jungwirth,\nNature Mater. 10, 347 (2011).\n[5] Y.Y. Wang, C. Song, B. Cui, G.Y. Wang, F. Zeng, and\nF. Pan, Phys. Rev. Lett. 109, 137201 (2012).\n[6] X. Marti, I. Fina, C. Frontera, J. Liu, P.Wadley, Q. He,\nR. J. 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K¨ ubler, Theory of Itinerant Electron Mag-\nnetism(Clarendon Press, Oxford, 2000).\n[13] T. Graf, C. Felser, S. S. P. Parkin, Prog. Solid State\nChem.39, 1 (2011).\n[14] C. Felser and A. Hirohata, Heusler Alloys, Springer Se-\nries in Materials Science, Vol. 222 (Springer Internationa l\nPublishing, 2016).\n[15] J. C. Slater, Phys. Rev. 49, 931 (1936).\n[16] L. Pauling, Phys. Rev. 54, 899 (1938).\n[17] I. Galanakis, P. H. Dederichs, and N. Papanikolaou,\nPhys. Rev. B 66, 174429 (2002).\n[18] G. H. Fecher, H. C. Kandpal, S.Wurmehl, C. Felser, and\nG. Sch¨ onhense, J. Appl. Phys. 99, 08J106 (2006).\n[19] H. C Kandpal, G. H Fecher, and C. Felser, J. Phys. D:\nAppl. Phys. 40, 1507 (2007).\n[20] J. K¨ ubler, A. R. Williams, and C. B. Sommers, Phys.\nRev. B28, 1745 (1983).\n[21] A. K. Nayak, M. Nicklas, S. Chadov, P. Khuntia, C.\nShekhar, A. Kalache, M. Baenitz, Y. Skourski, V. K.\nGuduru, A.Puri, U.Zeitler, J. M. D.Coey, andC. Felser,\nNature Mater. 14, 679 (2015).\n[22] H. Kurt, K. 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Rev.\nB66, 161319(R) (2002)."    },    {        "title": "1908.07286v2.Peculiarities_in_pseudo_transitions_of_a_mixed_spin___1_2_1___Ising_Heisenberg_double_tetrahedral_chain_in_an_external_magnetic_field.pdf",        "content": "Peculiarities in pseudo-transitions of a mixed spin- (1=2;1)Ising-Heisenberg\ndouble-tetrahedral chain in an external magnetic field\nOnofre Rojas,1Jozef Strečka,2Oleg Derzhko,3and S. M. de Souza1\n1Departamento de Física, Universidade Federal de Lavras, CP 3037, 37200-000, Lavras-MG, Brazil\n2Department of Theoretical Physics and Astrophysics, Faculty of Science,\nP. J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovakia\n3Institute for Condensed Matter Physics, National Academy of\nSciences of Ukraine, Svientsitskii Str. 1, 79011 L’viv, Ukraine\nAbstract\nRecently, it has been rigorously verified that several one-dimensional (1D) spin models may exhibit a peculiar pseudo-\ntransition accompanied with anomalous response of thermodynamic quantities in a close vicinity of pseudo-critical temperature.\nIn the present work we will introduce and exactly solve a mixed spin-(1/2,1) Ising-Heisenberg double-tetrahedral chain in an\nexternal magnetic field as another particular example of 1D lattice-statistical model with short-range interactions that displays\na pseudo-transition of this type. The investigated model exhibits at zero temperature three ferrimagnetic phases, three frus-\ntrated phases, and one saturated paramagnetic phase. The ground-state phase diagram involves five unusual interfaces (phase\nboundaries), at which the residual entropy per site equals to a larger entropy of one of two coexisting phases. Four such inter-\nfaces are between a non-degenerate ferrimagnetic phase and a macroscopically degenerate frustrated phase, while one interface\nis between two non-degenerate ferrimagnetic phases. Though thermal excitations typically destroy all fingerprints of zero-\ntemperature phase transitions of 1D lattice-statistical models with short-range forces, the mixed spin-(1/2,1) Ising-Heisenberg\ndouble-tetrahedral chain is quite robust with respect to thermal excitations and it displays peculiar pseudo-transitions close to\nall five aforementioned interfaces.\nPACS numbers: 05.70.Fh, 75.10.-b, 75.10.Jm, 75.10.Pq\nKeywords: Residual entropy; Quasi-phases; Pseudo-transitions; Ising-Heisenberg\nI. INTRODUCTION\nThere are a few paradigmatic examples of one-\ndimensional (1D) lattice-statistical models with short-\nrange couplings, which exhibit a discontinuous (first-\norder) phase transition at finite temperature. Perhaps\nthe most famous example is 1D KDP model of hydrogen-\nbonded ferroelectrics invented by Nagle [1], which dis-\nplays a discontinuous phase transition between the fer-\nroelectric and paraelectric phases due to assignment of\nan infinite energy to all ionized configurations. Another\nparticular example of this type is the Kittel model [2]\ndefined through a finite transfer matrix, which involves a\nconstraint on zipper corresponding to an infinite poten-\ntial being responsible for a non-analyticity of the free en-\nergy. Owing to a singular character of the potential, the\nKittel model also exhibits a first-order phase transition.\nThe next paradigmatic example is the 1D solid-on-solid\nmodel considered by Chui and Weeks [3], which is ex-\nactlysolvableinspiteofaninfinitedimensionofitstrans-\nfer matrix. By imposing suitable pinning potential the\n1D solid-on-solid model may also display a roughening\nphase transition of first order [3]. Furthermore, Daux-\nois and Peyrard [4] have examined another 1D lattice-\nstatistical model with an infinite dimension of the trans-\nfer matrix, which exhibits a phase transition at finite\ntemperature. Last but not least, Sarkanych et al. [5]\nproposed 1D Potts model with so-called invisible states\nand short-range couplings. It could be thus concluded\nthat all five aforementioned 1D lattice-statistical models\nbreak the Perron-Frobenius theorem, because some off-\ndiagonal transfer-matrix elements become null and thefree energy may consequently become non-analytic at a\ncertain critical temperature.\nVan Hove [6] proposed a theorem that proves absence\nof a phase transition in 1D lattice-statistical models with\nshort-range couplings. Later, Cuesta and Sanchez [7]\ngeneralized the non-existence theorem for a phase tran-\nsition at finite temperatures. Surely, this is not yet\nthe most general non-existence theorem, because mixed-\nparticle chains or more general external fields fall beyond\nthe scope of this theorem.\nThe term \"pseudo-transition\" and \"quasi-phase\" was\nintroduced by Timonin [8] in 2011 when studying the\nspin-ice model in a field. These terms refer to a sudden\nchange in first derivatives and vigorous peaks in second\nderivatives of the free energy although these marked sig-\nnatures are not in reality true discontinuities and diver-\ngences, respectively. Note furthermore that the pseudo-\ntransitions do not violate the Perron-Frobenius theorem,\nbecause the free energy is always analytic. A com-\nmon feature of the pseudo-transitions is that some off-\ndiagonal transfer-matrix elements (Boltzmann factors)\nbecome very small (almost zero), since very high albeit\nfinite energy is assigned to the corresponding states.\nObvious fingerprints of pseudo-transitions were re-\ncently found in several 1D spin or spin-electron models.\nFor instance, the pseudo-transitions were detected in the\nspin-1/2 Ising-Heisenberg diamond chain [9, 10], two-leg\nladder [11], as well as triangular tube [12]. Similarly,\nthe emergence of pseudo-transitions was verified in the\nspin-1/2 Ising diamond chain [13] and the coupled spin-\nelectron double-tetrahedral chain [14–16]. In general, the\nfirst derivatives of the free energy such as entropy, inter-\n1arXiv:1908.07286v2  [cond-mat.stat-mech]  27 Sep 2019nal energy or magnetization show a steep change around\npseudo-critical temperature. This feature is similar to\nthe first-order phase transition, but all thermodynamic\nresponse functions are in fact continuous. Contrary to\nthis, second derivatives of the free energy such as spe-\ncific heat and magnetic susceptibility resemble typical\nbehavior of a second-order phase transition at a finite\ntemperature. Therefore, this peculiar pseudo-critical be-\nhavior drew attention to a more comprehensive study of\nthisphenomenonaimedatelucidatingallitsessentialfea-\ntures [17–19]. Recently, a further attention has been paid\nto uncover the mechanism triggering pseudo-transitions\nbased on a rigorous analysis of the correlation function\n[10] and pseudo-critical exponents [20].\nThe goal of the present study is to investigate a mixed\nspin-(1/2,1) Ising-Heisenberg tetrahedral chain in an ex-\nternal magnetic field, which has a pretty rich ground-\nstate phase diagram and exhibits a number of finite-\ntemperature pseudo-transitions close to some inter-phase\nboundaries.There are some 3D compounds in which,\nwhen we consider one columnar stripe, we could ob-\nserve a double tetrahedral chain structure. Such as\ncobalt oxide RBaCo 4O7, where Rdenotes a rare earth\natom, which has a swedenborgite lattice structure[21].\nAnother compound with a similar structure could be\nthe salt with 3D corrugated packing frustrated spin\n[22] of C\u000f\u0000\n60in (MDABCO+)(C\u000f\u0000\n60)[MDABCO+=N-\nmethyldiazabicyclooctanium cation and C\u000f\u0000\n60radical an-\nions], a stripe of this salt can be viewed also as a double-\ntetrahedral chain.\nThis article is organized as follows. In Sec. II we\nconsider and exactly solve the mixed spin-(1/2,1) Ising-\nHeisenberg tetrahedral chain in a magnetic field. Ther-\nmodynamics in a close vicinity of the pseudo-transition\nis examined in Sec. III, where an influence of the residual\nentropy upon basic thermodynamic quantities is investi-\ngated in detail. Finally, several concluding remarks are\npresented in Sec. IV.\nII. MIXED SPIN-( 1=2;1)ISING-HEISENBERG\nDOUBLE-TETRAHEDRAL CHAIN\nThe coupled spin-electron model on a double-\ntetrahedral chain [14–16], which involves localized Ising\nspins at nodal lattice sites and mobile electrons delo-\ncalized over triangular plaquettes, represents a promi-\nnent example of 1D lattice-statistical model mimicking a\ntemperature-driven phase transition [14]. However, ear-\nlier investigations of the analogous spin-1/2 Heisenberg\n[23–25] and Ising-Heisenberg [26, 27] models on a double-\ntetrahedral chain did not verify anomalous thermody-\nnamic response closely related to a pseudo-transition un-\ntil the latter Ising-Heisenberg model was revisited and\nmore thoroughly studied [18].\nJ0\nJJzSa,i \nSb,i Sb,i +1 Sa,i +1 \nSc,i +1 Sc,i \nσi σi+1 \nIsing-Heisenberg interaction Heisenberg interaction Heisenberg spin-1 Ising spin-1/2 Figure 1: A schematic representation of the mixed spin-\n(1/2,1) Ising-Heisenberg double-tetrahedral chain. Small\nballs correspond to the Ising spins \u001biand large balls corre-\nspond to the Heisenberg spins S\r;i(\r=a;b;c ).\nIn the present work we will examine in particular the\nmixed spin-( 1=2;1) Ising-Heisenberg double-tetrahedral\nchain, which is schematically depicted in Fig. 1 and de-\nfined through the following Hamiltonian\nH=NX\ni=1Hi; (1)\nwith\nHi=\u0000[J(Sb;i;Sc;i)z+J(Sc;i;Sa;i)z+J(Sa;i;Sb;i)z]\n\u0000\u0000\nSz\na;i+Sz\nb;i+Sz\nc;i\u0001\n[hz+J0(\u001bi+\u001bi+1)]\u0000h\n2(\u001bi+\u001bi+1):\n(2)\nIn above,S\u000b\n\r;i(\u000b=fx;y;zg,\r=fa;b;cg) denote the\nspin-1 Heisenberg atoms, \u001bi=\u00061\n2denotes the Ising spin,\nandJ(S\r;i;S\u000e;i)z=JSx\n\r;iSx\n\u000e;i+JSy\n\r;iSy\n\u000e;i+JzSz\n\r;iSz\n\u000e;i.\nThe Hamiltonian (2) is written as a sum of cell Hamilto-\nniansHi, which correspond to spin clusters with the geo-\nmetric shape of two face-sharing tetrahedra (i.e., trigonal\nbipyramid).\nThe overall Hilbert space of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain splits into sev-\neral disjoint (orthogonal) subspaces, because the Hamil-\ntoniansHifrom different unit cells commute with each\nother. The Hilbert subspace corresponding to the spin-\n1 Heisenberg triangle from the i-th unit cell is given by\nthe Hamiltonian matrix of dimension 27\u000227and it can\nbe further split into several smaller block-diagonal matri-\nces depending on the z-component of the total spin: for\nSz\nt= 0one has one 7\u00027block matrix, for jSz\ntj= 1two\n6\u00026matrices, forjSz\ntj= 2two3\u00023matrices, and for\njSz\ntj= 3two1\u00021matrices. All eigenvalues and eigenvec-\ntors of spin-1 Heisenberg triangle Hamiltonian are listed\nin Table I. The first column stands for the eigenvalues of\ntheSz\ntoperator, while the counter kis used just to dis-\ntinguish the states with same eigenvalues and the respec-\ntive state degeneracy gkin fourth column. With the help\nof eigenvalues and eigenvectors of the spin-1 Heisenberg\ntriangle reported in Table I one can express the full en-\nergy spectrum per Hiunit cell of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain as follows\n\"k(\u001bi;\u001bi+1) =\u000fk\u0000\u0012\nJ0Sz\nt+h\n2\u0013\n(\u001bi+\u001bi+1):(3)\n2Here,\u000fkmarks the respective eigenvalue of the spin-1 Heisenberg triangle listed in Table I.\nTable I: Full spectrum of the spin-1 Heisenberg triangle specified according to the respective eigenvalue, state degeneracy, and\neigenvector. The eigenstates are grouped according to the z-component of the total spin Sz\nt=Sz\na+Sz\nb+Sz\nc. The first column\nstands for the eigenvalues of the Sz\ntoperator, and the second column is just to distinguish the eigenvector with the same Sz\nt.\nThe definition of mixing angles: cot (2\u001e1) =Jz\u0000J\n2J,cot (2\u001e2) =Jz+2J\n4J, and cot (2\u001e3) =Jz\u00002J\n2p\n6J.\njSz\ntjkEnergy (\u000fk) gkState\n00J+Jz 2|0,0i=8\n>><\n>>:1\n2\u0012\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n0\n1E\u0013\np\n3\n6\u0012\n2\f\f\f1\n\u00001\n0E\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n\u0000\f\f\f\u00001\n0\n1E\n+ 2\f\f\f\u00001\n1\n0E\u0013\n1Jp\n6 cot\u001e3 1|0,1i=p\n6\n6cos\u001e3\u0012\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n-sin\u001e3\f\f\f0\n0\n0E\n2\u0000Jp\n6 tan\u001e3 1j0;2i=p\n6\n6sin\u001e3\u0012\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n+cos\u001e3\f\f\f0\n0\n0E\n3\u0000J+Jz 2j0;3i=8\n>><\n>>:1\n2\u0012\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n0\n1E\u0013\np\n3\n6\u0012\n\u00002\f\f\f1\n\u00001\n0E\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f\u00001\n0\n1E\n+ 2\f\f\f\u00001\n1\n0E\u0013\n4 2J+Jz 1j0;4i=p\n6\n6\u0012\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n\u0000\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n15\n6\u00002J(1\u0000cot\u001e2)\u0006hz1j\u00061;0i=p\n3\n3\u0014\ncos\u001e2\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n+\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n+\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n\u0000sin\u001e2\u0012\f\f\f\u00061\n0\n0E\n+\f\f\f0\n\u00061\n0E\n+\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n7\n8\u00002J(1 + tan\u001e2)\u0006hz1j\u00061;1i=p\n3\n3\u0014\nsin\u001e2\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n+\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n+\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n+ cos\u001e2\u0012\f\f\f\u00061\n0\n0E\n+\f\f\f0\n\u00061\n0E\n+\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n9\n10J(1 + cot\u001e1)\u0007hz 2j\u00061;2i=8\n>><\n>>:p\n2\n2\u0014\nsin\u001e1\u0012\f\f\f\f0\n0\n\u00061\u001d\n\u0000\f\f\f0\n\u00061\n0E\u0013\n+ cos\u001e1\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\u0013\u0015\np\n6\n6\u0014\ncos\u001e1\u0012\n2\f\f\f\f\u00071\n\u00061\n\u00061\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00061\n\u00061\n\u00071\u001d\u0013\n+ sin\u001e1\u0012\n2\f\f\f\u00061\n0\n0E\n\u0000\f\f\f0\n\u00061\n0E\n\u0000\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n11\n12J(1\u0000tan\u001e1)\u0007hz 2j\u00061;3i=8\n>><\n>>:p\n2\n2\u0014\ncos\u001e1\u0010\f\f\f\u00061\n0\n0E\n\u0000\f\f\f0\n\u00061\n0E\u0011\n+ sin\u001e1\u0012\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\u0015\np\n6\n6\u0014\nsin\u001e1\u0012\n2\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n\u0000cos\u001e1\u0012\n2\f\f\f\f0\n0\n\u00061\u001d\n\u0000\f\f\f0\n\u00061\n0E\n\u0000\f\f\f\u00061\n0\n0E\u0013\u0015\n213\n14J\u0000Jz\u00072hz 2j\u00062;0i=8\n>><\n>>:p\n2\n2\u0012\f\f\f\u00061\n\u00061\n0E\n\u0000\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\np\n6\n6\u0012\f\f\f\u00061\n\u00061\n0E\n\u00002\f\f\f\f\u00061\n0\n\u00061\u001d\n+\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\n15\n16\u00002J\u0000Jz\u00072hz 1j\u00062;1i=p\n3\n3\u0012\f\f\f\u00061\n\u00061\n0E\n+\f\f\f\f\u00061\n0\n\u00061\u001d\n+\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\n317\n18\u00003Jz\u00073hz 1j\u00063;0i=\f\f\f\f\u00061\n\u00061\n\u00061\u001d\nA. Ground-state phase diagram\nThe ground-state phase diagram shown in Fig. 2(a) to-\ntallyinvolvessevenphasesspecifiedbelow. First, thesat-\nuratedparamagneticphase( SA)hasaccordingtoEq.(3)\nthe following energy per unit cell\nESA=\u00003J0\u00003Jz\u00003hz\u00001\n2h; (4)\nwhich corresponds to the eigenstate defined through the\neigenvectorj3;0iispecified in Table I\njSAi=NY\ni=1j3;0iij+ii: (5)\nObviously, both Ising spin magnetization per unit cell\n(mI=1\n2)andHeisenbergspinmagnetizationperunitcell(mH= 3) are fully polarized, and total magnetization\nper unit cell attains the following value mt=mI+mH=\n7\n2.\nThe ground-state phase diagram shown in Fig. 2(a)\nalso displays three different ferrimagnetic ( FI) phases.\nThe ground-state energy of the first ferrimagnetic phase\nFI1reads\nEFI1=2J+Jz\u00001\n2h; (6)\nwhereas its corresponding eigenvector is given by\njFI1i=NY\ni=1j0;4iij+ii (7)\nwith the eigenvector j0;4iidefined in Table I. In the first\nferrimagnetic phase FI1the Ising spin magnetization is\n3Jz h\n0 10 5 20 15 10 \n020 30 40 50 60 70 \nqF I 1qF I 2qF I 3\nqFR 1qSA \nqFR \n2qFR \n3S(a) (b) \nSA \nFR 1FR \n2FR \n3\nF I 1F I 2F I 3\nJz h\n10 \n020 30 40 50 60 70 \n0 10 5 20 15 Figure 2: (a) Ground-state phase diagram in the Jz\u0000hplane\nby assuming the fixed parameters J=\u000010,J0=\u000010, and\nhz=h; (b) Density-plot of entropy in the Jz\u0000hplane for\nthe same set of parameters as in (a) at T= 0:4.\nmI=1\n2, the Heisenberg spin magnetization equals zero\nmH= 0, and the total magnetization thus becomes mt=\n1\n2.\nThe ground-state energy for the second ferrimagnetic\nphaseFI2can be expressed as\nEFI2=\u0000J0\u00002J(1\u0000cot\u001e2)\u0000hz\u00001\n2h;(8)\nwhere cot (2\u001e2) =Jz+2J\n4Jwith\u0000\u0019\n4<\u001e 2<\u0019\n4. The corre-\nsponding eigenvector reads\njFI2i=NY\ni=1j1;1iij+ii (9)\nwith the eigenvector j1;1iidefined in Table I. The Ising\nspinmagnetizationinthesecondferrimagneticphase FI2\nbecomesmI=1\n2, the Heisenberg spin magnetization is\nmH= 1, and the total magnetization is mt=3\n2.\nThe ground-state energy for the third ferrimagnetic\nphaseFI3is given by\nEFI3=3J0\u00003Jz\u00003hz+1\n2h; (10)\nwhereas its corresponding eigenvector reads\njFI3i=NY\ni=1j3;0iij\u0000ii (11)\nwiththeeigenvector j3;0iibeingdefinedinTableI.Anal-\nogouslytothepreviouscase, theIsingspinmagnetization\nis given by mI=\u00001\n2, the Heisenberg spin magnetization\nequalstomH= 3, andthetotalmagnetizationis mt=5\n2.\nIt should be pointed out that the saturated paramagnetic\nphase as well as all three ferrimagnetic phases are non-\ndegenerate, whichmeansthatthereisnoresidualentropy\nS= 0at zero temperature within those ground states.\nHowever, the ground state of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain may be one of\nthree frustrated ( FR) phases with a nonzero residual en-\ntropy. The ground-state energy of the first frustrated\nphaseFR1is given by\nEFR1=J0\u0000J(1 + cot\u001e1)\u0000hz+1\n2h;(12)where cot (2\u001e1) =Jz\u0000J\n2Jwith\u0000\u0019\n4<\u001e 1<\u0019\n4. The corre-\nsponding ground-state eigenvector reads as follows\njFR1i=NY\ni=1j1;3iij\u0000ii; (13)\nwhere two-fold degenerate eigenstate j1;3iiis specified\nin Table I. Owing to this fact, the frustrated phase FR1\nis macroscopically degenerate with the residual entropy\nS= ln(2)per unit cell when the entropy is measured in\nunits of the Boltzmann constant kB. Note that the Ising\nspin magnetization is being mI=\u00001\n2, the Heisenberg\nspin magnetization is mH= 1, and the total magnetiza-\ntion becomes mt=1\n2.\nTheground-stateenergyofthesecondfrustratedphase\nFR2can be expressed as follows\nEFR2= 2J0+J\u0000Jz\u00002hz+1\n2h (14)\nand its respective eigenvector is given by\njFR2i=NY\ni=1j2;0iij\u0000ii: (15)\nThe definition of two-fold degenerate eigenstate j2;0iiis\nreported in Table I, which implies that the second frus-\ntrated phase FR2also has residual entropy S= ln(2).\nThe Ising spin magnetization is mI=\u00001\n2, the Heisen-\nberg spin magnetization is mH= 2, and the total mag-\nnetization results in mt=3\n2.\nThe ground-state energy of the third frustrated phase\nFR3follows from the relation\nEFR3=\u00002J0+J\u0000Jz\u00002hz\u00001\n2h; (16)\nwhereas its respective eigenvector reads\njFR3i=NY\ni=1j2;0iij+ii: (17)\nThe two-fold degenerate eigenvector j2;0iiis defined\nin Table I and hence, the third frustrated phase FR2\nis macroscopically degenerate with the residual entropy\nS= ln(2) per unit cell. The corresponding Ising spin\nmagnetization achieves the value mI=1\n2, the Heisen-\nberg spin magnetization equals to mH= 2, and the total\nmagnetization is given by mt=5\n2.\nUsually, plots can be drawn in units of some parame-\nters likeJ, and then the temperature can be measured\nin unitsJ. However, here for convenience, we set the pa-\nrameters to be J=\u000010andJ0=\u000010, just for scale the\ntemperature by a factor 10. From now on, we will con-\nsider this set of parameters to study the pseudo-critical\ntemperature throughout the article.\nAll dashed lines in Fig. 2(a) represent usual ground-\nstate phase boundaries between two phases. The resid-\nual entropy per unit cell at the phase boundary between\nFR1andFR2becomesS= ln(4). Similarly, the residual\n4entropy at the interface between FR2andFI3equals to\nS= ln(3), while the residual entropy at the phase bound-\nary between FI3andSAequals toS= ln(2). Analo-\ngously, the residual entropy attains the value S= ln(3)\nat phase boundaries between SA\u0000FR3andFR3\u0000FI2.\nFinally, the residual entropy becomes S= ln(2)at the\ninterface between FI2andFI3. In all aforementioned\ncases the residual entropy per unit cell is always higher\nthan the entropy of both individual phases, which coexist\ntogether at a relevant ground-state boundary. By con-\ntrast, solid lines represent all unusual phase boundaries\nbetween two phases. The residual entropy per unit cell\nS= ln(2)can be found at interfaces between the phases\nFR1-FI1,FR2-FI1,FR2-FI2, andFR3-FI3, whereas\ntheresidualentropyperunitcellvanishes S= 0atthein-\nterface between two non-degenerate ferrimagnetic phases\nFI2andFI3.\nIII. THERMODYNAMICS\nThe mixed spin-(1/2,1) Ising-Heisenberg double-\ntetrahedral chain can be mapped onto the effective spin-\n1/2 Ising chain given by the Hamiltonian\nH=\u0000NX\ni=1\u0002\nK0+Ksisi+1+1\n2B(si+si+1)\u0003\n;(18)\nwhereK0,K, andBare effective temperature-dependent\nparameters. Bearing this in mind, thermodynamics of\nthe effective spin-1/2 Ising chain can be expressed in\nterms of the transfer matrix V=\"\nw1w0\nw0w\u00001#\naccord-\ning to the procedure previously discussed in Ref. [17].\nEach element of the transfer matrix (Boltzmann factor)\nwnwithn=f\u00001;0;1g, which will be further referred to\nas the sector, can be defined as\nwn=18X\nk=0gn;ke\u0000\f\"n;k; (19)\nwhere\f= 1=(kBT),kBisBoltzmann’sconstant, Tisthe\nabsolute temperature and the eigenvalues \"n;kare given\nby Eq. (3).\nTo be more specific, the Boltzmann factors are explic-\nitly given by\nwn=un(\nq3;nz6+\u0012\nx4+2\nx2\u0013\nz2q2;n+\u0000\n2t+x\u00004\u0001\nz2\n+1\nz\u0014\u00122y1\nx+x2y2\u0013\nq1;n+x2y3\u0015\u001b\n;(20)\nwherex= e\fJ=2,z= e\fJz=2,u= e\fh=2,t= 2 cosh (\fJ),\nwhile the coefficients yrandqr;nwithr=f1;2;3gare\ndefined as follows\nyr=2 cosh [\fJcsc (2\u001er)]; (21)\nqr;n=2 cosh [r\f(nJ0+hz)]: (22)The transfer-matrix eigenvalues are determined by the\nfollowing equation\n\u0015\u0006=1\n2\u0010\nw1+w\u00001\u0006q\n(w1\u0000w\u00001)2+ 4w2\n0\u0011\n:(23)\nConsidering the effective spin-1/2 Ising chain under a pe-\nriodic boundary condition gives the partition function\nZN=\u0015N\n++\u0015N\n\u0000. Consequently, the free energy can be\nobtained in the thermodynamic limit ( N!1) accord-\ning to the formula\nf=\u00001\n\fln\u0014\n1\n2\u0010\nw1+w\u00001+q\n(w1\u0000w\u00001)2+ 4w2\n0\u0011\u0015\n:\n(24)\nSubstituting Boltzmann’s factors wninto Eq. (24), we\ncan exactly calculate the free energy of the mixed spin-\n(1/2,1) Ising-Heisenberg double-tetrahedral chain at fi-\nnite temperature.\nIt has been recently demonstrated [17] that some 1D\nlattice-statistical models satisfy the following condition\njw1\u0000w\u00001j\u001dw0at low enough temperatures. Under\nthis condition, the free energy of the mixed spin-(1/2,1)\nIsing-Heisenberg double-tetrahedral chain reduces to\nf=\u0000Tlnfmax [w1(T);w\u00001(T)]g:(25)\nThe final formula for the free energy per unit cell (24)\ntakes the following simple form at a phase boundary be-\ntween the individual phases with the same energy \"c\nf=\"c\u0000Tln [max (g1;0;g\u00001;0)]: (26)\nConsequently, the residual entropy per unit cell at a rel-\nevant phase boundary reads\nSc= ln [max (g1;0;g\u00001;0)]: (27)\nKnowing this quantity is sufficient for prediction of a\npseudo-transition at finite temperatures [18].\nIn Fig. 2(b) we illustrate the density plot of the en-\ntropy as a function of Jzandhfor the fixed temperature\nT= 0:4by using the same scale as in the ground-state\nphase diagram shown in Fig. 2(a). It is quite evident\nthat the entropy follows the vestige of zero-temperature\nphase diagram at finite temperatures. The notation for\nthe ground state is changed at finite temperatures by\nadding a prefix \" q\" to the name of respective ground\nstates, which will denote the respective quasi-phase [8]\nbecause of a lack of true spontaneous long-range order at\nfinite temperatures. It could be expected that thermal\nexcitations basically influence the phase boundaries. It\nhas been argued previously that all dashed curves dis-\nplayed in Fig. 2(a) describe standard interfaces, which\nare manifested through an increase of the entropy ex-\nceedingtheentropyvalueofbothcoexistingphases. Con-\ntrary to this, the phase boundaries depicted by solid lines\nin Fig. 2(a) behave quite differently, since they show at\nthe respective interface a sharp rise of the entropy to a\ngreater entropy of one of two coexisting phases.\n5(f) (e) (d) (c) (b) (a) qFR 1\nhT\nqF I 1\nqSA qFR 2\nqF I 3\nhT\nhT\nhT\nhT\nhTqF I 1\nqSA qFR 2\nqF I 3\nqF I 1\nqSA qFR 2\nqF I 3qFR 1qF I 1\nqSA qF I 3qF I 2qFR 2\nqF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2\nqFR 3\nmIFigure 3: Density plot of Ising spin magnetization in the\nT\u0000hplane for the fixed values of the coupling constants\nJ=\u000010,J0=\u000010, and several values of Jz: (a)Jz=\u000011;\n(b)Jz=\u000013; (c)Jz=\u000015; (d)Jz=\u000015:65; (e)Jz=\u000019;\n(f)Jz=\u000019:85.\nThedensityplotofIsingspinmagnetizationisdepicted\nin Fig. 3 in the T\u0000Jzplane for the following set of\nparameters J=\u000010andJ0=\u000010. In this figure, yellow\nregion corresponds to spin ’up’ ( mI= 1=2), cyan region\ncorresponds to spin ’down’ ( mI=\u00001=2), and red region\ncorresponds to null Ising magnetization ( mI= 0). Surely\nthe temperature in units of T=jJjwould be divided by a\nfactor 10 in Fig. 3 and the following figures.\n(f) (e) (d) (c) (b) (a) \nqF I 1\nqSA qFR 2\nqF I 3qFR 1 qF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2qFR 2\nqFR 3hT\nhT\nqF I 1\nqSA qFR 2\nqF I 3qFR 1\nhT\nqF I 1\nqSA qFR 2\nqF I 3\nhT\nhT\nhTmH\nFigure 4: Density plot of Heisenberg spin magnetization in\ntheT\u0000hplane for the fixed values of the coupling constants\nJ=\u000010,J0=\u000010, and several values of Jz: (a)Jz=\u000011;\n(b)Jz=\u000013; (c)Jz=\u000015; (d)Jz=\u000015:65; (e)Jz=\u000019;\n(f)Jz=\u000019:85.\nThe density plot of the Heisenberg spin magnetization\nis depicted in Fig. 4 in the T\u0000Jzplane for the same\nset of parameters J=\u000010andJ0=\u000010. The color\ncode for the density plot is as follows: yellow region\ncorresponds to the saturated Heisenberg magnetization\nmH= 3, cyan region corresponds to the null Heisenberg\nmagnetization mH= 0, orange region corresponds to the\nmoderate Heisenberg magnetization mH= 2, and dark\nTξξξξ2\nξ\nqF I 1 qFR 1(a) \nqF I 1 qFR 2(b) \n10 3\n10 2qF I 2 qF I 3 (d) \n5\n0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0qF I 3\nqFR 3(e) qFR 2 qF I 2(c) Figure 5: Correlation length against temperature for the\nfixed parameters J=\u000010,J0=\u000010and several values of Jz\nandhz=h: (a)h= 4,Jz=\u000011; (b)h= 11;Jz=\u000011:5;\n(c)h= 26,Jz=\u000015:6; (d)h= 36:76,Jz=\u000017; (e)h= 52;\nJz=\u000019:9.\nred region corresponds to the moderate Heisenberg mag-\nnetizationmH= 1. It can be seen from Figs. 3 and 4\nthat the pseudo-transitions between the quasi-phases is\naccompanied with abrupt change in the magnetization of\nthe Ising spins and/or the magnetization of the Heisen-\nbergspins. ThedensityplotsshowninFig.4(a)-(f)imply\na full alignment of the Heisenberg spins just within the\nquasi-phases qFI 3andqSA.\nNow, let us analyze the correlation length, which can\nbe calculated according to the following simple relation\n\u0018=\u0014\nln\u0012\u0015+\n\u0015\u0000\u0013\u0015\u00001\n: (28)\nThe correlation length is depicted in Fig. 5 as a func-\ntion of temperature for the fixed parameters J=\u000010,\nJ0=\u000010, andhz=h. It is advisable to follow the\nzero-temperature phase diagram to interpret the relevant\ndependences of the correlation length. In Fig. 5(a) we il-\nlustrate the correlation length for h= 4andJz=\u000011,\nwhereas the shark peak delimits the quasi-phases qFI 1\nandqFR 1in agreement with the ground-state phase\nphase diagram shown in Fig. 2(a). Although the cor-\nrelation length seems to diverge at a pseudo-critical tem-\nperature, it is in fact just a sharp finite peak. In Fig. 5(b)\none observes a similar curve for h= 11andJz=\u000011:5,\nbut now the peak indicates a pseudo-transition between\nthe quasi-phases qFI 1andqFR 2. Fig. 5(c) depicts the\ncorrelation length for h= 30andJz=\u000015:65, whereas\nthe sharp peak determines a pseudo-transition between\nthe quasi-phases qFI 2andqFR 2. Similarly, the corre-\nlation length plotted in Fig. 5(d)-(e) demonstrates that\n6a pseudo-transition between the quasi-phases qFI 3-qFI 2\nandqFI 3-qFR 3are accompanied with a sharp robust\npeak of the correlation length. It is worthy to men-\ntion that the quasi-phases melt smoothly upon increas-\ning temperature when the temperature is higher than the\npseudo-critical temperature.It is quite clear from Eq. (25) that the pseudo-critical\ntemperature Tpcan be alternatively obtained by solving\nthe equation\nw1(Tp) =w\u00001(Tp): (29)\nhpTp\nJz =−10 .3Jz =−12 Jz =−14 Jz \n=\n−15 \n.5\n−15 .565 −15 .58 \n−15 .6\n−15 .65 −15 \n.7−16 .5−18 −18 \n−19 .35 \n−19 .6\n−19 .8\n−19 .9−19 \n.35 \nFigure 6: Pseudo-critical temperature as a function of the\nmagnetic field for the fixed values of interaction parameters\nJ=\u000010,J0=\u000010,hz=hand several values of Jz.\nThe numerical solution of Eq. (29) allows us to plot the\npseudo-critical temperature Tpagainst the magnetic field\nhpforseveralvaluesof Jz(seeFig.6). Forsufficientlylow\nmagnetic fields 0<hp<10the pseudo-critical tempera-\nturedelimitsthequasi-phases qFI 1(belowthecurve)and\nqFR 1(above the curve), whereas for the moderate fields\n10.hp.21the pseudo-transition line delimits the\nquasi-phases qFI 1(left from the curve) and qFR 2(right\nfrom the curve). Furthermore, the investigated model\nundergoes a pseudo-transition between the quasi-phases\nqFI 2andqFR 2forTp.0:6and21.hp.31, while\nthepseudo-transitionbetweenthequasi-phases qFI 2(left\nside of the curve) and qFI 3(right side of the curve) takes\nplace for 31.hp.51. Finally, the pseudo-transition\nline delimits the quasi-phases qFI 3(below the curve) and\nqFR 3(above the curve) for high enough magnetic fields\n51.hp<60. Although the condition (29) may still give\nrelatively high values of the pseudo-critical temperature\n(e.g.,T&1), it turns out that the pseudo-critical line\nmelts smoothly for sufficiently high temperatures T\u00181\n(in some particular cases even at lower temperatures). In\ngeneral, there is no way to identify the maximum value\nof the pseudo-critical temperature. Besides, the pseudo-\ncritical temperature also melts for hp!0andhp!60\nas evidenced by Figs. 3 and 4.\nTemperature variations of some thermodynamic quan-\ntities are plotted in Fig. 7 close to a pseudo-transitionbetween the quasi-phases qFR 1andqFI 1for the fixed\nvalues of the interaction parameters J=\u000010,J0=\u000010,\nJz=\u000011, and several values of the magnetic field\nh=f4;6;8;9;10goutlined by {black solid, orange solid,\nred solid, blue dashed, and green dot dashed} curves, re-\nspectively. A strong thermally-induced change of the en-\ntropyS(T)is observable in Fig. 7(a) around the pseudo-\ncritical temperature Tp\u00190:3275. It is worthy to mention\nthat the pseudo-critical temperature remains almost con-\nstant for 0<h< 10. It is quite evident from Fig. 7(b),\nmoreover, that the Ising spins are mostly aligned parallel\ntothemagneticfield( mI= 0:5)belowthepseudo-critical\ntemperature T <Tpand antiparallel ( mI=\u00000:5) above\nitT >Tp. Contrary to this, the Heisenberg spins almost\ndo not contribute to the total magnetization ( mH= 0)\nbelowthepseudo-criticaltemperature T <Tp, whilethey\nprovide a significant contribution (mH= 1)above it\nT > Tp. Last but not least, the specific heat and mag-\nnetic susceptibility displayed in Fig. 7(d)-(e) in a semi-\nlogarithmic scale serve in evidence of a pseudo-transition\nthrough a strong narrow peak observable at the pseudo-\ncritical temperature.\nTemperature dependences of selected thermodynamic\nquantities are depicted in Fig. 8 by assuming the fixed\nvalues of the interaction parameters J=\u000010,J0=\n\u000010, and (h;Jz) ={(11;\u000011:5),(13;\u000012),(15:5;\u000013),\n(16:9;13:6),(18:81;\u000014:5)} outlined by {black solid, or-\n7TC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 7: Temperature dependences of some thermodynamic\nquantities by considering the fixed parameters J=\u000010,\nJ0=\u000010,Jz=\u000011, and several values of the magnetic field\nh=f4;6;8;9;10g(black solid, orange solid, red solid, blue\ndashed, and green dot dashed): (a) entropy S; (b) Ising spin\nmagnetization; (c) Heisenberg spin magnetization; (d) spe-\ncific heat (semi-logarithmic plot); (e) magnetic susceptibility\n(semi-logarithmic plot).\nange solid, red solid, blue dashed, and green dot dashed}\ncurves, respectively. The present choice of the interac-\ntion parameters is consistent with the pseudo-transition\nbetween the quasi-phases qFI 1andqFR 2, which varies\nwith the interaction parameter Jzand magnetic field h.\nIt is obvious from Fig. 8(a) that the entropy S(T)ex-\nhibits a steep increase close to a pseudo-critical tempera-\ntureTp, while the magnetization of Ising spins shown in\nFig. 8(b) is pointing upward ( mI= 0:5) forT <Tpand\ndownward ( mI=\u00000:5) forT >Tp. Similarly, the mag-\nnetization of Heisenberg spins illustrated in Fig. 8(c)\nis zero (mH= 0) forT < Tp, while there is a sudden\nchange atT=Tpabove which it strongly depends on\nthe magnetic field hand the coupling constant Jz. Fi-\nnally, sharp narrow peaks can be repeatedly detected at\na pseudo-critical temperature in the respective tempera-\nture dependences of the specific heat [Fig. 8(d)] and the\nmagnetic susceptibility [Fig. 8(e)].\nA pseudo-transition between the quasi-phases qFI 2\nandqFR 2is illustrated in Fig. 9 by considering the fixed\nparameters J=\u000010,J0=\u000010,Jz=\u000015:65, and sev-\neral values of the magnetic field of h=f25;27;28;29;30g\noutlined by {black solid, orange solid, red solid, blue\nTC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 8: Temperature dependences of some thermodynamic\nquantities by considering the fixed parameters J=\u000010,\nJ0=\u000010, and (h;Jz) ={(11;\u000011:5),(13;\u000012),(15:5;\u000013),\n(16:9;13:6),(18:81;\u000014:5)} (black solid, orange solid, red\nsolid, blue dashed, and green dot dashed): (a) entropy S;\n(b) Ising spin magnetization; (c) Heisenberg spin magnetiza-\ntion; (d) specific heat (semi-logarithmic plot); (e) magnetic\nsusceptibility (semi-logarithmic plot).\ndashed, and green dot dashed} curves, respectively.\nFig. 9(a) shows the entropy S(T)as a function of tem-\nperature: for T < Tpthe entropy increases significantly\nbut is virtually independent of h(for22>h>30), then\na sudden rise occurs at T=Tpfollowed by a successive\nsmooth increase for T > Tp. The Ising magnetization\ndepicted in Fig. 9(b) is nearly constant mI= 0:5for\nT < Tp, but it becomes almost \u00000:5forT&Tpbefore\nshowing a continuous rise approaching null upon further\nincrease of temperature. Analogously, the Heisenberg\nspin magnetization illustrated in Fig. 9(c) tends to zero\nmH!1forT <Tp, while it approaches to mH!2for\nT&Tp. The specific heat and magnetic susceptibility\nplotted in Fig. 9(d)-(e) in a semi-logarithmic scale dis-\nplay vigorous narrow peaks verifying a pseudo-transition\nbetween the quasi-phases qFI 2andqFR 2.\nNext, the pseudo-transition at the interface between\nthe quasi-phases qFI 2andqFI 3is illustrated in Fig. 10\nby considering set of the parameters J=\u000010,J0=\u000010,\nand (h;Jz) ={(36:76;\u000017),(38:7;\u000017:5),(40:6;\u000018),\n(42:55;18:5),(44:45;\u000019)} drawn by {black solid, or-\nange solid, red solid, blue dashed, and green dot dashed}\ncurves, respectively. Itshouldbestressedthatbothcoex-\nistingquasi-phases qFI 2andqFI 3arenon-frustratedand\n8TC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 9: Temperature dependences of some thermodynamic\nquantities by considering the fixed parameters J=\u000010,\nJ0=\u000010,Jz=\u000015:65, and several values of the magnetic\nfieldh=f25;27;28;29;30g(black solid, orange solid, red\nsolid, blue dashed, and green dot dashed): (a) entropy S;\n(b) Ising spin magnetization; (c) Heisenberg spin magnetiza-\ntion; (d) specific heat (semi-logarithmic plot); (e) magnetic\nsusceptibility (semi-logarithmic plot).\nconsequently, the residual entropy per unit cell should\nalsobecomenullaccordingtoEq.(27). Theentropy S(T)\nas a function of temperature shown in Fig. 10(a) is for\nT <Tpnearly zero, then it shows a small but sudden rise\natT=Tp, which is followed by a roughly linear increase\nforT >Tp. The magnetization of Ising spins [Fig. 10(b)]\ndisplays an opposite behavior to the previous one: the\nIsing spins are aligned in opposite to the magnetic field\n(mI=\u00000:5) forT < Tpand they are aligned in the\nmagnetic-field direction ( mI= 0:5) forT > Tp. Simi-\nlarly, the magnetization of Heisenberg spins [Fig. 10(c)]\nis close to its maximal value mH= 3forT < Tpand\nit suddenly drops to mH= 1forT > Tp. Finally, one\nobserves a typical narrow peak in thermal variations of\nthe specific heat and magnetic susceptibility displayed in\nFig. 10(d)-(e).\nLast but not least, let us discuss a pseudo-transition\nbetween the quasi-phases qFR 3andqFI 3exemplified in\nFig. 11 for the fixed values of the interaction parameters\nJ=\u000010,J0=\u000010,Jz=\u000019:9, and several mag-\nnetic fields h=f50;52;54;56;56:5gsketched by {black\nsolid, orange solid, red solid, blue dashed, and green dot\ndashed} curves, respectively. It is noteworthy that ther-\nmal variation of the entropy S(T)displayed in Fig. 11(a)\n2\nTC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 10: Temperature dependences of some thermody-\nnamic quantities by considering the fixed parameters J=\n\u000010,J0=\u000010, and (h;Jz) ={(36:76;\u000017),(38:7;\u000017:5),\n(40:6;\u000018),(42:55;18:5),(44:45;\u000019)} (black solid, orange\nsolid, red solid, blue dashed, and green dot dashed)): (a) en-\ntropyS; (b) Ising spin magnetization; (c) Heisenberg spin\nmagnetization; (d) specific heat (semi-logarithmic plot); (e)\nmagnetic susceptibility (semi-logarithmic plot).\nis quite reminiscent of the entropy dependence illustrated\nin Fig. 7(a). In addition, the temperature dependences\nof the magnetization of the Ising and Heisenberg spins\nshown in Fig. 11(b) and (c) are quite similar to the pre-\nvious cases shown in Fig. 10(b) and (c), respectively. Al-\nthough the specific heat shows a strong narrow peak at\nthe pseudo-critical temperature, it often becomes negli-\ngible further away from the pseudo-critical temperature\n[see Fig. 11(d)]. The similar situation can be also found\nin the temperature dependences of the magnetic suscep-\ntibility shown in Fig. 11(e).\nIV. CONCLUSIONS\nThepseudo-transitionsofthemixedspin-(1/2,1)Ising-\nHeisenberg double-tetrahedral chain are examined in de-\ntail at non-zero temperature and magnetic field. The\nground-statephasediagramoftheinvestigatedspinchain\ntotally involves seven phases, three of which can be clas-\nsified as the non-degenerate ferrimagnetic phases, three\nas the macroscopically degenerate frustrated phases, and\none as the saturated paramagnetic phase. Interestingly,\nfive different ground-state boundaries of the mixed spin-\n9TC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 11: Temperature dependences of some thermody-\nnamic quantities by considering the fixed parameters J=\n\u000010,J0=\u000010,Jz=\u000019:9, and several values of the mag-\nnetic fieldh=f50;52;54;56;56:5g(black solid, orange solid,\nred solid, blue dashed, and green dot dashed): (a) entropy S;\n(b) Ising spin magnetization; (c) Heisenberg spin magnetiza-\ntion; (d) specific heat (semi-logarithmic plot); (e) magnetic\nsusceptibility (semi-logarithmic plot).(1/2,1) Ising-Heisenberg double-tetrahedral chain repre-\nsent peculiar interfaces, at which the residual entropy\nper unit cell is simply given by the larger entropy of\none of two coexisting phases. This condition seems\nto be sufficient criterion whether or not the pseudo-\ntransition does emerge in a close vicinity of the ground-\nstate phase boundary. In fact, the residual entropy per\nunit cell at the usual ground-state phase boundaries is\nstrictly larger than the residual entropy of both coexist-\ning phases. Although thermal fluctuations usually de-\nstroy in 1D lattice-statistical models with short-range\ninteractions all fingerprints of the ground-state phase\nboundaries, the aforementioned five interfaces are quite\nrobust with respect to thermal fluctuations. In conse-\nquence of that, the mixed spin-(1/2,1) Ising-Heisenberg\ndouble-tetrahedral chain may exhibit in a vicinity of five\naforedescribed ground-state phase boundaries a marked\npseudo-transition manifested by vigorous narrow peaks\nof the specific heat and magnetic susceptibility besides a\nsudden change of the entropy and magnetization.\nAcknowledgments\nThisworkwaspartiallysupportedbyBrazilianAgency\nCNPq and FAPEMIG.\n[1] J. F. Nagle, Am. J. Phys. 36, 1114 (1968).\n[2] C. Kittel, Am. J. Phys. 37, 917 (1969).\n[3] S. T. Chui and J. D. Weeks, Phys. Rev. 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Rojas, Solid State Commun. 269,\n131 (2017).\n[18] O. Rojas, arXiv:1810.07817.\n[19] T. Krokhmalskii, T. Hutak, O. Rojas, S. M. de Souza,\nand O. Derzhko, Towards low-temperature peculiarities\nof thermodynamic quantities for decorated spin chains,\narXiv:1908.06419.\n[20] O. Rojas, J. Strečka, M. L. Lyra, and S. M. de Souza,\nPhys. Rev. E 99, 042117 (2019).\n[21] S.BuhrandtandL.Fritz, Phys.Rev.B 90, 094415(2014)\n[22] A. Otsuka, D. V. Konarev, R. N. Lyubovskaya,\nS. S. Khasanov, M. Maesato, Y. Yoshida, and G. Saito,\nCrystals 8, 115 (2018).\n[23] M. Mambrini, J. Trébosc, and F. Mila, Phys. Rev. B 59,\n13806 (1999).\n[24] O. Rojas and F. C. Alcaraz, Phys. Rev. B 67, 174401\n(2003).\n[25] M. Maksymenko, O. Derzhko, and J. Richter, Acta\nPhysica Polonica A 119, 860 (2011); M. Maksymenko,\nO. Derzhko, and J. Richter, Eur. Phys. J. B 84, 397\n(2011).\n[26] V. Ohanyan, Physics of Atomic Nuclei 73, 494 (2010).\n10[27] D. Antonosyan, S. Bellucci, and V. Ohanyan, Phys. Rev.\nB79, 014432 (2009).\n11"    },    {        "title": "2208.00756v2.Effect_of_magnetism_and_phonons_on_localized_carriers_in_the_ferrimagnetic_kagome_metals_GdMn__6_Sn__6__and_TbMn__6_Sn__6_.pdf",        "content": "Effect of magnetism and phonons on localized carriers in the ferrimagnetic kagome\nmetals GdMn 6Sn6and TbMn 6Sn6\nM. Wenzel,1,∗A. A. Tsirlin,2, 3O. Iakutkina,1Q. Yin,4H.C. Lei,4M. Dressel,1and E. Uykur1, 5\n11. Physikalisches Institut, Universit¨ at Stuttgart, 70569 Stuttgart, Germany\n2Felix Bloch Institute for Solid-State Physics, Leipzig University, 04103 Leipzig, Germany\n3Experimental Physics VI, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, 86135 Augsburg, Germany\n4Laboratory for Neutron Scattering, and Beijing Key Laboratory of Optoelectronic Functional Materials MicroNano Devices,\nDepartment of Physics, Renmin University of China, Beijing 100872, China\n5Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany\n(Dated: December 22, 2022)\nKagome metals possess peculiar optical spectra consisting of contributions from free charge carri-\ners in a Drude-type response, localized carriers seen as a strongly temperature-dependent localization\npeak, and, in some cases, phonons displaying strong anomalies. The rare-earth kagome metal series,\nRMn6Sn6, provides a marvelous playground to study the electronic properties of kagome metals in\nthe presence of variable magnetic order. Here, we report temperature-dependent reflectivity studies\non two members of the RMn6Sn6family, GdMn 6Sn6(in-plane ferrimagnet) and TbMn 6Sn6(out-\nof-plane ferrimagnet), in a broad energy range (50 - 18000 cm−1, equivalent to 6.2 meV - 2.23 eV)\ndown to 10 K. At high temperatures, a phonon mode at approximately 160 cm−1is observed, which\nbecomes screened out in TbMn 6Sn6below ∼150 K as the localization peak linearly passes through\nthe mode. In GdMn 6Sn6, the disappearance of the phonon is accompanied by the onset of saturation\nof the peak position, suggesting an unusual interplay between the two features.\nProposed by Syˆ ozi in 1951, the kagome lattice quickly\nbecame popular among both theoretical and experimen-\ntal physicists due to its unique magnetic and electronic\nproperties [1, 2]. Consisting of spatially separated metal-\nlic kagome planes, kagome metals are model compounds\nfor studying strong electronic correlations, magnetism,\nand topologically non-trivial states [3]. Here, the itiner-\nant carriers give rise to the peculiar kagome electronic\nband structure hosting dispersionless flat bands, saddle\npoints, as well as linearly dispersing Dirac bands [4–9].\nThe ternary rare-earth series, RMn6Sn6(R= Sc, Y,\nGd-Lu), opens new ways to investigate the influence of\nmagnetism on the electronic properties of kagome met-\nals and hence, distinguish between magnetic-driven and\nkagome layer-driven properties. While these compounds\nhave been studied extensively over the last three decades\nregarding their unusual magnetic structure, they recently\ngained attention in the framework of kagome metals [10–\n12]. These compounds crystallize in the P6/mmm space\ngroup featuring spatially decoupled magnetic Mn-kagome\nplanes stacked along the c-axis, which are stabilized by\nSn1 atoms. Within one unit cell, the kagome layers are\nseparated by non-magnetic Sn2 atoms forming a hon-\neycomb lattice, while RSn3 layers separate the kagome\nplanes from one unit cell to another, as sketched in\nFigs. 1(a) and 1(b). The underlying magnetic structure\nstrongly depends on the rare-earth element separating\nthe layers, resulting in a large variety of ferrimagnetic ( R\n= Gd, Tb, Dy, Ho) and antiferromagnetic ( R= Sc, Y,\nEr, Tm, Yb, Lu) ground states across the series [10, 13].\n∗maxim.wenzel@pi1.physik.uni-stuttgart.deAngle-resolved photoemission spectroscopy (ARPES)\nand Landau level measurements reveal the signatures\nof the kagome lattice, including topologically non-\ntrivial Dirac bands and flat bands in these materi-\nals [7, 9, 12, 14]. Comprising spin-polarized Mn 3 d\nstates with a strong intrinsic spin-orbit coupling, these\ntwo-dimensional kagome bands exhibit non-trivial Chern\nnumbers [6, 7, 15] giving rise to an intrinsic anomalous\nHall effect [16–21]. While the different magnetic struc-\ntures do not seem to affect the main band dispersions\nnear the Fermi energy EF, significantly, a gap at the\nDirac points has been proposed only for the ferrimag-\nnetic systems [12, 22–24]. Moreover, this Chern gap, as\nwell as the energy of the Dirac points ED, can be tuned\nwith the rare-earth element [22]. Here, the number of\nunpaired 4 felectrons of the rare-earth element plays an\nimportant role as a coupling between the 4 fand the 3 d\nelectrons is observed.\nThe key implications of these topological features lie in\nunusual transport properties that crucially rely on charge\ncarriers and their dynamics [12, 27–29]. Especially the ef-\nfect of magnetism is one of the central issues [30]. There-\nfore, here, we study these dynamics and their depen-\ndence on the magnetic order with temperature-dependent\nbroadband Fourier transform infrared spectroscopy stud-\nies on RMn6Sn6systems, namely on GdMn 6Sn6and\nTbMn 6Sn6. While both systems possess an almost iden-\ntical crystal structure and a ferrimagnetic ground state\nbelow room temperature, in the former one, the spins are\naligned within the kagome plane, whereas in the Tb com-\npound, a perpendicular alignment to the kagome layers is\nreported [10, 13, 31–33]. This was confirmed prior to our\noptical study by dc transport and magnetic susceptibil-\nity measurements shown in Figs. 1(c) and 1(d). We fur-arXiv:2208.00756v2  [cond-mat.str-el]  21 Dec 20222\n0.00.40.81.21.61\n01 001 000100000.00.40.81.21.60\n1 0020030001234565.05.56.06.57.0E\n II ab 10 K \n50 K \n100 K \n150 K \n175 K(\nf)  σ1 (104Ω-1cm-1)(\ne) \nFrequency (cm-1) 200 K \n225 K \n250 K \n275 K \n300 KTbMn6Sn6 σ1 (104Ω-1cm-1) \nE II abGdMn6Sn6(c)G dMn6Sn6 \nTemperature (K)(d)T bMn6Sn6 χ (µB/f.u.)H\n = 0.1 TH\n II abTr0\n50100150ρ\n (µΩcm)3\n003300.30.60.9dρ/dTT\n (K)H = 0.1 TH\n II ab χ (µB/f.u.)0\n50100150ρ\n (µΩcm)0.010 .11 Energy (eV)\nFIG. 1. (a) and (b) Crystal and magnetic structure below 300 K of GdMn 6Sn6and TbMn 6Sn6, respectively [13, 25]. (c) and (d)\nMagnetic susceptibility and dc resistivity curves measured in the ab-plane. The Curie temperature of both systems lies above\nthe measured temperature range; however, a spin reorientation from the basal plane near to the c-axis around Tr∼310 K\nis visible for TbMn 6Sn6. For GdMn 6Sn6, no anomalies are observed in the measured temperature range. Open circles are\nthe dc resistivity values obtained from the Hagen-Rubens fits of the optical measurements as explained in the Supplemental\nMaterial [26]. (e) and (f) Temperature-dependent in-plane optical conductivity with the dotted lines being the Hagen-Rubens\nextrapolation to low energies.\nther performed density functional theory plus Hubbard\nU(DFT+ U) calculations to evaluate the electronic struc-\ntures, revealing the correlated character of the RMn6Sn6\nseries. Due to localization effects, the optical response\nof the charge carriers splits into the conventional Drude\npart and a prominent low-energy peak. This peak shows\na clear dependence on the magnetic order and underlies\nthe magnetic tunability of this compound family.\nFigures 1(e) and 1(f) display the temperature-\ndependent real part of the in-plane optical conductiv-\nity of GdMn 6Sn6and TbMn 6Sn6, respectively. At first\nglance, the spectra are remarkably similar and resem-\nble the spectrum of the ferromagnetic Fe 3Sn2[34, 35].\nConsistent with the metallic nature of these compounds,\na very narrow Drude component is observed at low en-\nergies, which becomes even sharper upon cooling. For\nGdMn 6Sn6, only the tail of this feature is visible even at\n300 K. Two step-like absorption features can be identified\nin the otherwise relatively flat conductivity at high ener-\ngies. Very similar steps were interpreted as the signature\nof two-dimensional Dirac fermions in Fe 3Sn2. In addi-\ntion to the sharp Drude component and interband tran-\nsitions, a phonon mode around 160 cm−1is observed.\nFurthermore, we have realized that the low-energy dy-\nnamics cannot be reproduced only with a single Drude\ncomponent, but an additional peak-like absorption fea-\nture is required as shown in Fig. 2 (a) and (b) for thedata at 300 K. With this peak showing a strong red-shift\nupon cooling, it puts the RMn6Sn6series on common\nground with other kagome metals and clearly separates\nthis feature from other low-energy transitions, which are\ninterband in nature [34, 36–38].\nA closer look at the low-energy regime reveals sub-\nstantial differences between the two ferrimagnetic com-\npounds. Figures 2 (b) and 2(d) show the temper-\nature evolution of this so-called localization peak in\nGdMn 6Sn6and TbMn 6Sn6after subtracting the fitted\nDrude, phonon, and interband contributions from the\nexperimental optical conductivity. Not only is the lo-\ncalization peak more pronounced in the in-plane ferri-\nmagnetic system GdMn 6Sn6, but the peak position sat-\nurates at low temperatures, as shown in Fig. 2(a). In\ncontrast, a linear red-shift over the whole temperature\nrange is observed in TbMn 6Sn6[see Fig. 2(c)]. Hence,\nthe peak moves out of the measured range at low tem-\nperatures, and its position has to be estimated from its\nhigh-frequency tail, as well as by considering the linear\nbehavior of the shift at higher temperatures, leading to\nincreasing error bars of the fits.\nVisually, the temperature evolution of the peak posi-\ntion in GdMn 6Sn6looks strikingly similar to the behavior\nin Fe 3Sn2. For the latter, a possible coupling between the\nlocalization peak and the underlying magnetic structure\nis discussed since the linear scaling breaks down after a3\n024681\n0100100010000024681\n01 001 0001 00000.00.20.40.60.81.01.20.00.20.40.60.81.01.20\n100200300(\nf)(e)(\nd) Peak Position (cm-1)G dMn6Sn6ω\nPhonon(c)0\n1 002003000100200300T\nbMn6Sn6TbMn6Sn6 Peak Position (cm-1)T\nemperature (K)ωPhonon \nσ1 (103Ω-1cm-1) \n300 K200 K100 K10 KGdMn6Sn60.010.11 Energy (eV) \n σ1 (103Ω-1cm-1)F\nrequency (cm-1)10 K1\n00 K2\n00 K3\n00 K(b) \nFrequency (cm-1) experiment \ntotal fit \nDrude \nlocalization \nphonon \ninterband \nσ1 (104Ω-1cm-1)3\n00 KT\nbMn6Sn6 σ1 (104Ω-1cm-1) \nG\ndMn6Sn6(a)3\n00 K0.010 .11 Energy (eV)1\n01 0010000.70.80.91.0ReflectivityF\nrequency (cm-1)1\n01 0010000.70.80.91.0ReflectivityF\nrequency (cm-1)\nFIG. 2. (a) and (b) Decomposed optical conductivity at 300 K, consisting of a Drude component (purple), a localization peak\n(blue), a phonon mode (green), and several interband transitions (orange). The insets show the total fit to the measured\nreflectivity. Details on the fitting process as well as the decomposed spectra at lower temperatures can be found in the\nSupplemental Material [26]. (c) and (d) Temperature dependence of the localization peak position. The red dashed line marks\nthe phonon mode, while the red arrow indicates the temperature where the mode disappears. (e) and (f) Temperature evolution\nof the localization peak, obtained by subtracting the fitted Drude, phonon mode, and interband contributions from the spectra.\nThe solid lines are the Fratini model fits to the total experimental conductivity as described in the Supplemental Material [26].\nreorientation of the Fe-spins at 120 K [34, 39]. Addition-\nally, the shape of the peak transforms into a sharp Fano\nresonance. The saturation as observed in GdMn 6Sn6was\nalso reported in the non-magnetic KV 3Sb5, suggesting\nthat the origin of this effect may be other than mag-\nnetic. Additionally, no change of the in-plane ferrimag-\nnetic structure of GdMn 6Sn6is reported below room tem-\nperature; hence, the primary cause for the observed satu-\nration must be something else. Nevertheless, a common-\nality between the two magnetic systems is the in-plane\ndirection of the magnetic moments in both Fe 3Sn2below\nits spin-reorientation transition and GdMn 6Sn6.\nOne plausible explanation for the observed saturation\nuniting magnetic and non-magnetic kagome metals is the\ninvolvement of a phonon mode. Indeed, phonons and\ntheir importance for the electronic structure of kagome\nmetals have been studied in multiple compounds. In the\nAV3Sb5family, phonons are discussed to be the driv-\ning force behind the charge-density-wave formation and\nthe low-temperature superconductivity [40, 41]. Opti-\ncal measurements revealed strong phonon anomalies as-\nsociated with a coupling of the phonon modes to the\nelectronic background in KV 3Sb5and RbV 3Sb5[36, 37].\nFurthermore, a strong interplay between phonons and\nfermionic degrees of freedom was revealed by scanningtunneling microscopy (STM) studies of paramagnetic\nCoSn [42].\nDFT calculations, shown in the Supplemental Mate-\nrial [26], reveal a total number of nine IR-active phonon\nmodes in each compound. Four of these modes have the\nA2usymmetry involving out-of-plane atomic displace-\nments and hence, cannot be detected by our in-plane\nmeasurements. While in highly metallic systems phonon\nmodes are often too weak to be detected and/or screened\nby the free carriers, our measurements were able to cap-\nture a prominent E 1umode around 160 cm−1at room\ntemperature. At low temperatures, this mode disap-\npears in both compounds. At first glance, this anomalous\nbehavior might be explained by a structural distortion;\nhowever, low-temperature XRD studies report almost no\nchanges in the crystal structure of RMn6Sn6down to 2 K\n[13, 25]. Hence, an interplay between the phonon mode\nand the localization peak has to be considered as a possi-\nble scenario, not least because both features are located\naround the same energy range.\nFor a further comparison of the two features, the posi-\ntion of the phonon mode is marked with the red dashed\nline in Figs. 2(a) and 2(c), while the red arrow points\nat the temperature at which the phonon mode disap-\npears in each compound. In TbMn 6Sn6, the phonon4\nmode disappears as soon as the localization peak passes\nthrough it, suggesting that the localization peak screens\nout the phonon mode. On the other hand, a more com-\nplex relationship between the two features is observed in\nGdMn 6Sn6. Here, the phonon mode shows an enhance-\nment and a slight broadening as the localization peak\npasses through it, and is retained even below the cross-\ning over a narrow temperature range. Eventually, the\nmode disappears around the temperature where the po-\nsition of the localization peak saturates. This behavior\nsuggests an unusual coupling between the phonon mode\nand the localization peak in GdMn 6Sn6. Based on the ob-\nservation that the strong localization peak anomalies ap-\npear in the in-plane ferromagnetic system, one plausible\nexplanation would be a magneto-elastic coupling to the\nin-plane infrared-active phonon mode. Additionally, the\nrare-earth element could directly influence the phonon\nmode and hence its interplay with the localization peak.\nUltimately, an interplay with some other bosonic ex-\ncitations such as magnons, for instance, could as well\nlead to the distinct behavior of the localization peak in\nGdMn 6Sn6compared to TbMn 6Sn6. Indeed, magnon\nbands extending to energies up to ∼100 meV have\nbeen reported in several members of the RMn6Sn6family\n[43, 44].\nThe presence of a red-shifting localization peak is a\ncommon occurrence in systems with slow electron dy-\nnamics, such as organic conductors, cuprates, and man-\nganites [45, 46], many of them being strongly correlated\nmaterials. Hence, we now turn to analyzing the elec-\ntronic correlations in the RMn6Sn6series. Figures 3(a)\nand 3(b) show the comparison between the experimental\nand the calculated optical conductivities using DFT tak-\ning into account the different magnetic structures. For\nall calculations, a Hubbard UR= 10 eV was added\nto the rare-earth element with the DFT+ Umethod us-\ning the double-counting correction in the fully localized\nlimit to treat the strongly correlated 4 felectrons [9, 47–\n49]. In the case of GdMn 6Sn6, a good agreement with\nthe experiment is found, while for TbMn 6Sn6, the low-\nenergy spectral weight cannot be reproduced with this\nmethod. The agreement is improved by adding a Hub-\nbardUMn= 0.4 eV to the Mn-atoms. Another possibility\nis shifting the Fermi energy down by 47 meV; however,\nthis requires removing one electron from the structure,\nwhich is hard to reconcile with the system.\nAlthough with different adjustments, one can bring the\ncalculations to the experiment’s level, in either case, the\nenergy of the calculated conductivity needs to be rescaled\nby a factor of 2.5 in GdMn 6Sn6(2 in TbMn 6Sn6). A very\nsimilar scaling factor was previously reported for ARPES\nmeasurements of GdMn 6Sn6[9]. This suggests that these\nsystems are clearly beyond DFT, and electronic correla-\ntions therein can not be fully treated on the mean-field\nDFT+ Ulevel.\nWe further observed the step-like absorption features,\ncombined with the relatively flat optical conductivity, as\nthe potential signatures of the Dirac points in these sys-\n0.00 .20 .40 .60 .81 .00123456 \nσ1 (103Ω-1cm-1) experiment at 10 K \nstoichiometricG\ndMn6Sn6(\nb)(\nc)1001 0001 00000123456 \nFrequency (cm-1) σ1 (103Ω-1cm-1) \n experiment at 10 K \nstoichiometric \nµ = - 47 meV \nUMn = 0.4 eVT\nbMn6Sn6(a) \nS\nWDrude / SWbandReMn6Sn6KV3Sb5RbV3Sb5CsV3Sb5correlationsC\no3Sn2S2ZrSiSeZrSiSW\nTe20.11 Energy (eV)FIG. 3. (a) and (b) Experimental interband transitions along\nwith the DFT+ Ucalculated optical conductivity. For all cal-\nculations a Hubbard UR= 10 eV was added to the rare-earth\nelement. Furthermore, the energy scale of the calculated con-\nductivity is rescaled for a better comparison with the experi-\nment. (c) Correlation scaling for different kagome metals and\nother topological materials taken from ref. [50].\ntems. Considering that there are two Dirac points, one\nabove and one below the Fermi energy (see Supplemen-\ntal Material [26]), one would expect these step-like ab-\nsorption features to appear [34]. This interpretation be-\ncomes even more tempting when the energies of the steps\nare compared with the ARPES measurements. However,\nconsidering the relatively high energy range of these fea-\ntures and the significant number of bands crossing the\nFermi energy, the step-like absorption is most likely just\na cumulative effect of different contributions; hence, one\nshould be careful in its assignment. On the other hand,\nabsorption features at lower energies ( ω < 1000 cm−1)\ncan be related to transitions between bands very close\nto the Fermi energy, most probably involving transitions\nbetween the saddle points nearby the Mpoint, as shown\nin our band structure calculations in the Supplemental\nMaterial [26].5\nAlthough the RMn6Sn6series lies beyond the limits\nof the DFT+ Umethods presented here, the calcula-\ntions can be used for an initial assessment of the cor-\nrelation strength. As proposed previously for different\ncompounds, including cuprates, iron pnictides, and topo-\nlogically nontrivial Dirac systems [50, 51], the ratio of\nthe spectral weight of the mobile carriers from the ex-\nperiment and the DFT calculations can be used as a\ngauge of electronic correlations. Here, SW Drude /SW band\nis close to 1 for uncorrelated materials, while the ra-\ntio becomes zero for Mott insulators showing the most\ncorrelated behavior. Figure 3(c) depicts this scaling\nfor the AV3Sb5series and topological semimetals taken\nfrom refs. [36, 50]. From the calculations, we can deter-\nmine a rough value of SW Drude /SW band≈0.2, point-\ning towards much stronger correlations in comparison\nwith the AV3Sb5series and other kagome metals re-\nported to date. Moreover, no significant difference be-\ntween GdMn 6Sn6and TbMn 6Sn6is observed, whereas\nthe correlation strength changes drastically between dif-\nferent members of the AV3Sb5family.\nIn summary, we establish the correlated nature of ferri-\nmagnetic kagome metals of the RMn6Sn6family and un-\ncover partial localization of charge carriers manifested by\nthe prominent low-energy peak in the optical conductiv-\nity. The temperature evolution of this peak is sensitive to\ndetails of the magnetic order. While in TbMn 6Sn6, the\nlocalization peak red-shifts linearly through the whole\ntemperature range upon cooling and screens out thephonon mode at ∼160 cm−1, it displays different char-\nacteristics in GdMn 6Sn6. Here, the peak is more pro-\nnounced, while its position saturates at low tempera-\ntures. This dissimilar behavior indicates a major dif-\nference in low-energy degrees of freedom that damp elec-\ntron dynamics and, consequently, should affect transport\nproperties at low temperatures. Both compounds dis-\nplay a strongly correlated character, as a good agreement\nwith the experimental interband transitions is only found\nafter rescaling the energy of the calculated optical con-\nductivity, and the experimental Drude spectral weight is\ndrastically lower than the DFT prediction.\nThe authors acknowledge the fruitful discussion with\nSimone Fratini, and technical support by Gabriele Un-\ntereiner. We also thank Falk Lissner and Rainer Niewa\nfor the XRD measurements. H.C.L. was supported\nby National Key R&D Program of China (Grant No.\n2018YFE0202600), the Beijing Natural Science Foun-\ndation (Grant No. Z200005), the Fundamental Re-\nsearch Funds for the Central Universities and Research\nFunds of Renmin University of China (RUC) (Grant\nNos. 18XNLG14, 19XNLG13, and 19XNLG17), and\nthe Beijing National Laboratory for Condensed Matter\nPhysics. The work has been supported by the Deutsche\nForschungsgemeinschaft (DFG) via Grants No. UY63/2-\n1, No. DR228/48-1, and No. DR228/51-1. E. U. ac-\nknowledges the European Social Fund and the Baden-\nW¨ urttemberg Stiftung for the financial support of this\nresearch project by the Eliteprogramme.\n[1] I. Syˆ ozi, Statistics of Kagom´ e Lattice, Progress of Theo-\nretical Physics 6, 306 (1951).\n[2] M. Mekata, Kagome: The Story of the Basketweave Lat-\ntice, Physics Today 56, 12 (2003).\n[3] D. F. Liu, A. J. Liang, E. K. Liu, Q. N. Xu, Y. W.\nLi, C. Chen, D. Pei, W. J. Shi, S. K. Mo, P. Dudin,\nT. Kim, C. 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Sofo, Linear optical properties\nof solids within the full-potential linearized augmented\nplanewave method, Comput. Phys. Commun. 175, 1\n(2006).1\nSupplemental Material for ”Effect of magnetism and phonons on localized carriers in\nferrimagnetic kagome metals GdMn 6Sn6and TbMn 6Sn6”\nM. Wenzel, A. A. Tsirlin, O. Iakutkina, Q. Yin, H.C. Lei, M. Dressel, and E. Uykur\nI. CRYSTAL GROWTH\nSingle crystals of GdMn 6Sn6and TbMn 6Sn6were grown by the Sn flux method with Gd/Tb : Mn : Sn = 1 : 6 : 20\nmolar ratio. Gd/Tb (ingots), Mn (pieces) and Sn (grains) were put into an alumina crucible and sealed in a quartz\nampule under partial argon atmosphere. The sealed quartz ampule was heated up to 1373 K and kept there for 20 h\nto ensure the homogeneity of melt. After that, for GdMn 6Sn6, the temperature was rapidly cooled down to 1023 K\nfor 20 h and subsequently cooling down to 873 K at 2 K/h. For TbMn 6Sn6, the temperature was cooled down directly\nto 873 K with the rate of 5 K/h. Finally, the ampules were taken out of furnace and the single crystals were separated\nfrom the flux by a centrifuge.\nII. EXPERIMENTAL DETAILS\nPrior to our optical study, we carried out four-point dc resistivity and magnetic susceptibility measurements within\ntheab-plane to monitor possible magnetic transitions and confirm the stoichiometry. For the magnetic susceptibility\nmeasurements, a small magnetic field of H= 0.1 T was applied. The obtained data agrees well with the literature\nand confirms the spin reorientation in TbMn 6Sn6around 310 K from the basal plane near to the c-axis [S1]. For\nGdMn 6Sn6, all magnetic transitions are above the measured temperature range; hence, we observed no anomalies in\nour data [S2].\nFreshly cleaved samples with the dimensions of 2 x 2 mm2surface area and thickness of about 100 µm were used for\nthe optical study. Here, temperature-dependent reflectivity measurements were performed in the ab-plane covering\na broad frequency range from 50 to 18000 cm−1(6.2 meV - 2.23 eV) down to 10 K, as shown in Fig. S1. For the\nhigh-energy range ( ω > 600 cm−1) a Bruker Vertex 80v spectrometer with an incorporated Hyperion IR microscope\nwas used, while the low-energy range was measured with a Bruker IFS113v spectrometer and a custom-built cryostat.\nFreshly evaporated gold mirrors served as reference in these measurements. The absolute value of the reflectivity was\nobtained by an in-situ gold-overcoating technique in the far-infrared range, as described in ref. [S3]\nConsidering the metallic nature of the samples, we used Hagen-Rubens extrapolation below 50 cm−1, while x-ray\nscattering functions were utilized for the high-energy range to extrapolate the data [S4]. The optical conductivity is\nthen calculated from the measured reflectivity by standard Kramers-Kronig analysis.\n101 001 0001 00000.50.60.70.80.91.01\n01 001 0001 0000 Reflectivity \n Frequency (cm-1) 10 K \n50 K \n100 K \n150 K \n175 K \n200 K \n225 K \n250 K \n275 K \n300 K \nG\ndMn6Sn6E II ab(b) \nFrequency (cm-1) \nT\nbMn6Sn6E II ab(a)0.010 .11 Energy (eV)0\n.010 .11 Energy (eV)\nFIG. S1. Temperature-dependent reflectivity over a broad frequency range (50 to 18000 cm−1) measured in the ab-plane. The\ndotted lines are the Hagen-Rubens extrapolations.arXiv:2208.00756v2  [cond-mat.str-el]  21 Dec 20222\nIII. DECOMPOSITION OF OPTICAL SPECTRA\nDifferent contributions to the total optical conductivity were modeled with the Drude-Lorentz approach. With ε∞\nbeing the high-energy contributions to the real part of the dielectric permittivity, the dielectric function [˜ ε=ε1+iε2]\nis expressed as\n˜ε(ω) =ε∞−ω2\np,Drude\nω2+iω/τ Drude+/summationdisplay\njΩ2\nj\nω2\n0,j−ω2−iωγj. (S1)\nHere,ωp,Drude and 1/τDrude are the plasma frequency and the scattering rate of the itinerant carriers, respectively.\nThe parameters ω0,j, Ωj, andγjdescribe the resonance frequency, width, and the strength of the jthexcitation,\nrespectively.\nFollowing the approach of previous optical studies of kagome metals, we base our analysis of the localization peak\non the displaced Drude formalism proposed in 2014 by Fratini et al. [S5]. Here, possible localization effects, due to\ninteractions of charge carriers with low-energy degrees of freedom, such as phonons, electric or magnetic fluctuations,\nare considered by modifying the classical Drude response with an additional backscattering of the electrons. This\n101 001 0001 00000.00.40.81.21\n01 001 0001 00000.00.40.81.20.00.40.81.20\n.00.40.81.20\n.00.40.81.20.00.40.81.2T\n = 10 K  σ1 (104Ω-1cm-1) \n  \nFrequency (cm-1)GdMn6Sn6G\ndMn6Sn6T = 100 KT = 200 KG\ndMn6Sn6 σ1 (104Ω-1cm-1) \n F\nrequency (cm-1)T = 10 KT = 200 KT\n = 100 K1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1) \n σ1 (104Ω-1cm-1)(a)(\nb)(\nc)(d)(\ne)(\nf)  σ1 (104Ω-1cm-1) σ1 (104Ω-1cm-1)TbMn6Sn6T\nbMn6Sn6T\nbMn6Sn6 σ1 (104Ω-1cm-1)1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)0.010 .11 Energy (eV)0\n.010 .11 Energy (eV)\nFIG. S2. Decomposed optical conductivity at 200 K, 100 K, and 10 K, consisting of a Drude component (purple), a localization\npeak (blue), a phonon mode (green), and several interband transitions (orange) modeled with the Drude-Lorentz approach.\nThe insets show the total fit to the measured in-plane reflectivity.3\nLorentzian 1 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 172.705 432.556 651.473 909.009 332.377 682.769\n50 148.140 432.556 619.329 850.895 332.377 758.926\n100 156.009 432.556 660.067 760.626 332.377 899.605\n150 198.806 432.556 853.54 865.581 332.377 807.755\n175 189.058 432.556 901.554 - - -\n200 202.635 432.556 767.635 1024.99 332.377 1041.35\n225 215.642 432.556 1034.66 1050.23 332.377 982.126\n250 192.204 432.556 1164.59 1180.61 332.377 1006.25\n275 163.49 432.556 1105.97 1265.02 332.377 1002.79\n300 206.121 432.556 1215.73 1251.79 332.377 1049.3\nLorentzian 2 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 95.7128 1698.15 2554.68 69.8246 1705.89 1924.48\n50 99.9937 1698.15 2788.11 72.9802 1705.89 1985.85\n100 100.325 1698.15 2776.85 76.0712 1705.89 2027.15\n150 107.838 1698.15 2994.56 88.1694 1705.89 2441.02\n175 110.115 1698.15 3048.64 - - -\n200 113.452 1698.15 3017.55 98.043 1705.89 2775.34\n225 108.198 1698.15 3016.64 103.049 1705.89 2831.12\n250 126.412 1698.15 3437.89 93.1684 1705.89 2754.83\n275 127.789 1698.15 3432.88 96.9267 1705.89 2883.27\n300 121.526 1698.15 3432.53 100.372 1705.89 3007.04\nLorentzian 3 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 50.7642 6479.28 11939 74.6754 6338.93 12808.7\n50 47.1342 6479.28 11939 75.5724 6338.93 12808.7\n100 49.0074 6479.28 11939 76.3284 6338.93 12808.7\n150 50.8744 6479.28 11939 76.336 6338.93 12808.7\n175 50.9762 6479.28 11939 - - -\n200 53.1189 6479.28 11939 77.1148 6338.93 12808.7\n225 53.6501 6479.28 11939 78.7435 6338.93 12808.7\n250 52.5772 6479.28 11939 76.814 6338.93 12808.7\n275 53.1029 6479.28 11939 76.8908 6338.93 12808.7\n300 55.8173 6479.28 11939 73.3722 6338.93 12808.7\nLorentzian 4 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 18.9776 31246.3 80843.9 19.2266 27418.1 61229.7\n50 19.2876 31246.3 80843.9 19.0763 27418.1 61229.7\n100 19.3669 31246.3 80843.9 18.9045 27418.1 61229.7\n150 19.1541 31246.3 80843.9 19.0954 27418.1 61229.7\n175 19.0385 31246.3 80843.9 - -\n200 19.0366 31246.3 80843.9 18.7211 27418.1 61229.7\n225 18.8462 31246.3 80843.9 18.9102 27418.1 61229.7\n250 19.0347 31246.3 80843.9 18.7916 27418.1 61229.7\n275 19.029 31246.3 80843.9 18.7916 27418.1 61229.7\n300 19.2212 31246.3 80843.9 19.0186 27418.1 61229.7\nTABLE I. Fit parameters of the total number of four Lorentzians used to model the interband optical transitions in GdMn 6Sn6\nand TbMn 6Sn6.\nleads to a shift of the zero-frequency response to a finite value:\n˜σlocalization (ω) =C\nτb−τtanh{/planckover2pi1ω\n2kBT}\n/planckover2pi1ω·Re/braceleftbigg1\n1−iωτ−1\n1−iωτb/bracerightbigg\n. (S2)\nHere,Cis a constant, /planckover2pi1is the reduced Planck constant, kBthe Boltzmann constant, τbthe backscattering time, and4\n0100200300040801200\n10020030002004006000\n1002003000204060800\n1002003000100200300400500 \nTemperature (K) 1/τDrude (cm-1) \n Temperature (K) \n 1/τlocalization (cm-1)1\n/τ1\n/τb0\n20406080100ρ\n (µΩcm)(d)( c)T\nbMn6Sn6  1/τDrude (cm-1)T\nemperature (K)GdMn6Sn60\n4080120(b)ρ\n (µΩcm)(a) \nTemperature (K)1/τlocalization (cm-1) \n 1\n/τ1\n/τb\nFIG. S3. Elastic scattering rate, 1/ τ(blue) and backscattering rate, 1/ τb(red) of the Fratini model fits. Additionally, the\nelastic scattering rate of the Drude contribution (green), overlaid with the dc resistivity (orange), is given.\nτthe elastic scattering time of the standard Drude model.\nThe total dielectric permittivity takes the form\n˜ε(ω) = ˜εDrude (ω) + ˜εLorentz (ω) + ˜εlocalization (ω). (S3)\nThe complex optical conductivity [˜ σ=σ1+iσ2] is then calculated as\n˜σ(ω) =−iω[˜ε(ω)−ε∞]/4π. (S4)\nFig. S2 shows the decomposed optical conductivity at various temperatures. The spectra were fitted in a consistent\nway for all temperatures using one Drude contribution (purple), a total number of four Lorentzians (see Table I for the\nparameters) to describe the interband optical transitions (orange), a sharp Lorentzian for the phonon mode (green),\nas well as the Fratini model to describe the localization peak (blue). At 300 K the localization peak is only weakly\npronounced and additionally screened by low-energy interband transitions in TbMn 6Sn6. On the other hand, the\npeak is clearly visible by the eye in the spectrum of GdMn 6Sn6due to the absence of strong low-energy interband\nabsorptions and the sharper Drude contribution.\nIn Fig. S3, we show the elastic scattering rate and the backscattering rate obtained from the Fratini model fits to\nthe optical spectra, as well as the scattering rate of the classical Drude model. When overlaying the Drude scattering\nrate with the dc resistivity, a remarkably similar temperature evolution is found in TbMn 6Sn6, indicating that the dc\ntransport is governed by the free electrons. On the other hand, a clear deviation of this behavior above ∼200 K is\nobserved in GdMn 6Sn6. Considering the akin temperature dependence of the resistivity to the elastic scattering rate\nof the localization peak at high temperatures, this signals a significant contribution of the incoherent carriers to the\ndc transport in GdMn 6Sn6.\nIV. PHONON CALCULATIONS\nPhonon calculations were performed on the density-functional theory (DFT) level in VASP [S6, S7] using the refined\nstructural parameters given in Table II and the Perdew-Burke-Ernzerhof (PBE) flavor of the exchange-correlation\npotential [S8]. Spin-orbit coupling was included, and different directions of the magnetic moment were chosen.\nFerromagnetic order was introduced for Mn atoms, whereas f-electrons of Gd and Tb were placed into the core, and\nonly a small residual magnetic moment due to d-electrons appeared on these atoms. This simplification was necessary\nin order to achieve good convergence of total energies and forces, as required in phonon calculations. The 8 ×8×4\nk-mesh was used.\nFrequencies of Γ-point phonons were obtained from the built-in procedure with frozen atomic displacements of\n0.015 ˚A. Fig. S4 (a) and (b) depict the calculated IR-active phonon modes of GdMn 6Sn6and TbMn 6Sn6. In both\ncompounds, a total number of nine IR-active modes are expected, which do not significantly vary in frequency with\nchanges in the direction of the magnetic moments. Four of these are A 2uc-axis modes (dashed lines) and hence,\ncannot be observed in our in-plane measurements. The remaining five modes are E 1umodes (solid lines) involving\nin-plane atomic displacements. However, the appearance of phonon modes in reflectivity spectra strongly depends\non the intensity of the phonon mode, especially for highly metallic samples, as in the case of the ReMn6Sn6series.\nHence, it is possible that only the E 1umode around 160 cm−1is strong enough to be captured by our measurements.5\nGdMn 6Sn6\na=b= 5.5399(2) ˚A,c= 9.0318(5) ˚A\nV= 240.054(18) ˚A3\nP6/mmm\nλ= 0.71073 ˚A\nΘmin= 0.41◦, Θmax= 27.48◦\n−7≤h≤7,−7≤k≤7,−11≤l≤11\nRint= 0.0656\nAtomx/a y/b z/c U iso(˚A2)\nGd 0 0 0.5 0.01120(34)\nMn 0.5 0 0.25224(13) 0.01074(34)\nSn11\n32\n30 0.01147(33)\nSn21\n32\n30.5 0.01024(33)\nSn3 0 0 0.16206(11) 0.01184(33)TbMn 6Sn6\na=b= 5.5305(2) ˚A,c= 9.0223(5) ˚A\nV= 238.988(18) ˚A3\nP6/mmm\nλ= 0.71073 ˚A\nΘmin= 0.41◦, Θmax= 27.48◦\n−6≤h≤7,−7≤k≤7,−11≤l≤11\nRint= 0.0787\nAtomx/a y/b z/c U iso(˚A2)\nTb 0 0 0.5 0.01182(41)\nMn 0.5 0 0.25244(16) 0.01129(42)\nSn11\n32\n30 0.01155(40)\nSn21\n32\n30.5 0.01078(40)\nSn3 0 0 0.16276(13) 0.01211(40)\nTABLE II. Details of data collection and refined structural parameters for GdMn 6Sn6(left) and TbMn 6Sn6(right).\nThis mode can be represented with a sharp Lorentzian,\nσ1(ω) =∆εω2ω2\n0γ\n4π[(ω2−ω2\n0)2+γ2ω2]. (S5)\nHere, ∆εstands for the intensity, ω0for the resonance frequency, and γfor the linewidth. Consistent with the\nhardening of the lattice, we observe a slight blue shift of the mode upon cooling in both compounds. In GdMn 6Sn6, a\nsignificant enhancement of intensity and a slight broadening of the mode are observed as the localization peak crosses\nthe respective phonon mode. On the other hand, no such changes are observed in TbMn 6Sn6. Here, both the intensity\nas well as the linewidth stay constant within the error bars of our fits.\nV. CALCULATION OF THE OPTICAL CONDUCTIVITY\nDFT calculations of the band structure and optical conductivity were performed in the Wien2K code [S9] using\nthe same PBE functional [S8]. Spin-orbit coupling was included in all calculations. For a realistic implementation\nof the magnetic structures, the [100]-direction of the magnetic moments was chosen for GdMn 6Sn6, while the [001]-\ndirection was set for TbMn 6Sn6. Additionally, an antiferromagnetic coupling between the Mn- and rare-earth-atoms\nwas implemented. Moreover, a Hubbard UGd/Tb = 10 eV was added to the 4 fshell of the rare-earth element using\nthe DFT+Umethod with the FLL (fully localized limit) double-counting correction to push the minority 4 fstates to\nenergies well above the Fermi level. DFT calculations were converged on the 15 ×15×4k-mesh. Optical conductivity\nwas calculated within Wien2K [S10] on a denser 26 ×26×14k-mesh.\nFig. S5 shows the calculated band structures along high-symmetry paths of the first Brillouin zone. Both compounds\npossess flat bands around 0.5 eV and saddle points nearby the Mpoint. The Dirac points nearby Kare marked by\ncircles, and their energies are noted in Table III. In the case of a two-dimensional Dirac point, the optical conductivity\nis supposed to show a sharp Drude component along with a step-like onset at 2 |ED|, followed by a frequency-\nindependent behavior. Hence, the interpretation of the observed steps in the optical conductivity as the signature of\ntwo-dimensional Dirac fermions is very tempting. The obtained Dirac cone energies from our experiment are noted in\nTable III. A direct comparison with our calculations reveals a remarkable agreement of the determined energies. On\nthe other hand, a comparison with ARPES studies shows a larger deviation of the energies of the second Dirac point.\nHowever, it should be noted that ARPES only probes the states below the Fermi energy leading to less accurately\ndetermined values.\nDespite the good agreement between the experiment and our calculations, the step-like absorption features shown\nin Fig. S6 should be interpreted cautiously. A closer look at the calculated bandstructure reveals the large number\nof bands crossing the Fermi energy in these compounds. Thus, the multi-band nature of the ReMn6Sn6series should\nnot be disregarded.\nFor TbMn 6Sn6, the accuracy of the calculated optical conductivity in comparison with the experiment increases\nwhen either shifting the Fermi energy down by 47 meV or adding a Hubbard UMn= 0.4 eV to the Mn-atoms. However,6\n20022525027530002040602\n002252502753001571581591601612\n00225250275300468101214608 01 001 201 401 601 802 002 202 40 \nGdMn6Sn6 \nTbMn6Sn6  ΔεT\nemperature (K)(c) \n ω0 (cm-1)T\nemperature (K)(d) \n γ (cm-1)T\nemperature (K)(e) \n G\ndMn6Sn6(a)E1uA\n2u0.0100 .0150 .0200 .025Energy (eV)T\nbMn6Sn6 IR-active phonon modesF\nrequency (cm-1)(b)\nFIG. S4. (a) and (b) Calculated IR-active phonon frequencies of GdMn 6Sn6and TbMn 6Sn6. The solid lines represent the\nin-plane E 1umodes while dashed lines mark A 2umodes involving atomic displacements along the c-axis.(c)-(e) Fit parameters\nof the observed phonon mode in the optical spectra corresponding to the E 1umode marked by the red area in (a) and (b).\n-1.0-0.50.00.51.0-\n1.0-0.50.00.51.0  \nE - EF (eV)GdMn6Sn6/s61511\n                      M           K                         /s61511 initial \nUMn = 0.4 eV/s61511\n                      M           K                         /s61511E - EF (eV) \nTbMn6Sn6\nFIG. S5. DFT+ Uband structures for GdMn 6Sn6(left) and TbMn 6Sn6(right) shown along high-symmetry paths of the first\nBrillouin zone. The observed Dirac points at the Kpoint are marked with circles and their energies are noted in Table III.\nin both cases, the energy of the first Dirac point shifts above the Fermi level, which is not expected from ARPES\nstudies on the ReMn6Sn6series.7\n40008 00012000160000123456784\n0008 0001200016000012345678G\ndMn6Sn6 experiment at 10 K \nfit \nsteps   σ1 (103Ω-1cm-1)F\nrequency (cm-1)TbMn6Sn6(b)   σ1 (103Ω-1cm-1) \nF\nrequency (cm-1)(a)0.40 .81 .21 .62 .0Energy (eV)0\n.40 .81 .21 .62 .0Energy (eV)\nFIG. S6. Experimental optical conductivity after subtracting the localization peak and the low-energy interband transitions.\nThe remaining spectra resemble the optical conductivity of two-dimensional Dirac fermions. The steps at 2 |ED|are highlighted\nwith dots.\noptical study calculations ARPES estimates\nED1(meV)ED2(meV)ED1(meV)ED2(meV)ED1(meV)ED2(meV)\nGdMn 6Sn6 63 291 - 42 233 - 42 [S11] 170 [S12]\nTbMn 6Sn6 65 298 - 41 239 not reported 130 [S13]\nTABLE III. Energies of the Dirac points obtained from the optical study at T= 10 K, the DFT+ Ucalculations, and estimates\nfrom ARPES measurements.\n[S1] D. C. Jones, S. Das, H. Bhandari, X. Liu, P. Siegfried, M. P. Ghimire, S. S. Tsirkin, I. I. Mazin, and N. J. Ghimire,\nOrigin of spin reorientation and intrinsic anomalous Hall effect in the kagome ferrimagnet TbMn 6Sn6, arXiv:2203.17246.\n[S2] D. Gorbunov, M. Kuz’min, K. Uhl´ ıˇ rov´ a, M. ˇZ´ aˇ cek, M. Richter, Y. Skourski, and A. Andreev, Magnetic properties of a\ngdmn6sn6 single crystal, Journal of Alloys and Compounds 519, 47 (2012).\n[S3] C. C. Homes, M. Reedyk, D. A. Cradles, and T. Timusk, Technique for measuring the reflectance of irregular,\nsubmillimeter-sized samples, Appl. Opt. 32, 2976 (1993).\n[S4] D. B. Tanner, Use of x-ray scattering functions in Kramers-Kronig analysis of reflectance, Phys. Rev. B 91, 035123 (2015).\n[S5] S. Fratini, S. Ciuchi, and D. Mayou, Phenomenological model for charge dynamics and optical response of disordered\nsystems: Application to organic semiconductors, Phys. Rev. B 89, 235201 (2014).\n[S6] G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a\nplane-wave basis set, Computational Materials Science 6, 15 (1996).\n[S7] G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis\nset, Phys. Rev. B 54, 11169 (1996).\n[S8] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865\n(1996).\n[S9] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, R. Laskowski, F. Tran, and L. Marks, WIEN2k, An Augmented\nPlane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universit¨ at Wien,\nAustria), 2018. ISBN 3-9501031-1-2.\n[S10] C. Ambrosch-Draxl and J. Sofo, Linear optical properties of solids within the full-potential linearized augmented planewave\nmethod, Comput. Phys. Commun. 175, 1 (2006).\n[S11] Z. Liu, N. Zhao, M. Li, Q. Yin, Q. Wang, Z. Liu, D. Shen, Y. Huang, H. Lei, K. Liu, and S. Wang, Electronic correlation\neffects in the kagome magnet GdMn 6Sn6, Phys. Rev. B 104, 115122 (2021).\n[S12] W. Ma, X. Xu, J.-X. Yin, H. Yang, H. Zhou, Z.-J. Cheng, Y. Huang, Z. Qu, F. Wang, M. Z. Hasan, and S. Jia, Rare\nEarth Engineering in RMn6Sn6(R= Gd−Tm, Lu) Topological Kagome Magnets, Phys. Rev. Lett. 126, 246602 (2021).\n[S13] J.-X. Yin, W. Ma, T. A. Cochran, X. Xu, S. S. Zhang, H.-J. Tien, N. Shumiya, G. Cheng, K. Jiang, B. Lian, Z. Song,\nG. Chang, I. Belopolski, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, B. Swidler, H. Zhou, H. Lin, T. Neupert,\nZ. Wang, N. Yao, T.-R. Chang, S. Jia, and M. Zahid Hasan, Quantum-limit Chern topological magnetism in TbMn 6Sn6,8\nNature 583, 533 (2020)."    },    {        "title": "1202.6166v1.Gallium_Substituted__114__YBaFe4O7__From_a_ferrimagnetic_cluster_glass_to_a_cationic_disordered_spin_glass.pdf",        "content": " 1 Gallium Substituted “114”  YBaFe 4O7: From a ferrimagnetic cluster glass to \na cationic disordered spin glass  \n \nTapati  Sarkar *, V. Caignaert, V. Pralong and B. Raveau  \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal Juin, 14050 CAEN, France  \n \nAbstra ct \n \n The study of the ferrites  YBaFe 4-xGaxO7 shows that the substitution of Ga for Fe in \nYBaFe 4O7 stabilizes the hexagonal symmetry for 0.40 \n x \n 0.70, at the expense of the cubic \none. Using combined measurements of a. c. and d. c. magnetization, we estab lish that Ga \nsubstitution for Fe in YBaFe 4O7 leads to an evolution from a geometrically frustrated spin \nglass (for x = 0) to a cationic disorder induced spin glass (x = 0.7 0). We also find  an \nintermediate narrow range of doping where the samples are clearl y phase separated having \nsmall ferrimagnetic clusters embedded in a spin glass matrix. The origin of the ferrimagnet ic \nclusters  lies in the change in symmetry  of the samples  from cubic to hexagonal (and a \nconsequent lifting of the geometric al frustration) as a result of  Ga doping.  We also show the \npresence of exchange bias and domain wall pinning in these samples. The cause of both these \neffects can be traced back to the inherent ph ase separation present in the  samples.  \n \n \n \n \n \n \n \n \nKeywords : “114” oxides, magne tic frustration , phase separation .   \n \n \n \n \n* Corresponding author: Dr. Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 Introduction  \n \n The recent studies of the “114” cobaltites (Ln,Ca) 1BaCo 4O7 [1 – 5] and ferrit es \n(Ln,Ca) 1BaFe 4O7 [6 – 8] have generated a lot of interest in the scientific community because \nof their complex magnetic, electronic  and thermoelectric properties  [9]. These cobaltites and \nferrites have the same basic structure, and are closely related to  spinels and barium \nhexaferrites by their close packing of “O 4” and “BaO 3” layers.  This close packing forms a 3 -\ndimensional framework [Fe 4O7]\n (or [Co 4O7]\n) consisting of corner -sharing FeO 4 (or CoO 4) \ntetrahedra , with the lanthanide elements occupying the octahedral sites of this framework.  The \ntriangular geometry of the cobalt (or iron) sublattices ( Fig. 1 ) plays a dominant role in their \nmagnetic properties. It was indeed shown that for hexagonal LnBaCo 4O7 cobaltites ( Fig. 1 \n(a)), there exists a strong com petition between the 1 D magnetic ordering along the \n  \ndirection in the “Co 5” trigonal bipyramids, and the magnetic frustration in the (001) plane \nbuilt up of “Co 3” triangles [10, 11]. In fact, the magnetic frustration can be lifted by an \northorhombic dist ortion of the structure. This is illustrated by the concomitant structural and \nmagnetic transitions that appear at low temperature in these cobaltites [1, 2, 12], and by the \nferrimagnetic structure of CaBaCo 4O7 [13]. Similarly, the “114” ferrites exhibit a  competition \nbetween 1 D magnetic ordering and 2 D magnetic frustration, as has been shown for the \nhexagonal phases CaBaFe 4O7 [6], and for CaBa Fe4-xLixO7 [14]. But importantly, the “114” \nferrites differ from the “114” cobaltites by the fact that the LnBaFe 4O7 oxides exhibit a cubic \nstructure [7]. Though the latter is closely related to the hexagonal structure, the iron sublattice \nis very different ( Fig. 1 (b) ), consisting of “Fe 4” tetrahedra instead of “Fe 5” bipyramids and \n“Fe 3” triangles. No structural tra nsition appears at low temperature, and consequently, the \ncubic ferrites exhibit a spin glass behaviour due to a perfect  geometrical frustration.  Further, \nthe LnBaFe 4O7 series exhibit s an oxidation state disorder. Unlike the case of CaBaCo 4O7 [13], \nno char ge ordering is observed in LnBaFe 4O7, and this disorder is also important for the \nobserved glassiness.  \nRecently, we showed that the substitution of a divalent cation, Zn2+, for iron in \nYBaFe 4O7, allowed the hexagonal symmetry to be stabilized at the detrim ent of the cubic one \n[15]. Paradoxically, it was observed that the substitution of this diamagnetic cation for Fe2+ \ninduces ferrimagnetism, in contrast to the spin glass behaviour of the undoped phase \nYBaFe 4O7. In fact, a competition between ferrimagnetism  and magnetic frustration was \nobserved for the hexagonal phase YBaFe 4-xZnxO7. This was interpreted as the effect of two  3 antagonist phenomena: the partial lifting of the geometrical frustration due to the appearance of \nthe hexagonal symmetry inducing a 1 D magnetic ordering, and the existence of cationic \ndisordering favouring the glassy state.  \n Bearing in mind that the Fe2+:Fe3+ ratio is a crucial factor governing  the magnetic \nproperties of iron oxides, it must be emphasized that the substitution of Zn2+ for Fe2+ increases \nthe average valence of iron , i.e. the Fe2+:Fe3+ ratio decreases from 3 in the spin glass phase \nYBaFe 4O7 to 2.6 – 1.5 in the solid solution YBaFe 4-xZnxO7 when x changes from 0.4 to 1.5 \n[15]. In order to further understand the role of the ave rage valence of iron in the magnetic \nproperties of these ferrites, we have investigated the possibility of substitution of a diamagnetic \ncation such as gallium for Fe3+ in the YBaFe 4O7 structure. In the present study of the ferrite \nYBaFe 4-xGaxO7, we show t hat the introduction of gallium in the structure stabilizes the \nhexagonal symmetry, similar to the zinc substitution, but differently from the latter, the lifting \nof the geometrical frustration induces the formation of ferrimagnetic clusters embedded in a \nspin glass matrix, which tend to disappear as the gallium content increases, leading to a pure \nspin glass for higher Ga content, with a higher T g compared to YBaFe 4O7. \n \nExperimental  \n \nPhase -pure samples of YBaFe 4-xGaxO7  [x = 0.4 0 – 0.70] were prepared by solid state \nreaction technique. The precursors used were Y 2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe \npowder. First, the precursor BaFe 2O4 was prepared from a stoichiometric mixture of BaCO 3 \nand Fe 2O3 annealed at 1200°C for 12 hrs in air. In a second step, a stoichiometric mixture of \nY2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe powder was intimately ground and pressed in \nthe form of rectangular bars. The bars were then kept in an alumina finger, sealed in silica \ntubes under vacuum and annealed at 1100°C for 1 2 hrs. Finally, the samples were quenched to \nroom temperature in order to stabilize the “114” phase.  \nThe X -ray diffraction patterns were registered with a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2\n  range 10° - 120° an d step size \n 2\n \n= 0.017°.  The cationic composition was confirmed by means of Energy Dispersive X -Ray \nSpectroscopy  (EDS) technique using a Scanning Electron Microscope (ZEISS Supra 55). The \nd. c. magnetization measurements were performed using a superconduc ting quantum \ninterference device (SQUID) magnetometer with variable temperature cryostat (Quantum \nDesign, San Diego, USA). The a. c. susceptibility, \n ac(T) was measured with a PPMS from \nQuantum Design with the frequency ranging from 10 Hz to 10 kHz. H ac was kept fixed at 10  4 Oe, while H dc was varied from 0 Oe to 2000 Oe. All the magnetic properties were registered on \ndense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \nResults and discussion  \n \n \n Similar  to Zn substitution, Ga substitution also favours the forma tion of the hexagonal \nphase at the expense of the cubic one. Nevertheless, the homogeneity range of the hexagonal \nYBaFe 4-xGaxO7 solid solu tion is significantly different  (0.40 \n  x \n 0.70) vis – à – vis that of \nYBaFe 4-xZnxO7 [15]. The cubic symmetry of YBaF e4O7 is retained for 0 \n  x \n 0.20, whereas \nthe domain 0.20 < x < 0.40 is biphasic, corresponding to a mixture of the cubic and hexagonal \nphases.  On the other hand,  for x > 0.70 , several impurity phases appear , namely Y 2O3 and \nGa2O3. The cationic compositio n of the single phase obtained for the range 0.40 \n  x \n 0.70 \nusing EDS analysis are also shown in Table 1 .  \n  \nStructural characterization  \n \n In Fig. 2, we show the X -ray diffraction (XRD) pattern s of the two end members, (a) \nYBaFe 3.6Ga0.4O7 and (b) YBaFe 3.3Ga0.7O7 as representative example s. As stated before, t he \nsample s are seen to stabilize in the hexagonal symmetry with the space group P63mc. The \nRietveld analysis of the lattice structure was done using the FULLPROF refinement program \n[16] and the fit s are also shown in Fig. 2. All the samples in the range x = 0.4 0 – 0.70 were \nseen to stabilize in the same hexagonal symmetry.  \nThe extracted cell parameters have been tabulated in Table 1 . The ionic radius of Fe3+ (0.49 \nÅ) is very similar to that of Ga3+ (0.47 Å). As can be seen from the extracted cell parameters \nshown in Table 1, a increases very slightly as x increases (an increase of only ~ 0.08 % as x \nincreases from 0.4 to 0.7), while c shows a slight decrease (~ 0.11 %). This causes the cell \nvolume to re main practically unchanged as a function of doping in accordance with the \nsimilar ionic radii of Fe3+ and Ga3+. \n \nD. C. magnetization studies  \n \n In the “114” ferrites, it has been established earlier [8, 15] that ferrimagnetism is \ninherently linked with the cross -over from cubic to hexagonal symmetry. The doping -induced \ntransition to the hexagonal symmetry involves a partial lifting of the 3D geometrical  5 frustration, which is the root cause of the appearance of ferrimagnetism. Thus, we restrict our \ndiscussion  of the magnetic data to the YBaFe 4-xGaxO7 samples exhibiting hexagonal symmetry \n(0.4 \n  x \n 0.7). We note here that the cubic samples (x < 0.2) are spin glasses similar to the \nundoped YBaFe 4O7, and will not be discussed further.  \n The temperature dependence  of d. c. magneti c susceptibility (\n dc = M/H)  was registered \naccording to the standard zero field cooled (ZFC) and field cooled (FC) procedures. A \nmagnetic field of 0.3 T was applied during the measurements. The measurements were done in \na temperature rang e of 5 K to 300 K. The \n ZFC(T) and \n FC(T) curves of all the samples are \nshown in Fig. 3. \n Undoped YBaFe 4O7 is a spin glass with T g = 50 K [ 7]. The ZFC \n dc versus T curve for \nYBaFe 4O7 shows a pure cusp -like shape [ 7], typical of canonical spin glasses.  A close look at \nFig. 3 reveals that the YBaFe 4-xGaxO7 series of samples shows two different kinds of low \ntemperature M ZFC(T) curves vis -à-vis the shape of the curves. While the \n dc(T) curve of the \nhighest substituted  sample (x = 0.7 0) is very similar to that o f canonical spin glasses (with a \npeak at ~ 50 K and a gradual decrease of the magnetization value below 50 K),  for the lowest \ndoped sample (x = 0.4 0), there is a sharp drop in the susceptibility  value below the temperature \nat which \n ZFC reaches its maximum  value (75 K). The susceptibility  value drops sharply till ~ \n50 K (marked by a black arrow in Fig. 3 (a)), below which the decrease in \n ZFC is more \ngradual.  We note here that the measuring field that we have chosen (0.3 T) is smaller than the \ncoercive fiel d of the YBaFe 3.6Ga0.4O7 sample at T = 5 K (data shown later in Fig. 5 ). Thus, it is \nquite possible that the sharp drop in the susceptibility value occurs at the temperature where the \ncoercive field of the sample becomes smaller than 0.3 T. However, follow ing this argument, \nwe should have obtained similar sharp drops in the \n ZFC(T) curves for the x = 0.5 and x = 0.6 \nsamples also, as the coercive fields of the x = 0.5 and x = 0.6 samples at T = 5 K are also larger \nthan 0.3 T. Instead , it is observed that the  sharp drop in the \n ZFC(T) curve seen in the x = 0.40 \nsample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.50 and x = 0.60 samples.  \nMore importantly, a study of the temperature dependence of the coercive field (H C) of the x = \n0.4  sample (d ata not shown here) reveals that H C becomes smaller than 0.3 T  at ~ 17 K (i.e. at \na temperature much below 50 K).  This suggests  that this feature is not a simple effect of the \ncoercive  field, rather it may have  a more complex origin.  \nAnother possibility is  that this sudden decrease in \n ZFC(T) is due to domain wall \npinning effects, which  has, in fact,  been observed  previously  in manganites [1 7 – 19]. Due to \npinning, the domains would not  freely rotate below the pinning temperature unless a high  6 enough extern al field is present to overcome the pinned state. Upon zero field cooling, the \ndomains would be  pinned into random orientations. Whe n a low field is applied (0.3 T  in this \ncase), the pinning effect still dominates over the effect of the applied magnetic fi eld, and the \nmagnetization is lower than what would be expected in the absence of pinning. However, the \npinned domain walls can be thermally activated by increasing the temperature. This could be \nthe cause of the  visible jump in the \n ZFC(T) curve at the te mperature where the pinning effects \nare overcome by temperature (~ 50 K).  As can be seen in Fig. 3, the pinning effect gradually \ndecreases as the doping concentration is increased (the sharp drop in the \n ZFC(T) curve seen in \nthe x = 0.4 0 sample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.5 0 and x = \n0.60 samples, and completely vanishes for the x = 0.7 0 sample).  This indicates  that the domain \nwall pinning is more prominent for small doping and vanishes for higher doping. This is \ncounter -intuitive if the pinning is thought to arise due to the presence of Ga in the lattice. Thus, \nthe fact that the domain wall pinning decreases with an increase in the doping concentration \nleads us to believe that this pinning does not arise from the disorder in  the system. Rather, it \nhas a more complex origin, which we discuss later.  \n Before we proceed further, we perform some additional measurements to make sure \nthat the sharp drop in the \n ZFC(T) curve observed in the lowest doped sample is indeed due to \ndomain  wall effects , and not arising from some additional (antiferro) magnetic transition in the \nsample.  Thus, we subject the x = 0.4 0 sample to a degaussing experiment [1 7], wherein the \nsample was initially cooled from 300 K down to 5 K in a zero external magne tic field. At 5 K, \na large magnetic field (5 T) was applied. The magnetic field was then reduced to zero, and the \nsample was degaussed at 5 K by cycling a field of reducing intensity so that the remanent \nmagnetization of the sample was reduced to zero. A m agnetic field of 0.3 T was then applied, \nand the \n dc(T) curve was recorded while warming the sample, in the same wa y as ZFC \nmagnetization is recorded. The results are shown in Fig. 4. We find that the sudden sharp drop \nobserved in the normal ly obtained  ZFC curve ( Fig. 4 (a)) vanishes when the sample is \nsubjected to a high enough magnetic field, and then degaussed  (Fig. 4 (b)). This experiment, \nthus, provides supplementary evidence that the sudden drop in magnetization seen below 75 K \nis not due to any kind of (antiferro) magnetic transition  in the sample , but is probably  \nassociated with domain wall pinning effects. The high magnetic field (5 T) to which the sample \nwas subjected was sufficient for domain wall displacements thereby destroying  the pinning. In \nfact, a ZFC magnetization recorded under a high enough field of 5 T (see inset of Fig. 4 (a)) \ndoes not show any sharp drop in the magnetization of the sample.   7 The d. c. magnetization M(H) curves of all the samples  registered at T = 5 K  are shown \nin Fig. 5. The virgin curves of the M(H) loops are represented by black circles while the rest of \nthe M(H) loops are shown by red lines. The first notable point is that for higher Ga substitution \n(x = 0.7 0), the M(H) loop is narrow and S – shaped ( Fig. 5 (d)), which is quite typical of spin \nglasses  and superparamagnets . On the other hand, for lower Ga substitution, the samples have \nlarger loops with higher values of the c oercivity and remanent magnetization, which keep \nincreasing as the doping concentration is decreas ed. This indicates the presence of a higher \ndegree of magnetic ordering in the lower doped samples as compared to the higher doped ones.  \n Another feature  which strongly supports the presence of domain wall pinning  is that the \nvirgin curve of the x = 0.4 0 sample lies slightly outside the main M(H) loop ( Fig. 5 (a)). This \nunusual feature of the virgin curve lying outside the main hysteresis loop has earlier been \nassociated with irreversible domain wall motion in spinel oxides [ 20]. We also note that the \nvirgi n curve starts to shift inside the main M(H) loop as the doping concentration (x) is \nincreased, and for the x = 0.7 0 sample, the entire virgin curve lies inside the main M(H) loop \n(Fig. 5 (d)). We again note that the domain wall pinning is more prominent i n the samples with \nlower doping concentration.  \n In Fig. 6, we once again show the d. c. magnetization M(H) curves of all the samples \nregistered at T = 5 K, but in three different modes: (i) normal ZFC mode, (ii) FC mode with a \nmagnetic field of 2 T  and (iii) FC mode with a magnetic field of H = - 2 T. In the ZFC mode, \nthe samples were cooled from 300 K to 5 K in zero external magnetic field, following which M \nversus H curves were registered. In the FC mode, on the other hand, the samples were cooled \nfrom 30 0 K to 5 K in the presence of an external magnetic field (H = 2 T  or – 2 T), and then M \nversus H curves were registered.  For the highest substituted  sample (x = 0.7 0), all three  M(H) \ncurves overlap each other ( Fig. 6 (d)). However, for lower doping concent ration, the field \ncooled M(H) loops exhibit shifts both in the field as well as in the magnetization axes.  This is \nthe exchange bias phenomenon [ 21, 22] that results from exchange interaction between \nferromagnetic and antiferromagnetic materials. In our YB aFe 4-xGaxO7 samples with low Ga \nconcentration, the observed exchange bias can be explained in terms of interfacial exchange \ncoupling between the coexisting ferrimagnetic cluster glass and the disordered spin glass -like \nphases.  This exchange bias effect ari sing from the inherent phase separation in the YBaFe 4-\nxGaxO7 samples is similar to that seen in some disordered manganites [ 23]. As can be seen \nfrom Fig. 6, the exchange bias effect keeps decreasing as the doping concentration is increased, \nand as stated b efore, it completely disappears for the doping concentration x = 0 .70. We can \nexplain this observation by considering that for lower Ga concentration, the samples consist of  8 coexisting ferrimagnetic clusters embedded in a spin glass -like matrix, but as the  doping \nconcentration by the diamagnetic cation (Ga) is increased, the ferrimagnetic clusters are \nprogressively reduced and we ultimately get a homogeneous spin glass (x = 0.7 0). The absence \nof phase separation in the x = 0.7 0 sample, thus, results in an a bsence of the exchange bias \neffect. The fact that the YBaFe 4-xGaxO7 samples with lower Ga concentration are intrinsically \nphase separated, while the x = 0.7 0 sample is not, also affords us an alternative explanation for \nthe domain wall pinning effects seen  in the lower doped samples. As stated previously, the fact \nthat the domain wall pinning is seen in the lower doped samples and not in th e x = 0.7 0 sample \nmeans that it cannot arise from the disorder in the system. Rather, we believe that the pinning \narise s from an interplay between the two magnetic phases in the phase separated samples. Such \na domain wall pinning process arising from the interplay between two coexisting magnetic \nphases has been seen earlier in intermetallic alloys [ 24]. We also note that a part from the \nexchange bias effect in the lower doped samples, field cooling also results in an overall \nincrease in the coercivity and remanence magnetization values. This can be interpreted as an \nincrease in the volume fraction of the magnetically ordered  phase when the samples are cooled \nin the presence of an external magnetic field.  Since field cooling improves the remanence in \nthe lower doped samples, hence it was important to register M -H curves after field cooling  \nwith positive as well as negative coo ling fields and check whether the M -H loops shift in \nopposite directions in order to confirm that there is indeed a genuine exchange bias effect in \nthe lower doped samples.  \n \nA. C. magnetic susceptibility studies  \n \n The measurements of the a. c. magnetic sus ceptibility \n ac(T, f, H) were performed at \ndifferent frequencies ranging from 10 Hz to 10 kHz, and different external magnetic field s \n(Hdc) ranging from 0 T to 0.2 T using a PPMS facility. The amplitude of the a.  c. magnetic \nfield was ~ 0.001 T In Fig. 7 and Fig. 8, we show the temperature dependence of the real (in -\nphase) component of the a. c. susceptibility  in the temperature range 10 K – 160 K  of the \nlowest doped sample (x = 0.4 0) and the highest doped sample (x = 0.7 0) respectively, with a  \nmeasuring f requency of 10 kHz and in zero magnetic field ( Hdc = 0).  \nFrom Fig. 7, it is clear that the \n '(T) curve of the x = 0.4 0 sample shows two features, \none at T = 86 K (marked by a black arrow), and the second at T = 44 K (marked by a red \narrow). While the high  temperature feature at 86 K is a clear peak, the low temperature one at \n44 K is a shoulder like feature and is more clearly evidenced in the imaginary (out -of-phase)  9 component of the a. c. susceptibity (shown in inset (a) of Fig. 7). Repeating the measure ments \nusing four measuring frequencies (ranging from 10 Hz – 10 kHz) reveals that both the features \nare frequency dependent (this is shown in inset (b) of Fig. 7). The x = 0.7 0 sample, on the other \nhand, shows only one feature (at T = 60 K) in the \n '(T) an d \n''(T) curves, shown in the main \npanel and inset (a) of Fig. 8 respectively. Inset (b) of Fig. 8 shows the \n '(T) curves measured \nusing four different frequencies, and reveals that this peak  at 60 K is also strongly frequency \ndependent.  We note that the i maginary part of \n  for YBaFe 3.6Ga0.4O7 is ~ 8 % of the real part \nof \n . This is commonly found for systems where the spin domains are relatively large. \nYBaFe 3.6Ga0.4O7 can thus be described as a cluster glass. On the other hand, the imaginary part \nof \n for YBaFe 3.3Ga0.7O7 is significantly smaller (about 2.4% of the real part of \n ) which \nindicates that YBaFe 3.3Ga0.7O7 is closer to a canonical spin glass.  \n Although all the features in the a. c. susceptibility curves of the two samples described \nabove have a si ngle commonality , in that they are all strongly frequency dependent, but the \nnature a nd origin of these peaks can be quite  different.  Specifically, we need to establish the \nnature of the low temperature shoulder in the x = 0.4 sample at T ~ 50 K. Since it occurs close \nto the temperature where we ha ve evidenced domain wall pinning from  the d. c. magnetic data, \nit is tempting to attribute this low temperature shoulder in the a. c. susceptibility data to the \nsame phenomenon. However, it is also possible that t his low temperature feature is a \nsuperparamagnetic effect of the ferrimagnetic clusters.  To investigate this , we perform further \nmeasurements of \n '(T) and \n ''(T) of the two limiting samples  (x = 0.4 0 and x = 0.7 0) in the \npresence of different external magn etic fields H dc ranging from 0 to 0.2 T. The results for the \nYBaFe 3.3Ga0.7O7 sample are shown in Fig. 9. It is seen that both \n ' and \n '' are strongly \nsuppressed by the magnetic field.  The peak also shows a continual shift towards lower \ntemperature as the e xternal magnetic field is increased  (see the black arrows in Fig. 9). This is \ntypical of the behaviour of a spin glass freezing temperature under the influence of magnetic \nfield. \n In Fig. 10, we show the results for the YBaFe 3.6Ga0.4O7 sample. It is seen t hat w hile the \nhigh temperature peak was significantly suppressed in the presence of external magnetic field, \nthe low temperature peak was largely unaffected relative to the case H dc = 0. This rules out the \nscenario of superparamagnetism being responsible f or this low temperature feature, and \nconfirms that the 50 K anomaly arises due  to enhanced domain wall pinning,  signatures of \nwhich have been observed and commented upon earlier in the d. c. magnetization \nmeasurements also.  We also note that the high tempe rature peak shifts towards higher  10 temperature as the external magnetic field is increased (see the black arrow in Fig. 10 (a)). \nThis is not expected for a pure  spin glass freezing transition, where the peak should shift \ntowards lower temperature as the mag netic field is increased. We believe that this anomaly \narises because the x = 0.4 0 sample is not a pure spin glass, rather it is a phase separated sample \nconsisting of ferrimagnetic clusters embedded in a spin glass matrix.  \n \nConclusion  \n \n These results show  that the substitution of Ga3+ for Fe3+ in YBaFe 4O7 induces a \nstructural transition from cubic to hexagonal, similar to the substitution of Zn2+ for Fe2+ in this \ncompound. Though the two types of substitutions induce a lifting of the geometrical frustratio n \nthrough a change of the structure, the effect of these diamagnetic cations upon the magnetic \nproperties is different. A strong ferrimagnetic component is induced by zinc substitution [15], \nwhereas Ga substitution leads to the formation of ferrimagnetic c lusters embedded in a spin \nglass matrix, essentially leading to phase separation in the samples. The difference originates \nfrom the opposite evolution of the Fe3+:Fe2+ ratio as the substitution rate increases in the two \ncases. Both Fe3+ and Fe2+ exhibit th e high spin configuration since they have a tetrahedral \ncoordination in these ferrites. Thus, the magnetic moment induced by the eg2t2g3 Fe3+ cations \nshould be much higher than that induced by the eg3t2g3 Fe2+ cations. Hence, an increase in the \nFe3+:Fe2+ ratio should favour stronger magnetic interactions.  In the case of Zn2+ doping, the \nFe3+:Fe2+ ratio increases , thereby  favouring the appearance of ferrimagnetism. On the other \nhand, for Ga3+ doping, the Fe3+:Fe2+ ratio decreases , thereby inducing only weak \nferrimagnetism and cluster formation . In both series, YBaFe 4-xGaxO7 and YBaFe 4-xZnxO7, a \ndilution effect is observed with an increase in the doping concentration. As a consequence, \nferrimagnetism is weakened for higher concentrations in the Zn – phase. In the Ga – phase, the \nferrimagnetic clusters are magnetically coupled by exchange interactions mediated through the \nsurrounding spin glass matrix. For higher Ga concentrations, the exchange coupling between \nthe ferrimagnetic clusters becomes less efficient, ultimately leading to the formation of a pure \nspin glass phase for x = 0.70, which is similar to the pristine sample (x = 0), but with a slightly \nhigher T g (60 K). The Ga – substituted phase also differs from the Zn – phase by the presence \nof exchange bias  and domain wall pinning. The cause of both these effects can be traced back \nto the inherent phase separation present in the samples.  \n \n  11 Acknowledgements  \n \nWe acknowledge the CNRS and the Conseil Regional of Basse Normandie for financial \nsupport in the frame  of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \nReferences  : \n \n      [1]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002 , 4, 923  \n      [2]     Martin Valldor, J. Phys.: Con dens. Matter ., 2004 , 16, 9209  \n      [3]     A. Huq, J. F. Mitchell, H. Zheng, L. C. Chapon, P. G. Radaelli, K. S. Knight and P.  \n               W. Stephens, J. Solid State Chem ., 2006 , 179, 1136  \n      [4]     D. D. Khalyavin, L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys.  \n                Rev. B , 2009 , 80, 144107  \n      [5]     V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Communications ,  \n                2009 , 149, 453  \n      [6]     B. Raveau, V. Caignaert, V. Pralong, D.  Pelloquin and A. Maignan, Chem. Mater .,  \n                2008 , 20, 6295  \n      [7]     V. Caignaert, A. M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van  \n                Tendeloo and B. Raveau, Chem. 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Sarkar, V. Pralong , V. Caignaert and B. Raveau , Chem. Mater ., 2010, 22, 2885  \n      [16]  J. Rodriguez -Carvajal, An Introduction t o the Program FULLPROF 2000; Laboratoire  \n               Léon Brillouin, CEA -CNRS: Saclay, France (2001)  \n      [17]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 220, 106  \n      [18]   C. R. Sankar and P. A. Joy, Phys. Rev. B , 2005 , 72, 024405  \n      [19]   T. Gao, S. X. Cao, K. Liu, B. J. Kang, L. M. Yu, S. J. Yuan and J. C. Zhang, Journal  \n                of Phys: Conf. Series , 2009 , 150, 042038  \n      [20]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 210, 31 \n      [21]   R. L. Stamps,  J. Phys. D, 2000 , 33, R247  \n      [22]   W. H. Meiklejohn and C. P. Bean , Phys. Rev ., 1956 , 102, 1413  \n      [23]   S. Karmakar, S. Taran, E. Bose, B. K. Chaudhuri, C. P. Sun, C. L. Huang and H. D.  \n                Yang, Phys. Rev. B , 2008 , 77, 144409  \n      [24]   A. Bracchi, K. Samwer, S. Schneider and J. F. Löffler, Appl. Phys. Lett ., 2003 , 82,  \n                721 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  13 Table caption s \n \nTable 1 : Cell parameters as obtained from the Rietveld refinement of X -ray powder diffraction \ndata.  \n \nFigure  captions  \n \nFigure 1:  Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, see text.  \n \nFigure 2:  X-ray diffraction pattern along with the fits for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \nFigure 3: Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected \naccording to zero field cooling (ZFC) and field cooling (FC) processes for YBaFe 4-xGaxO7 (a) \nx = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at B = 0.3 T.  \n \nFigur e 4: \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode without \ndegaussing, and (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see \ntext for details). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field o f 5 T.  \n \nFigure 5:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \nFigure 6:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field cooling (red open circles), field cooling in \na field of 2 T (black lines) and field cooling in a field of - 2 T (blue line s).  \n \nFigure 7 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.   14  \nFigure 8 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \nFigure 9 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed a t f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \nFigure 10 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 Table 1  \n \nDoping \nconcentration \n(x)  \nCrystal \nsystem \n(Space \ngroup)  \n  \nUnit cell parameters  \n c / a \n2 x as \nobtained \nfrom EDS \nanalysis  a (Å)  c (Å)  \n0.40 Hexagonal \n(P63mc)  \n6.320 (1) \n 10.383 (1) 1.6428 3.02 0.43 (2)  \n0.50 Hexagonal \n(P63mc)  \n6.322 (1) \n 10.376 (1) 1.6413 3.15 0.50 (1)  \n0.60 Hexagonal \n(P63mc)  \n6.323 (1) \n 10.37 4 (1) 1.6407 2.94 0.59 (1)  \n0.70 Hexagonal \n(P63mc)  \n6.325 (1)  \n 10.37 2 (1)  1.6398 3.46 0.74 (3)  \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 1 . Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, s ee text.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 2. X-ray diffraction pattern along with the fit s for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \n \nFig. 3. Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected according \nto zer o field cooling (ZFC) and field cooling (FC) processes for  YBaFe 4-xGaxO7 (a) x = 0.40, \n(b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at H = 0.3 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  19 \n \n \nFig. 4. \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode  without dega ussing , \nand (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see text for \ndetails). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field of 5 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  20 \n \n \nFig. 5. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  21 \n \n \nFig. 6. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50,  (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field  cooling (red open circles), field cooling in \na field of 2 T (black lines)  and field cooling in a field of - 2 T (blue lines) .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  22 \n \n \nFig. 7. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibi lity, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  23 \n \n \nFig. 8. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a f unction of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  24 \n \n \nFig. 9. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and  Hac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  25 \n \n \nFig. 10. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n "    },    {        "title": "1311.3537v2.Nonequilibrium_dynamics_of_a_mixed_spin_1_2_and_spin_3_2_Ising_ferrimagnetic_system_with_a_time_dependent_oscillating_magnetic_field_source.pdf",        "content": "arXiv:1311.3537v2  [cond-mat.stat-mech]  26 Aug 2014Nonequilibrium dynamics of a mixed spin-1/2 and spin-3/2 Is ing ferrimagnetic system\nwith a time dependent oscillating magnetic field source\nErol Vatansever\nDokuz Eyl¨ ul University, Graduate School of Natural and App lied Sciences, TR-35160 Izmir, Turkey\nHamza Polat∗\nDepartment of Physics, Dokuz Eyl¨ ul University, TR-35160 I zmir, Turkey\n(Dated: June 14, 2018)\nNonequilibrium phase transition properties of a mixed Isin g ferrimagnetic model consisting of\nspin-1/2 and spin-3/2 on a square lattice under the existenc e of a time dependent oscillating mag-\nnetic field have been investigated by making use of Monte Carl o simulations with single-spin flip\nMetropolis algorithm. A complete picture of dynamic phase b oundary and magnetization profiles\nhave been illustrated and the conditions of a dynamic compen sation behavior have been discussed\nin detail. According to our simulation results, the conside red system does not point out a dynamic\ncompensation behavior, when it only includes the nearest-n eighbor interaction, single-ion anisotropy\nand an oscillating magnetic field source. As the next-neares t-neighbor interaction between the spins-\n1/2 takes into account and exceeds a characteristic value wh ich sensitively depends upon values of\nsingle-ion anisotropy and only of amplitude of external mag netic field, a dynamic compensation\nbehavior occurs in the system. Finally, it is reported that i t has not been found any evidence of\ndynamically first-order phase transition between dynamica lly ordered and disordered phases, which\nconflicts with the recently published molecular field invest igation, for a wide range of selected system\nparameters.\nI. INTRODUCTION\nThe phenomenon of ferrimagnetism is related to the counteraction of opposite magnetic moments with unequal\nmagnitudes located on different sublattices. Ferrimagnetic materia ls have, under certain conditions, a compensation\ntemperature at which the resultant magnetization vanishes below it s critical temperature1. Recently, it has been both\nexperimentally and theoretically shown that the coercive field exhibit s a rapid increase at the compensation point2,3.\nIt is obvious that such kind of point has a technological importance4,5, because at this point only a small driving\nfield is required to change the sign of the resultant magnetization. D ue to the recent developments in experimental\ntechniques, scientists begin to synthesize new classes of molecular -based magnets6–8. For instance, it has been shown\nthat the saturation magnetization, chemical analysis and infrared spectrum analysis of V(TCNE) x.y(solvent), where\nTCNE is tetracyanoethylene, are consistent with a ferrimagnet wit h spin-3/2 at the vanadium site and a spin of 1/2\nat the TCNE sites with x ∼27. In this regard, it is possible to mention that the theoretical models referring the\nmixed systems are of great importance since they are well adopted to study and to provide deeper understanding of\ncertain type of ferrimagnetism1.\nFrom the theoretical point of view, a great deal number of studies have been realized to get a clear idea about the\nmagnetic properties of mixed spin-1/2 and spin-3/2 ferrimagnetic I sing systems. In order to have a general overview\nabout it, it is beneficial to classify the studies in two categories base d on the investigation of equilibrium and nonequi-\nlibrium phase transition properties of such type of mixed spin system s. In the former group, static or equilibrium\nproperties of these type of systems have been analyzed within the several frameworks such as exact9,10, effective field\ntheory with correlations11–22, Bethe lattice23–26, exact star-triangle mapping transformation27, high temperature se-\nries expansion method29, multisublattice Green-function technique30, Oguchi approximation31as well as Monte Carlo\nsimulation32. It is underlined in the somestudies noted abovethat when the syst emincludes onlythe nearest-neighbor\ninteraction between spins and the single-ion anisotropy, the tempe rature variation of resultant magnetization does not\nexhibit a compensation behavior. In contrary to this, when the nex t-nearest neighbor interaction between spins-1/2\ntakes into account and exceeds a minimum value which depends upon t he other system parameters, the ferrimagnetic\nsystem reveals a compensation treatment which can not be observ ed in single-spin Ising systems.\nMagnetically interacting system under the influence of a magnetic fie ld varying sinusoidally in time exhibits two\nimportant striking phenomena: Nonequilibrium phase transitions and dynamic hysteresis behavior. Nowadays, these\ntypes of nonequilibrium systems are in the center of scientists beca use they have exotic, unusual and interesting\nbehaviors. For example, the universality classes of the Ising model and its variations under a time dependent driving\nfield are different from its equilibrium counterparts33–35. It is possible to emphasize that nonequilibrium phase\ntransitions originate due to a competition between time scales of the relaxation time of the system and oscillating\nperiodoftheexternalappliedfield. Forthehightemperaturesand highamplitudesoftheperiodicallyvaryingmagnetic\nfield, the simple kinetic ferromagnetic system exists in dynamically diso rdered phase where the time dependent2\nmagnetization oscillates around value of zero and is able to follow the e xternal applied magnetic field with some delay,\nwhereas it oscillates around a non-zero value which indicates a dynam ically ordered phase for low temperatures and\nsmall magnetic field amplitudes36. The physical mechanism described briefly above points out the exis tence of a\ndynamic phase transition (DPT)33,36,37.\nDPTs and hysteresis behaviors can also be observed experimentally . For example, by benefiting from surface\nmagneto-optic Kerr effect (MOKE), dynamic scaling of magnetic hys teresis in ultrathin ferromagnetic Fe/Au(001)\nfilms has been studied, and it is reported that the dispersion of hyst eresis loop area of studied system obeys to a\npower law behavior38. A comprehensive study, which includes the hysteresis loop measur ement of well-characterized\nultrathin Fe films grown on flat and stepped W(110) surfaces, has b een done by using MOKE, and prominent\nexperimental observations are reported in Ref.39. In addition to these pioneering works mentioned briefly above, to\nthe best of our knowledge, there exist a number of experimental s tudies regarding the nonequilibrium properties of\ndifferenttypesofmagneticmaterialssuchasCofilmsonaCu(001)su rface40, polycrystallineNi 80Fe20films41, epitaxial\nFe/GaAs(001) thin films42, Fe0.42Zn0.58F243, finemet thin films with composition Fe 73.5Cu1Nb3Si13.5B944, [Co/Pt]3\nmagnetic multilayers with strong perpendicular anisotropy45as well as assembly of paramagnetic colloids46. Based\nupon the detailed experimental investigations, it has been discover ed that experimental nonequilibrium dynamics of\nconsidered real magnetic systems strongly resemble the dynamic b ehavior predicted from theoretical calculations of a\nkinetic Ising model. From this point of view, it is possible to see that the re exists an impressive evidence of qualitative\nconsistency between theoretical and experimental investigation s.\nOn the other hand, in the latter group there exists a limited number o f nonequilibrium studies concerning the\ninfluences of time varying magnetic field on the mixed spin-1/2 and spin -3/2 Ising ferrimagnetic model. For instance,\nthermal and magnetic properties of a mixed Ising ferrimagnetic mod el consisting of spin-1/2 and spin-3/2 on a square\nlattice have been analyzed by making use of Glauber-type stochast ic process47. It has been reported that the studied\nsystem always exhibits a dynamic tricritical point in amplitude of exter nal applied field and temperature plane, but\nit does not show in the single-ion anisotropy and temperature plane f or low values of amplitude of field48. Following\nthe same methodology, a similar study has been done to shed some ligh t on what happens when an oscillating\nmagnetic field is applied to the mixed spin-1/2 and spin-3/2 Ising model on alternate layers of hexagonal lattice. It\nhas been found that depending on the Hamiltonian parameters, the system presents dynamic multicritical as well\nas compensation behaviors49. However, the aforementioned studies are mainly based on molecula r field theory. It\nis a well known fact that, in molecular field theory, spin fluctuations a re ignored and the obtained results do not\nhave any microscopic information details of system. From this point v iew, in order to obtain the true dynamics of a\nmixed spin-1/2 and spin-3/2 Ising ferrimagnetic system on a square lattice under the presence of a time dependent\noscillating magnetic field, we intend to use of Monte Carlo simulation tec hnique which takes into account the thermal\nfluctuations, and in this way, non-artificial results can be obtained .\nThe outline of the paper is as follows: In section II we briefly present our model. Section III is dedicated to the\nresults and discussion, and finally section IV contains our conclusion s.\nII. FORMULATION\nWe consider a two-dimensional kinetic Ising ferrimagnetic system wit h mixed spins of σ= 1/2 andS= 3/2 defined\non a square lattice, and the system is exposed to a time dependent m agnetic field source. The Hamiltonian describing\nour model is given by\nH=−J1/summationdisplay\n/angbracketleftnn/angbracketrightσA\niSB\nj−J2/summationdisplay\n/angbracketleftnnn/angbracketrightσA\niσA\nk−D/summationdisplay\nj(SB\nj)2−H(t)\n/summationdisplay\niσA\ni+/summationdisplay\njSB\nj\n (1)\nwhere the σi=±1/2, andSj=±3/2,±1/2 are the Ising spins on the sites of the sublattices A and B, respect ively.\nFirst and second sums in Eq. (1) are over the nearest- and next-n earest neighbor pairs of spins, respectively. We\nassumeJ1<0 such that the exchange interaction between nearest neighbour s is antiferromagnetic. J2is the exchange\ninteraction parameter between pairs of next-nearest neighbors of spins located on sublattice A, and Dis single ion-\nanisotropy term which affects only S= 3/2 spins located on sublattice B. The time varying sinusoidal magnetic fi eld\nis as following\nH(t) =h0sin(ωt) (2)\nhere,h0andωare amplitude and angular frequency of the external field, respec tively. The period of the oscillating\nmagnetic field is given by τ= 2π/ω.3\nThe linear dimension of the lattice is selected as L = 40 through all simula tions, and Monte Carlo simulation based\non Metropolis algorithm50is applied to the kinetic mixed Ising ferrimagnetic system on a 40 ×40 square lattice\nwith periodic boundary conditions in all directions. Configurations we re generated by selecting the sites sequentially\nthrough the lattice and making single-spin-flip attempts, which were accepted or rejected according to the Metropolis\nalgorithm. Data were generated over 50 independent samples realiz ations by running the simulations for 60000 MC\nsteps per site after discarding the first 20000 steps. This amount of transient steps is found to be sufficient for\nthermalization for the whole range of the parameter sets. Error b ars are found by using Jacknife method51. Because\nthe calculated errors are usually smaller than the sizes of the symbo ls in the obtained figures, they have not been\ngiven in this study.\nThe instantaneous values of the sublattice magnetizations MAandMB, and also the total magnetization MTat\nthe time t are defined as\nMA(t) =2\nL2/summationdisplay\ni∈AσA\ni, M B(t) =2\nL2/summationdisplay\nj∈BSB\nj, M T(t) =MA(t)+MB(t)\n2. (3)\nBy benefiting from the instantaneous magnetizations over a full pe riod of oscillating magnetic field, we obtain the\ndynamic order parameters as follows\nQA=1\nτ/contintegraldisplay\nMA(t)dt, Q B=1\nτ/contintegraldisplay\nMB(t)dt, Q t=1\nτ/contintegraldisplay\nMT(t)dt, (4)\nwhereQA,QBandQtdenote the dynamic order parameterscorrespondingto the subla tticesAandB, and the overall\nlattice, respectively. To determine the dynamic compensation temp eratureTcompfrom the computed magnetization\ndata, the intersection point of the absolute values of the dynamic s ublattice magnetizations was found using\n|QA(Tcomp)|=|QB(Tcomp)|, (5)\nsign(QA(Tcomp)) =−sign(QB(Tcomp)), (6)\nwithTcomp< Tc, whereTcis the dynamic critical temperature. We also calculate the time avera ge of the cooperative\npart of energy of the kinetic mixed Ising ferrimagnetic system over a full cycle of the magnetic field as follow52\nEcoop=−1\nL2τ/contintegraldisplay\nJ1/summationdisplay\n/angbracketleftnn/angbracketrightσA\niSB\nj+J2/summationdisplay\n/angbracketleftnnn/angbracketrightσA\niσA\nk+D/summationdisplay\nj(SB\nj)2\ndt. (7)\nThus, the specific heat of the system is defined as\nCcoop=Ecoop\ndT, (8)\nwhereTrepresents the temperature. We should mention here that DPT po ints separating the dynamically ordered\nand disordered phases are determined by benefiting from the peak s of heat capacities. We also verified that the peak\npositions of heat capacities do not significantly alter when larger Lis selected.\nIII. RESULTS AND DISCUSSION\nIn this section, we will focus our attention on the nonequilibrium dyna mics of the mixed spin-1/2 and spin-3/2 Ising\nferrimagnetic system under a time dependent magnetic field. First o f all, we will discuss the dynamic nature of the\nsystem when the system includes only the nearest-neighbor intera ction between spins, the single-ion anisotropy and\nexternal applied field. Next, we will give and argue the global dynamic phase diagrams including the both dynamic\ncritical and compensation temperatures in the case of the existen ce the next-nearest neighbor interaction between\nspins-1/2 located on sublattice A. Before we discuss the DPT featu res of the considered system, we should notice that\nthe situation of h0/|J1|= 0.0 indicates the equilibrium case, and our Monte Carlo simulation findings for this value\nofh0/|J1|are completely in accordance with the recently published work32where the equilibrium properties of the\npresent system were analyzed by following a numerical methodology of heat-bath Monte Carlo algorithm.4\nThe considered system exhibits three types of magnetic behaviors depending on the Hamiltonian parameters. These\nare dynamically ferrimagnetic ( i), ferromagnetic ( f) and paramagnetic ( p) phases, respectively. In the first type of\nphase, namely in iphase,|QA| /negationslash=|QB|, and, the time dependent sublattice magnetizations, MA(t) andMB(t) oscillate\nwith time around a non-zero value whereas they alternate around a non-zerovalue and |QA|=|QB|in the second type\nof phase, namely in fphase. In pphase which corresponds to the third type of phase, |QA|=|QB|andMA(t) and\nMB(t) oscillate around zero value, and they are delayed with respect to t he external applied magnetic field. Keeping\nin this mind, we illustrate the dynamic phase diagrams in a ( D/|J1|−kBTc/|J1|) plane with three oscillation periods\nτ= 50,100 and 200 and for some selected values of the applied field amplitude s (h0/|J1|) in Figs. 1(a)-(c). One of\nthe main findings is that DPT temperature decreases as the value of applied field amplitude increases. The physical\nmechanism underlying this observation can be much better underst ood by following a simple way: If one keeps the\nsystem in one well of a Landau type double well potential, a certain am ount of energy coming from magnetic field\nis necessary to achieve a dynamic symmetry breaking. If the amplitu de of the applied field is less than the required\namount then the system oscillates in one well. In this situation, the ma gnetization does not change its sign. In\nother words, the system oscillates around a non-zero value corre sponding to a dynamically ordered phase. As the\ntemperature increases, the height of the barrier between the tw o wells decreases. As a result of this, the less amount\nof magnetic field is necessary to push the system from one well to an other and hence the magnetization can change\nits sign for this amount of magnetic field. Consequently, the time ave raged magnetization over a full cycle of the\noscillating magnetic field becomes zero. For the relatively high oscillatio n period values, dynamic magnetizations\ncorresponding to the instantaneous sublattice order parameter s can respond to the oscillating magnetic field with\nsome delay, whereas a competition occurs between the period τof the field and the relaxation time of the system\nas the period of the external magnetic field decreases. Hence, th e dynamic magnetizations can not respond to the\nexternal magnetic field due to the increasing phase lag between the field and the time dependent magnetization. This\nmechanism makes the occurrence of the DPT difficult for the conside red system. Another important observation is\nthat an unexpected sharp dip occurs between dynamically ordered and disordered phases in the ( D/|J1|−kBTc/|J1|)\nplane with increasing value of the amplitude of the external applied fie ld, and our results show that observation such\nkind of treatment explicitly depends upon the value of τof field. Furthermore, it is necessary to state that for both\nlarge negative and positive values of single-ion anisotropies, the pha se transition points saturate a certain temperature\nregions, and they tend to shift to the lower temperature regions w ith increasing amplitude and period of the external\napplied field.\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s48/s46/s53\n/s32/s61/s32/s53/s48\n/s32/s32/s107\n/s66/s84\n/s99/s47/s124/s74\n/s49/s124\n/s68/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s102\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105\n/s102/s40/s99/s41\n/s32/s61/s32/s50/s48/s48\n/s68/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105\n/s102\n/s32/s32\n/s68/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s32/s61/s32/s49/s48/s48\nFIG. 1: (Color online) Dynamic phase boundaries of the syste m in the ( D/|J1|−kBTc/|J1|) plane with some selected values\nof external field amplitudes h0/|J1|= 0.0 and 0.5. The curves are plotted for three values of oscillating per iod: (a) τ= 50, (b)\nτ= 100 and (c) τ= 200. The dotted lines are boundary lines between two dynami cally ordered phases.\nIn Figs. 2(a)-(b), we depict the effect of the single-ion anisotropy on the thermal variations of dynamic order\nparameters corresponding to the phase diagram illustrated in Fig. 1 (a) for value of h0/|J1|= 0.5. It is clear\nfrom the figures that the treatments of the thermal variations o f sublattices as well as total magnetizations curves\nsensitively depend upon the value of single-ion anisotropy, for selec ted values of Hamiltonian parameters. In the5\nbulk ferrimagnetism of N` eel, it is possible to classify the thermal var iation of the total magnetization curve in\ncertain categories1. According to this nomenclature, for D/|J1| ≥0, the considered system clearly points out a Q-\ntype behavior, where the magnetizations of system begin to decre ase gradually starting from their saturation values\nwith increasing thermal agitation, and then they vanish at the DPT p oint. One can easily see that, in the range\n−1< D/|J1|<0, the magnetizations tend to fall prominently from their saturatio n values, and the system undergoes\na second order DPT as temperature increases. In addition to thes e, whenD/|J1|<−1, the system exhibits a L-type\nbehavior at which the total magnetization shows a temperature ind uced maximum which definitively depends on the\nvalue of single-ion anisotropy as well as other Hamiltonian parameter s. Based on the above simulation observations,\nit is possible to make an inference that the studied system has three types of dynamic magnetic behavior. It is also\nworthy of note that even though the magnitudes of spins are differ ent from each other, both QAandQBexhibit a\nDPT at the same critical temperature, which is a result of the neare st-neighbor exchange coupling J1.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s48/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s50/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s55/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s50/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s53\n/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124/s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s32/s61/s32/s53/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s65/s124/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\nFIG. 2: (Color online) Effects of the single-ion anisotropy o n the thermal variations of order parameters |Qt|,|QA|and|QB|\nfor a combination of Hamiltonian parameters corresponding to the phase diagram depicted in Fig. 1.\nThe influences ofthe applied field amplitude h0/|J1|onthe thermalvariationsoftotal and sublatticemagnetizations\nas well as dynamic heat capacity of the system are plotted in Figs. 3( a)-(c) corresponding to the phase diagram\nconstructed in Fig. 1(a) with value of single-ion anisotropy D/|J1|= 1.0.In Fig. 3(a), total magnetization curves\nof the system are shown. As seen in this figure, magnetization curv es exhibit Q-type behavior and dynamic critical\ntemperatures decreases with increasing h0/|J1|values. On the other hand, dynamic heat capacity curves which are\ndepicted in Fig. 3(c) show a sharp peak behavior indicating the phase transition temperature. Moreover, one can\nreadily see that increasing value of the applied field amplitude gives rise to decrease the maximum of the dynamic\nheat capacity curves. We should note here that such kinds of dyna mic heat capacity behaviors have been found in\na ferrimagnetic core-shell nanoparticle composed of a spin-3 /2 ferromagnetic core which is surrounded by a spin-1\nferromagnetic shell layer under the presence of a time dependent magnetic field53,54.\nIn Fig. 4, we investigate the effect of the applied field period on the DP T features of the system. Phase diagrams\nin Fig. 4(a) are constructed for a value of the applied field amplitude h0/|J1|= 0.5.It is possible to say that DPT\npoints are depressed with increasing applied field period especially in th e high values of the single-ion anisotropy.\nThe physics behind of these findings are identical to those emphasiz ed in Fig. 1. Therefore, we will not discuss these\ninterpretations here. Instead of this, in Fig. 4(b) we will give the infl uence of applied field period on the thermal\nvariations of sublattice magnetizations for a considered value of sin gle-ion anisotropy D/|J1|= 1.0 corresponding to\nthe phase diagram illustrated in Fig. 4(a). It is found that as the the rmal agitation increases starting from zero,\nthe values of sublattice magnetizations begin to gradually decrease and the system undergoes a DPT at the critical\ntemperature which sensitively depends on value of the τ.\nAccording to our simulation results, the kinetic mixed spin-1/2 and sp in-3/2 Ising ferrimagnetic system including\nonly nearest-neighborinteraction between spins, the single-ion an isotropyand external applied field does not display a\ndynamic multicritical behavior for a wide range of Hamiltonian paramet ers used in here, in contrary to the previously6\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s72/s101/s97/s116/s32/s67/s97/s112/s97/s99/s105/s116/s121/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124/s68/s47/s124/s74\n/s49/s124/s61/s49/s46/s48\n/s32/s61/s32/s53/s48/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s48/s46/s53\n/s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s40/s99/s41\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s66/s124\n/s124/s81\n/s65/s124\n/s32/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\nFIG. 3: (Color online) Temperature dependencies of (a) tota l magnetization |Qt|, (b) sublattice magnetizations |QA|and|QB|,\nand (c) dynamic heat capacity for D/|J1|= 1.0 andτ= 50 with h0/|J1|= 0.0 and 0.5.\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53\n/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32/s49/s48/s48\n/s32/s32/s50/s48/s48\n/s32/s32/s107\n/s66/s84\n/s99/s47/s124/s74\n/s49/s124\n/s68/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s66/s124\n/s124/s81\n/s65/s124/s40/s98/s41\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s32/s32/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124\nFIG. 4: (Color online) (a) Dynamic phase boundaries of the sy stem in ( D/|J1| −kBTc/|J1|) plane for h0/|J1|= 0.5 with\nτ= 50,100 and 200. (b) Effects of the external applied field period on the thermal variations of sublattice magnetizations |QA|\nand|QB|forD/|J1|= 1.0 andτ= 50,100 and 200.\npublished molecular field investigation where dynamic first order phas e transitions and dynamic tricritical points have\nbeen reported for the same model48.\nIn the following analysis, in order to shed some light on the effect of th e next-nearest neighbor interaction between\nspins-1/2 of the studied system, we give the dynamic phase bounda ries in (J2/|J1| −kBT/|J1|) plane for some\nconsidered values of applied field amplitudes with τ= 100 in Figs. 5(a-c). The phase boundaries containing both\ndynamic critical and compensation temperatures are plotted for t hree values of single-ion anisotropies D/|J1|=\n−1.0,0.0 and 1.0, respectively. At first sight, one can clearly see that the dynamic compensation temperatures do not\nemerge until J2/|J1|reaches a certain amount of value. After the aforementioned valu e ofJ2/|J1|, with an increment\ninJ2/|J1|does not lead to change the location of dynamic compensation point f or fixed values of D/|J1|andh0/|J1|.7\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s49/s46/s48\n/s32/s50/s46/s48\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s32/s32/s107\n/s66/s84/s47/s124/s74\n/s49/s124\n/s74\n/s50/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s105\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112\n/s105/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s40/s99/s41/s32\n/s32\n/s32/s32\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112\n/s105/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s84\n/s99\n/s32/s84\n/s99/s111/s109/s112\n/s74\n/s50/s47/s124/s74\n/s49/s124\n/s32/s32\n/s40/s98/s41\nFIG.5: (Color online)Dynamicphaseboundariesincludingb othcriticalandcompensation temperaturesin(J 2/|J1|−kBT/|J1|)\nplane for τ= 100 and with some selected values of external field amplitud es h0/|J1|= 0.0,1.0,and 2.0. The curves are plotted\nfor three values of single-ion anisotropies (a) D/|J1|=−1.0, (b)D/|J1|= 0.0 and (c) D/|J1|= 1.0.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48/s32\n/s32/s57/s46/s48\n/s32/s56/s46/s48\n/s32/s55/s46/s48\n/s72/s101/s97/s116/s32/s67/s97/s112/s97/s99/s105/s116/s121/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124 /s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s68/s47/s124/s74\n/s49/s124/s61/s45/s49/s46/s48\n/s104\n/s48/s47/s124/s74\n/s49/s124/s61/s49/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s40/s99/s41\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s124/s81\n/s65/s124\n/s32/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\nFIG. 6: (Color online) Influences of the next-nearest neighb or interactions on the thermal variations of (a) total magne tization\n|Qt|, (b) sublattice magnetizations |QA|and|QB|and (c) dynamic heat capacity for some selected values of Ham iltoanian\nparameters corresponding to the phase diagram illustrated in Fig. 5.\nIn contrary to this behavior, in accordance with the expectations , as the value of J2/|J1|gets bigger starting from\nzero, the much more thermal energy is necessary to reveal a DPT . On the other side, both the dynamic critical and\ncompensation temperatures strongly depend upon the selected v alues ofD/|J1|andh0/|J1|. An increase in the value\nofh0/|J1|gives rise to shift the dynamic critical and compensation points to low er temperatures and also allows the\nsystem to display a dynamic compensation behavior at the relatively lo w value of J2/|J1|. Additionally, it is possible\nto make an inference that with increasing value of single-ion anisotro py, the region where the dynamic compensation\nbehavior occurs shifts to upward for considered Hamiltonian param eters. This situation can be well understood by\ncomparing the Figs. 5(a), (b) and (c) with each other.8\nEffects of the next-nearest neighbor interactions on the therma l variations of total and sublattice magnetizations\nas well as on dynamic heat capacity of the studied system for some s elected values of D/|J1|=−1.0,h0/|J1|= 1.0\nwithτ= 100 corresponding to the phase diagram shown in Fig. 5(a) are see n in Figs. 6(a-c). We give the total\nmagnetizationcurvesforchangingvalueof J2/|J1|inFig. 6(a). Thesecurvesexplicitlyrefertheexistenceofadynamic\ncompensation behaviors, and they also exhibit a N-type magnetic be havior. As we discussed before, varying value of\nJ2/|J1|does not give rise to cause a change in value of dynamic compensation point in temperature plane. Besides, in\nordertoshowhowthe dynamiccompensation phenomenonrises, th ermalvariationsofthe absolutevaluesofsublattice\nmagnetizations are given in Fig. 6(b). It can be said that an increase in value of J2/|J1|leads to the existence of\na dynamically stronger ferromagnetic interaction between σspins. In this way, the σspins can remain ordered\nat relatively higher temperatures. When the thermal energy incre ases starting from zero, the values of sublattice\nmagnetizations begin to gradually decrease from their saturation v alues until both sublattices magnetizations are\nequail in magnitude at a certain temperatures at which dynamic comp ensation point emerges below the dynamic\ncritical transition temperature. J2/|J1|dependencies of dynamic heat capacities are presented in Fig. 6(c) . It is\nobvious from the figure that the dynamic heat capacity curves exh ibit a sharp peak at the transition temperature,\nand when the value of J2/|J1|increases, the dynamically ordered phase region gets wider, in othe r words, the location\nof the sharp peak slides to a higher value in the temperature plane. T he shape of the heat capacity curves are also\nnearly same for selected values of Hamiltonian parameters.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s49/s46/s48\n/s32/s50/s46/s48\n/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124 /s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s124/s81\n/s65/s124/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s32/s84\n/s99\n/s32/s84\n/s99/s111/s109/s112/s32 /s32/s61/s32/s53/s48\n/s32/s49/s48/s48\n/s32/s50/s48/s48/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s104\n/s48/s47/s124/s74\n/s49/s124/s40/s99/s41\n/s105/s112\nFIG. 7: (Color online) Effects of external applied field ampli tude on the thermal variations of order parameters |Qt|,|QA|and\n|QB|forD/|J1|=−1.0,J2/|J1|= 10.0 andτ= 100 with h0/|J1|= 0.0,1.0 and 2.0.\nAs a final investigation, we give and discuss the influences of varying value of applied field amplitude on the thermal\nvariations of total and sublattices magnetizations exhibiting dynam ic compensation as well as critical temperatures\nin Figs. 7(a-b) corresponding to the dynamic phase boundary seen in Fig. 5(a) for J2/|J1|= 10.0.It is possible\nto mention that both dynamic compensation and critical temperatu res strongly depend on value of the external\napplied field amplitude and they tend to shift to a lower region in temper ature plane, and compensation behavior\ndisappears with increasing value of h0/|J1|. In order to demonstrate the detailed magnetic behavior of syste m, we\ngive the dynamic phase boundaries in ( h0/|J1|−kBT/|J1|) planes for three values of the applied field periods such as\nτ= 50,100 and 200 for selected values of single-ion anisotropy D/|J1|=−1.0 and next-nearest neighbor interaction\nJ2/|J1|= 10.0 in Fig. 7(c). Based on the calculated phase diagrams, it can be said t hat the decreasing (increasing)\napplied field period has no effect on the dynamic behavior of compensa tion behavior whereas it affects prominently\nthe dynamic critical temperature of the system such that the dyn amically ordered phase region gets wider (narrower).9\nIV. CONCLUDING REMARKS\nIn conclusion, it has been carried out a detailed Monte Carlo investiga tion based on standard single-spin flip\nMetropolis algorithm to determine the true DPT properties of a mixed spin-1/2 and spin-3/2 Ising ferrimagnetic\nsystem under a time varying magnetic field. A complete picture of glob al dynamic phase diagrams separating the\ndynamically disordered and ordered phases has been constructed by benefiting from the peaks of thermal variations\nof dynamic heat capacities in order to have a better understanding of the physical background underlying of the\nconsidered system. The most important observations reported in the present study can be briefly summarized as\nfollows:\n•When the considered system only includes the nearest-neighbor int eraction, single-ion anisotropy and a time\ndependent sinusoidally oscillating magnetic field, it does not point out a dynamic compensation point.\n•Stationary state solutions of the system strongly depend on the s elected system parameters. As discussed in\ndetail in previous section, with increasing values of amplitude and per iod of the external applied field, the\ndynamic phase boundaries tend to shift to the lower temperature r egions in related planes, and a sharp dip\noccurs between dynamically ordered and disordered phases.\n•In contrary to the previously published investigation for the same m odel where dynamic first-order phase tran-\nsitions and tricritical points have been reported48, it has not been found any evidence of the dynamic first-order\nphase transitions in our present work. The reason is most likely the f act that the method we used completely\ntakes into account the thermal fluctuations, which allows us to obt ain non-artificial results.\n•When the next-nearest neighbor interaction between spins-1/2 is included and exceeded a characteristic value\nwhich sensitively depends on value of the single-ion anisotropy and am plitude of the external applied field, the\nsystem exhibits a dynamic compensation behavior below its critical te mperature. According to the our simula-\ntion results, the changing value of applied field period has no effect on the location of dynamic compensation\npoint.\nFinally, we should note that it is possible to improve the obtained result s by making use of a more realistic system\nsuch as Heisenberg type of Hamiltonian. 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The simulated spectra are compared to magneto-optical data avail-\nable in recent literature. A collinear ferrimagnetic phase with a small saturation magentization, a large perpendicular\nanisotropy, and Curie temperature above 700 K is found to be consistent with the measured spectra. We hypothesise\nthat an admixture of the noncollinear phase, which could explain the lower than predicted net moment and magnetic\nanisotropy observed experimentally, is also present.\nTraditionally, ferrimagnets have been applied in magneto-\noptical recording,1–3where a laser beam is used to increase the\ntemperature close to the Curie point so that the magnetization\ncan be reversed by a small applied magnetic field. Ferrimag-\nnetic alloys containing rare earth and transition metals, such as\nGdFeCo and TbCo, with large perpendicular anisotropy near\nroom temperature have show exceptional performance in this\narea.4\nRecently, ferrimagnetic materials have attracted much at-\ntention for applications in high-density magnetic random ac-\ncess memories (MRAM) and logic devices, which can com-\nbine high perpendicular magnetic anisotropy (PMA) with\nsaturation magnetization much lower than in typical fer-\nromagnets and with Curie temperature well above room\ntemperature.5–8\nFerrimagnets usually consist of two magnetic sublattices\nwith anfiterromagnetic coupling between them. This antifer-\nromagnetic exchange interaction allows for fast spin dynamics\nwhich has motivated the extensive research in antiferromag-\nnetic spintronics.9–11The absence of net magnetic moment\nin antifferomagnetic materials prevents dipolar coupling be-\ntween closely packed memory arrays. However, at the same\ntime, it precludes control of the magnetic domain structure by\nexternal magnetic filed. The small but finite net magnetizaiton\nof ferrimagnetic materials alleviates this problem. Moreover,\nclose to a compensation point the reduced net spin polariza-\ntion makes the state of a ferrimagentic layer in, e.g., a mag-\nnetic tunnel junction (MTJ), more susceptible to spin-transfer\ntorque (STT).12–14\nThe compensation of magnetic sublattices is commonly\nachieved in metallic alloys containing rare earths men-\ntioned above. However, one can avoid the reliance on\nrare earth elements by turning to metallic antiperovskite ni-\ntrides some of which have collinear or non-collinear ferri-\nmagnetic structure.15–17Mn-based antiperovskiet nitrides are\na broad family of materials hosting a range of phenom-\nena including magneto-transport,18–20magneto-caloric,21–23\nor magneto-optical properties tunable by chemical composi-\ntion or lattice strain.24–26We predicted that Mn 3GaN, which\nis a widely studied member of this family with a fully com-\npensated triangular antifferomagntic ground state (cubic lat-\ntice), can develop a collinear ferrimagnetic (FIM) phase when\napplying compressive biaxial strain at room temperature.27\nSubsequently, we used magneto-optical Kerr spectroscopy toanalyse the magnetic structure of Mn 3NiN thin film which is\na closely related antiperovskite with triangular antiferromag-\nnetic ground state. The measured data are consistent with the\npresence of a collinear FIM phase at room temperature.28\nMn 4N is another member of the antiperovskte family. Its\nmagnetic structure is even more complicated than that of\nMn 3NiN as Ni on the 1 asite is replaced by another Mn with\na large magnetic moment. Mn 4N has received much atten-\ntion in recent years mainly due to a large perpendicular mag-\nnetic anisotropy (PMA ≈105J/m3) and ultrafast response to\nexternal field.29–38In addition, the saturation magnetization,\nMsis relatively low as demonstrated by numerous studies of\nMn 4N thin films listed in Table I. Therefore, low critical STT-\nswitching current density, Jc∝αMstHk(where αis the damp-\ning constant, tis the magentic layer thickness, and Hkis the\nanisotropy field proportional to the PMA energy density, Ku)\nis expected.39Moreover, the shape anisotropy is negligible\ndue to low Msso the magnetic anisotropy is dominated by\nthe magnetocrystalline contribution.39\nIn order to utilize the potential of Mn 4N for spintronic ap-\nplications such as MTJs or skyrmionic devices,40it is crucial\nto attain thorough understanding of the microscopic origin\nof the large PMA combined with low magnetization and to\nbe able to grow thin films with well defined magnetic struc-\nture on substrates compatible with CMOS technology. (So\nfar, superior magnetic properties including sharp magnetiza-\ntion switching have been observed on STO substrates,41which\ncomplicates integration due to the requirement of post-growth\nannealing at high temperatures to achieve crystallinity of the\nSTO barrier.)\nIn studies listed in Table I, Mn 4N has been deposited\nsince 2014 on a range of substrates with different lattice mis-\nmatches, e.g., MgO, SrTiO 3(STO), and LaAlO 3(LAO) with\nmismatch of approximately −6%,−0.1%, and +2%, respec-\ntively, assuming (001) surfaces and lattice constant of cubic\nMn 4N equal to 0.3865 nm.47It has been observed using x-\nray diffraction that the films have c/a≈0.98−0.99, where\naandcare the in-plane and out-of-plane lattice constant, re-\nspectively, despite the different magnitude and sign of lattice\nmismatch.\nTherefore, experimental studies that keep track of c/aand\nPMA generally conclude that the origin of PMA in Mn 4N\nfilms is the tetragonal distortion.29,30,33,42Moreover, studies\nthat include ab initio simulations have associated the ob-arXiv:2307.15246v1  [cond-mat.mtrl-sci]  28 Jul 20232\nTABLE I. Comparison of Mn 4N films with thickness ( t), and mea-\nsured magnetic anisotropy energy (K u) and saturation magnetization\n(Ms).\nSubstrate c/at[nm] method Ku[MJ/m3]Ms[kA/m]\nMgO300.987 35 PLD 0.16 157\nMgO290.99 26 MBE 0.22 145\nMgO420.99 100 Sputtering 0.88 110\nSTO310.99 25 MBE 0.1 80\nMgO430.983 9 MBE 0.18 127\nMgO440.991 30 MBE 0.075 100\nMgO41- 10 MBE 0.11 118\nSTO41- 10 MBE 0.11 105\nSTO32- 10 MBE 0.11 71\nMgO330.993 18 MBE 0.06 63\nSTO330.989 17 MBE 0.126 73\nLAO330.998 19 MBE 0.045 53\nMgO390.99 30 Sputtering 0.1 80\nMgO450.99 28 Sputtering 0.17 156\nGlass360.993 45 Sputtering 0.022 36\nGlass460.988 48 Sputtering 0.073 99\nMgO/VN380.987 28 Sputtering 0.043 85\nserved PMA with a collinear ferrimagnetic phase, so called\nFIM Bphase, with total energy minimum at c/a=0.98.31,39\nCollinear ferrimagnetic phases FIM Aand FIM Bshown in\nFig. 1 were revealed by neutron diffraction experiments as\nearly as 1962.47Ito et al.31and Isogami et al.39demonstrated\nbyab inito simulations that FIM Bhas a significantly lower to-\ntal energy than FIM Ain the range c/a∈(0.96−1.1), in agree-\nment with our simulations shown in Fig. 1(d). Both theoretical\nstudies suggest that this intrinsic tetragonal phase with large\nPMA explains why c/a∈(0.98−0.99)has been reported in\nMn 4N films epitaxially grown on different substrates regard-\nless of the film thickness and lattice mismatch.\nHowever, negligible dependence of PMA on c/awas pre-\ndicted by Isogami et al.39which is in disagreement with PMA\nmeasured by Hirose et al.33in Mn 4N on MgO, STO, and\nLAO. Moreover, the theoretical studies mentioned so far31,39\nhave not considered any noncollinear magnetic phases de-\nspite the fact that a noncollinear ferrimagnetic phase (ncFIM)\nwas identified by neutron diffraction in bulk Mn 4N in 1979.17\nMagnetic moments of Mn atoms in this \"umbrella-like\" struc-\nture shown in Fig.1(a) do not mutually compensate as in cubic\nMn 3NiN or Mn 3GaN even though the lattice has cubic sym-\nmetry with space group Pm¯3m.17This is due to the fact that\n1acorner site is occupied by Mn with a large local magnetic\nmoment, m1a. The moments in face-center 3 cpositions, m3c,\nare tilted out of the (111) planes (where the Mn atoms form\na kagome lattice) by approximately 20◦to have a component\nalong the [111] axis, antiparallel to m1a. The ncFIM struc-\nture was confirmed computationally by Uhl et al.48and more\nrecently by Zhang et al.49including the net moment along\nthe [111] axis, mnet=1.1µB. The same team also proposed\nrelated noncollinear ferrimagnetic phases in Mn 4N film on\nMgO45based on ncFIM and the coplanar triangular antifer-\nromatic structure of Mn-based antiperovskite nitrides.\nFIG. 1. Comparison of ferrimagnetic structures considered: (a-c)\nunit cells with magnetic moments and chemical identities: Mn on 3 c\nsites in red, Mn on 1 asites in blue, N in green, and (d) corresponding\ncalculated total energies vs c/aratio.\nHere we compare the collinear and noncollinear FIM\nphases of strained Mn 4N based on total energy and Magneto-\noptical Kerr effect (MOKE) calculated using Density Func-\ntional Theory (DFT) and linear response theory. This is pri-\nmarily motivated by a recent study of MOKE in 23 nm thick\nMn 4N films sputtered on MgO substrate.50The authors inter-\npret their MOKE spectra based on projected densities of states\n(PDOS) simulated by Isogami et al.39and conclude that: \"The\nfine structures observed in Kerr rotation could be attributed to\na superposition of different magnetic phases from the dom-\ninant ferrimagnetism in Mn 4N, although theoretical calcula-\ntions may be necessary for further interpretation.\" Moreover,\nan in-plane component of magnetization has been detected in\none Mn 4N film deposited on MgO by MBE,51and the authors\nexplained it by presence of a FIM phase with magnetization\nalong the [111] axis.17,52\nTherefore, we simulate the MOKE spectra in this work\nfor the three main magnetic structures FIM A, FIM B, and nc-\nFIM. We employ noncollinear spin polarized DFT following\nthe approach of Ref. [53] and our earlier work.28We use\nthe projector augmented wave method as implemented in the\nV ASP code54with with generalized gradient approximation\n(GGA) parameterized by Perdew–Burke–Ernzerhof.55Our re-\nsults were obtained using a 500 eV energy cutoff and a 23 ×\n23 × 23 k-mesh (for a unit cell with 5 atoms) to ensure conver-\ngence (in agreement with numerical settings of Ref. [39]). The\nvalence configurations of manganese and nitrogen are 3 d64s1\nand 2 s22p3, respectively.3\nTABLE II. Comparison of magnetic space groups and corresponding\nanomalous Hall conductivity tensor (AHC) for each pahse.\nncFIM FIM A, FIM B\nSpace group 123, P4/mmm 123, P4/mmm\nMag. space group 12.62, C2’/m’ 123.345, P4/mm’m’\nAHC, σα,β\n0 σxyσxz\n−σxy 0σxz\n−σxz−σxz0\n\n0σxy0\n−σxy0 0\n0 0 0\n\nTaking into account our recent study of MOKE spectra\nin the closely related antiperovskite Mn 3NiN,28and earlier\nMOKE studies of some collinear antiferromagnets such as\nCuMnAs,56we modify the intra-atomic Coulomb interaction\nwithin GGA through the rotationally invariant approach to\nGGA+U as proposed by Dudarev et al.57We explore values\nof U from 0.7 eV (refined in Ref. [28]) to 2.2 eV on the Mn-\n3dorbital. (All data shown here in figures were simulated\nusing U = 0.7 eV .) This repulsion lifts the unoccupied man-\nganese d-states further away from the Fermi level, resulting in\na blueshift in the optical and magneto-optical responses which\nimproves the agreement with the available measured data.50\nThe unit cells and corresponding total energies as functions\nof the c/aratio are shown in Fig. 1. By analysing available x-\nray data of films listed in Table I we noticed a range of values\nof Poisson’s ratio, ν. Therefore, for each c/aratio, we calcu-\nlate the total energy assuming ν=0.3345as well as a=a0,\nwhere a0=0.389 nm is the equilibrium lattice parameter for\nncFIM from our DFT simulations. This is close to the exper-\nimental value a0=0.3865 nm and to an earlier DFT calcula-\ntion, a0=0.382 nm.49We note that our conclusion is inde-\npendent of this choice: The energy minimum for FIM Aphase\nobtained at c/a>1 is more than 0.2 eV higher than the energy\nminimum of FIM Batc/a=0.98, in agreement with earlier\nDFT studies,31,39which suggest that the Mn-Mn direct AFM\ninteraction might be stabilizing the FIM Bstructure. However,\nthe ground state of ncFIM phase ( c/a=1) is another 30 meV\nlower than the energy minimum of FIM B. (This energy differ-\nence is much bigger than 4 meV between ncFIM and a cubic\ncollinear FIM phase with m1a(m3c) parallel (antiparallel) to\nthe [111] axis predicted by Zhang et al.49)\nTherefore, it is conceivable that ncFIM phase with c/a∈\n(0.99−1.0)coexists with FIM Bphase in films where the lat-\ntice mismatch with the substrate does not induce a large biax-\nial strain thanks to dislocations30,31or a \"dead layer\"39im-\nmediately above the interface. Areas of the film with less\ndislocations (better epitaxy) are then more likely to stabilise\nthe tetragonal FIM Bphase. This suggestion further elabo-\nrates on the aforementioned explanation why Mn 4N films with\nc/a≈0.99 have been reported on various substrates regard-\nless of the film thickness and lattice mismatch. Furthermore,\na mixture of FIM Band ncFIM phases could explain the lower-\nthan-predicted M sand PMA as we will discuss below.\nMOKE spectra offer a valuable insight into the magnetic\nstructure of thin films so we take this opportunity to com-\npare spectra simulated for all there ferrimagnetic phases to\nthe measured data.50As in our recent study of MOKE spec-\ntra in Mn 3NiN,28we note that MOKE is an optical coun-\nFIG. 2. Simulated MOKE spectra: (a) Kerr rotation and (b) Kerr\nellipticity; for two values of c/aand U = 0.7 eV (without smoothing).\nterpart of the Anomalous Hall Effect (AHE). Both MOKE\nand the intrinsic contribution to Anomalous Hall Conductiv-\nity (AHC), σα,β, originate (within linear response thoery) in\nnon-vanishing integral of Berry curvature Ωn,α,β(k)over the\nBrillouin zone:\nσα,β=−e2\n¯hZdk\n2π3∑\nn(occ. )f[εn(k)−µ]Ωn,α,β(k),\nΩn,α,β(k) =−2I∑\nm̸=n⟨km|να(k)|kn⟩⟨kn|νβ(k)|km⟩\n[εkn−εkm]2,(1)\nwhere f[εn(k)−µ]denotes the Fermi distribution function\nwith εn(k)andµbeing the energy eigenvalues of occupied\n(unoccupied) Bloch band, n, and the Fermi energy, respec-\ntively, and where να(k)corresponds to the velocity operator\nin Cartesian coordinates.\nThe Kerr angle ( θK) and ellipticity ( ηK) in case of polar-\nMOKE geometry depend on AHC as follows28:\nθK+iηK=−σxy\nσxxp\n1+i(4π/ω)σxx, (2)4\nwhere ωis the photon energy and we assume that the z-axis\n(parallel to the incident beam) is perpendicular to the film sur-\nface.\nThe presence of MOKE and AHE can be determined by an-\nalyzing the transformation properties of the Berry curvature\nunder all symmetry operations of a particular magnetic space\ngroup. In Table II we list the space groups of the three FIM\nphases of Mn 4N (subject to tetragonal distortion) obtained by\nFINDSYM software.58,59The last row of Table II presents\nthe form of the AHC tensor in linear response regime ob-\ntained using software Symmetr60considering both set of sym-\nmetry operations and the spin-orbit coupling. We note that\nboth collinear FIM phases share the same form of AHC ten-\nsor with one independent nonzero element, σx,y, inducing the\npolar-MOKE. The AHC tensor of ncFIM has two independent\nnonzero elements, σx,yandσx,z=σy,zas in case of strained\nMn 3NiN28. Cubic Mn 4N would have a magnetic space group\n166.101, R ¯3m’ and σx,y=σx,z=σy,z. The listed forms of the\nAHC tensor are determined by the symmetry of the structure\nrather than the net magnetization so they would not change\n(AHE and MOKE would not vanish) even if the net magnetic\nmoment vanished, i.e., if full compensation of the ferrimag-\nnet was achieved, which is desirable when seeking ultrafast\nspintronic devices.40,49\nFig 2 presents the main result of the work. It shows Kerr\nrotation and Kerr ellipticity as a function of energy, ω∈\n(1−7)eV . Our model does not include the intraband con-\ntribution which dominates below 1 eV , hence the choice of\nenergy interval. We calculated the spectra for several ratios\nc/a∈(0.97−1.03)but the dependence on tetragonal distor-\ntion appears to be much smaller than the differences between\nthe three FIM phases so we plot only spectra for c/a=0.99\nand 1.01. The spectra in this figure include fine features as we\nuse small Gaussian smearing, σ=0.01 eV , to treat the partial\noccupancies in k-space integration. This approach would cor-\nrespond to experimental data measured in a crystalline film\nwith few defects at low temperature. However, the avail-\nable MOKE data were measured in sputtered films at ambient\ntemperature.50Therefore, we include also Fig. 4 where the\nfine features are smoothed out.\nIn order to interpret our spectra based on features of the\nband structure, we plot the projected DOS for all three phases\nin Fig. 3. We resolve PDOS only for Mn 3d orbitals as\nthe other contributions are small and too far from the Fermi\nenergy to play an important role in the visible magneto-\noptical response. Mn 1, Mn 2, and Mn 3occupy the 3 csites\nwith cartesian coordinates (0.0, 0.5, 0.5), (0.5, 0.0, 0.5), and\n(0.5, 0.5, 0.0), respectively, whereas Mn 4occupies the 1 asite\nwith coordinates (0, 0, 0) as shown in Fig. 1. We note that\nour PDOS for FIM Bphase is in agreement with Fig. 7(a) of\nRef. [39].\nFigure 3(a) for FIM Ashows PDOS with one dominant tran-\nsition indicated by a grey arrow between a peak in occupied\nstates of Mn 1−3and a peak in excited states of Mn 4. This tran-\nsition described by energy difference, dE≈2 eV , corresponds\nto the sharp peak in magneto-optical response at ω≈2 eV .\nFigure 3(b) for FIM Bshows PDOS with two dominant tran-\nsition indicated again by arrows described by dE≈2 eV and\nFIG. 3. Comparison of Projected Density of States (PDOS) for FIM A\n(a), FIM B(b), and ncFIM (c) phases with c/a=0.99 and U = 0.7 eV .\nGrey arrows indicate potentially dominant excitations.\n3 eV which correspond to a dip and a peak in Kerr rotation,\nrespectively.\nFigure 3(c) for ncFIM shows PDOS lacking prominent\npeaks around 2 eV below the Fermi level, which are present\nin case of FIM B. The peak in PDOS of Mn 4present in FIM A\naround 0.5 eV above Fermi energy is shifted to 1 eV in ncFIM.\nSuch band structure results in absence of prominent peaks in\nthe magneto-optical response at photon energies below 4 eV .\nWe conclude that the predicted spectral features can be inter-\npreted based on transitions from Mn-3d orbitals on 3 csites to\nMn-3 dorbitals on 1 asite and that the three FIM phases should\nrelatively easily distinguishable even after smearing their dis-\ntinctive features by effects such as lattice defects or elevated\ntemperature.\nIn order to compare our data to MOKE spectra measured\nby Sakaguchi et al.,50we interpolate our curves plotted in\nFig. 2 using the UnivariateSpline function from Python Scipy\npackage with smoothing parameter set to 0.0008 on a linear\ngrid with 200 points. The smeared spectra for c/a=0.99\nare shown in Fig. 4. Our Kerr rotation, φ(ω), and elliptic-\nity,η(ω), can be compared directly to Fig. 5(a) and (b) of5\nFIG. 4. Simulated MOKE spectra: (a) Kerr rotation and (b) Kerr\nellipticity; for c/a=0.99, U = 0.7 eV , and smoothing of DFT data in\nFig. 2.\nRef. [50], respectively, where the boron content in Mn 4N is\nzero.\nFirstly, we note the the amplitude of the measured spectra\nis approximately 0.1 degree which is an order of magnitude\nlarger than in the related noncollinear Mn-based systems such\nas Mn 3NiN28and Mn Sn.61The amplitude of the simulated\nspectra is only a factor of two larger which is an unexpectedly\ngood agreement indicating high quality of the Mn 4N film.\nSecondly, the measured φ(ω)is positive below 1.5 eV , has a\ndip around 2.2 eV , and seems to come back to positive values\nabove 3 eV , where the studied interval ends. Such trend is\nobserved only in data simulated for FIM Bphase, although the\ncrossing points, φ(ω) =0, are shifted to lower energies (1\nand 2.6 eV instead of the measured 1.5 and 3 eV). A shift of\noptical and magneto-optical responses is commonly achieved\nin DFT studied by increasing the Coulomb repulsion on sites\nwith more localized states (here the Mn-3 dstates). Therefore,\nwe simulated the spectra for a range of Hubbard parameters,\nU, from U=0.7 eV (shown here) to U=2.2 eV . Spectra with\nU≈2.2 eV have crossing points very close to 1.5 and 3 eV , as\nin the experiment. However, we believe that this value of Uistoo high compared to Mn 3NiN28and CuMnAs56where U=\n0.7 and U=1.7 eV was used, respectively, for Mn-3 dorbitals\nto predict MOKE spectra in semiquantitative agreement with\nmeasurement.\nThirdly, we speculate that the crossing point, φ(ω) =0, at\n2.6 eV for FIM Bcould be shifted to higher energy by a as-\nsuming an admixture of ncFIM, which is negative throughout\nthe energy interval. Such superposition of spectra would also\nshift the crossing point at 1 eV to lower energies in contrast to\nexperiment. However, our predictions are less reliable below\n1.5 eV as our model does not include the intraband contribu-\ntions (the Drude peak).\nFinally, we check the agreement in case of Kerr ellipticity,\nη(ω), which is Kramers-Kronig-related to φ(ω). Fig. 5(b)\nof Ref. [50] shows a monotonous decrease of η(ω)to zero\naround 3 eV . As expected, FIM Bis the only phase that shows\nsuch trend in the simulated spectrum but the crossing point is\nagain shifted to lower energy.\nTo complete the analysis and to formulate our hypothe-\nsis about dominant FIM Bphase with an admixture of nc-\nFIM phase, we discuss Figure 5 which shows the magneto-\ncrystalline anisotropy profile and the component of magneti-\nzation perpendicular to the film (direction of the applied field),\nMz, for the two relevant phases. In Fig. 5(a) the total energy\nis plotted as a function of angle θbetween the net magneti-\nzation and the [001] axis (perpendicular to film). The insets\nshow the orientation of the local moments of FIM Bphase for\nnet magnetization pointing along [001], [110], and [00 1]. This\nchoice is relevant for the polar-MOKE experiment, where the\nsample is measured in magnetic field applied parallel and an-\ntiparallel to the [001] axis and the two spectra are subtracted\nto eliminate nonlinear MOKE effects.50The film has to un-\ndergo a rotation of magnetic moments driven by the reversal\nof the perpendicular applied field between the two measure-\nments. So it has to overcome the energy barrier of the in-plane\norientation, θ=π/2, which is dE=1.4 meV per formula\nunit or 3.78 MJ/m3for the FIM Batc/a=0.99. This PMA\nis in perfect agreement with earlier calculations.39However,\nthe value is much larger than experimental PMA ≈0.1 MJ/m3\nas listed in Table I. We cannot compare PMA directly in case\nof Sakaguchi et al.50as they give only the anisotropy field,\nHk=1.5 T, which is a typical value on Mn 4on MgO but\nwe do not know the saturation magnetization and the size of\nthe applied field. So we have to assume that the sample was\nfully aligned with the applied field during MOKE measure-\nment. However, the discrepancy between the predicted and\nmeasured PMA remains an open question which complicates\nthe interpretation of MOKE spectra.\nThe energy barrier is lower for ncFIM at c/a=0.99,dE=\n0.67 eV/f.u. or 1.81 MJ/m3, which speaks in favour of our hy-\npothesis of ncFIM and FIM Bcoexistence. However, the spin\nreorientation mechanism becomes much more complicated in\ncase of ncFIM. There are 8 variants of this phase with the net\nmagnetization pointing parallel or antiparallel to the 4 body-\ndiagonals, in perfect analogy to Mn 3NiN.20The applied field\ncan align the net magnetization with the [001] axis but the\nrotation of the moments to the opposite field orientation can\ngo through different energy minima and energy barriers de-6\npending on the local conditions in the film. We have carried\nout an extensive DFT study of the total energy landscape as a\nfunction of coherent rotations of the 4 local moments.\nWe considered all rotations that belong to the Pm¯3mspace\ngroup of the cubic perovskite lattice: 2-fold and 4-fold ro-\ntations about the main axes, 3-fold rotations about the body\ndiagonals, and 2-fold rotations about the side diagonals. De-\ntails about our findings will be summarised elsewhere. Here\nwe show an example of a rotation between two energy minima\n(from [111] to [ ¯11¯1]) which incurs the lowest energy barrier,\ndE=0.67 eV/f.u. mentioned above. This rotation is a simul-\ntaneous rotation by π/2 about the [010] axis, by π/2 about\nthe [001] axis and by [2 π/3] about [ ¯11¯1] axis. (An intuitive\nsimple rotation of magnetization from [111] to [11 ¯1] about the\n[¯110] axis would not restore the ground state magnetic struc-\nture.) The dash-dotted line in Fig. 5 consists of: (i) a simple\nrotation about the [ ¯110] axis from a state with net magnetiza-\ntion along [001] to the ground state along [111] denoted by\nθ0, (ii) the simultaneous rotation to [ ¯11¯1] denoted by π−θ0),\n(iii) a simple rotation about [110] to a state with net magneti-\nzation along [00 ¯1]. All four significant states are depicted as\ninsets in Fig. 5(b). Notably, the structure in the first inset is of\nthe same type (and has the same direction of magnetization)\nas the noncollinear structure proposed for Mn 4N in Fig. 2(d)\nof Ref. [45].\nAs illustrated by Fig. 5, a film in ncFIM phase with cu-\nbic lattice or with slight tetragonal distortion, c/a∈(0.99,1),\ncan be in a multi-domain state even at saturation (when the net\nmagnetization is fully aligned with applied field) and the mag-\nnetization reversal can follow different paths for each domain.\nInvestigation of magnetic domain wall (WD) propagation in\ncase of ncFIM is beyond the scope of this study but we believe\nthat the availability of more energy minima and lower energy\nbarriers between them (compared to FIM B) would facilitate\nDW propagation, thereby lowering the effective anisotropy\nfield to values observed experimentally.\nOur hypothesis would have implications for the observed\nsaturation magnetization, Ms, so we include Fig. 5(b) to show\nthe out-of-plane magnetization component, M z. Isogami et\nal.39predict 180 mT for FIM Band observe 110 mT exper-\nimentally. They are able to attribute the discrepancy to a\ndead layer at the interface with substrate, and nitrogen de-\nficiency and top surface oxidation. Here we suggest that\nthe lower Mscould be due to the admixture of ncFIM with\nMs=0.727µb/f.u. = 143 mT. However, we admit that Zhang\net al.49predict Ms=1.24µB/f.u. = 244 mT for ncFIM using\nDFT. Our M sis lower due to the use of U = 0.7 eV which\nleads to larger local moments on 3 csites and consequently\nmore compensation of m1a.\nIn summary, we simulated the MOKE spectrum of strained\nMn 4N using DFT+U assuming three ferrimagnetic phases dis-\ncovered by earlier neutron diffraction as well as theoretical\nstudies. We compared our results to polar-MOKE spectra\nmeasured by Sakaguchi et al.50in Mn 4N films on MgO sub-\nstrate. We found that the key features of the simulated spec-\ntra are consistent with the measured spectrum only in case\nof the FIM Bphase. The agreement of the simulated Kerr ro-\ntation could be further improved if a fraction of the ncFIM\nFIG. 5. Comparison of FIM Band ncFIM phases based on: (a) total\nenergy and (b) magnetization projection, M z; angle θdescribes the\nswitching from magnetization along the [001] axis to the opposite\ndirection. The insets show the magnetic structure at significant points\nalong the selected switching paths.\nphase was added to the dominant FIM Bphase. At the same\ntime, the admixture of ncFIM could explain the lower PMA\nand M sobserved experimentally. We believe that our analysis\nwill motivate further MOKE studies where the applied field\ncan be inverted along a chosen path, using a vector magnet\nin particular. This could shed more light on the ncFIM phase\npreferred at lower tensile strains and enable sub-nanosecond\nspin dynamics at room temperature.\nACKNOWLEDGMENTS\nWe acknowledge fruitful discussions with Freya Johnson,\nLesley F. Cohen, Martin Veis, and Jakub Železný. This work\nwas supported by the Ministry of Education, Youth and Sports\nof the Czech Republic through the e-INFRA CZ (ID:90254)7\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\n1P. Chaudhari, J. Cuomo, and R. Gambino, “Amorphous metallic films for\nmagneto-optic applications,” Applied Physics Letters 22, 337–339 (1973).\n2R. Carey, D. Newman, and B. Thomas, “Magneto-optic recording,” Journal\nof Physics D: Applied Physics 28, 2207 (1995).\n3P. Hansen, C. Clausen, G. Much, M. Rosenkranz, and K. Witter, “Mag-\nnetic and magneto-optical properties of rare-earth transition-metal alloys\ncontaining gd, tb, fe, co,” Journal of applied physics 66, 756–767 (1989).\n4S. Tsunashima, “Magneto-optical recording,” Journal of Physics D: Ap-\nplied Physics 34, R87 (2001).\n5T. Suemasu, L. Vila, and J. P. 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Keimer,3and Yu. N. Khaydukov3,4,6\n1Institute of Metal Physics, 620180 Ekaterinburg, Russia\n2Ural Federal University, 620002 Ekaterinburg, Russia\n3Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstraße 1, D-70569 Stuttgart, Germany\n4Max Planck Society Outstation at the Heinz Maier-Leibnitz Ze ntrum (MLZ), D-85748 Garching, Germany\n5National Research Center ”Kurchatov Institute”, 123182 Mo scow, Russia\n6Skobeltsyn Institute of Nuclear Physics, Moscow State Univ ersity, Moscow 119991, Russia\n(Dated: February 2, 2021)\nSpin-flop transition (SFT) consists in a jump-like reversal of antiferromagnetic magnetic moments\ninto a non-collinear state when the magnetic field increases above the critical value. Potentially the\nSFT can be utilized in many applications of a rapidly develop ing antiferromagnetic spintronics.\nHowever, the difficulty of using them in conventional antifer romagnets lies in (a) too large switching\nmagnetic fields (b) the need for presence of a magnetic anisot ropy, and (c) requirement to apply\nmagnetic field along the correspondent anisotropy axis. In t his work we propose to use artificial\nferrimagnets in which the spin-flop transition occurs witho ut anisotropy and the transition field\ncan be lowered by adjusting exchange coupling in the structu re. This is proved by experiment on\nartificial Fe-Gd ferrimagnets where usage of Pd spacers allo wed us to suppress the transition field\nby two orders of magnitude.\nAntiferromagnetic (AF) spintronic is nowadays a\nrapidly developing area [1–5]. In addition to non-\nvolatility of conventional ferromagnetic spintronics the\nAF devices can offer immunity to external magnetic dis-\nturbances, absence of cross-talks between small-area de-\nvices and much faster dynamics (THz vs MHz). The\nantiferromagnetic systems are featured by spin-flop tran-\nsition (SFT) when there is the transition from antifer-\nromagnetic ordering to noncollinear (NC) state at mag-\nnetic field exceeding certain value HSP. Creation of non-\ncollinearmagneticstateandpossibilitytoswitchbetween\nAF and NC states may have useful applications by utiliz-\ning anomalous Hall or Nernst effects [6–11]. In addition,\nproximity of noncollinear magnetic texture to supercon-\nducting layer generates long-range triplet superconduc-\ntivity which may also find diverse applications in super-\nconducting spintronics [12–16].\nThe utilization of the spin-flop effect in AF systems\nis overly complicated due to at least two reasons. The\nfirst thing is the existence of SFT in AF requires uniaxial\nanisotropy and an external field applied along the corre-\nsponding axis. Secondly, typical transition fields HSP\nin bulk antiferromagnets are tens of Tesla [17–20] thus\nthey are too high for real applications. The need to have\nanisotropy inside the system can be circumvented by re-\nplacing antiferromagnets with ferrimagnets (FEMs). In\nthe FEMs one does not require presence of anisotropy\nand the SFT takes place at HSP=λ|m1−m2|[21],\nwherem1,2are the magnetic moment of first and second\nsublattices and λis the exchange parameter. In bulk sys-\ntems the HSPare still too high for applications and can\nhardly be tuned.\nIn contrast, artificial ferrimagnets based on magneticheterostructures give a possibility to tune the SFT field\nby varying parameters of ferromagnetic layers and by in-\ntroducing non-magnetic spacers. Heterostructures based\non 3d transition metals (TM) and heavy 4f rare-earth\n(RE) metals, like Fe/Gd, are model ferrimagnetic sys-\ntems demonstrating a rich magnetic phase diagram with\ncomplex types of magnetic ordering [22–27]. Coupling\nbetween 4f electrons of Gd and 3d electronsof Fe leads to\nthe antiferromagneticalignment of TM and RE magnetic\nmoments which due to the difference in magnetic mo-\nments of Fe( ∼2µB) and Gd ( ∼7µB) leads to the emer-\ngence of a one-dimensional ferrimagnetic lattice. The\nspin-flop transition was found in Gd/Fe systems at typ-\nical value HSP∼3kOe [28], which is much smaller than\nthat for bulk FEMs but still quite high for applications.\nFurther tuning of HSPcan be gained by suppression of\ninterlayer exchange coupling which can be performed by\nspacing of Fe and Gd with a non-magnetic material like\nCr [29, 30], Pt [31] or Si [32].\nThe SFT can be detected by integral magnetic tech-\nniques as a kink on a magnetic hysteresis loop at HSP.\nIn case of artificial FEMs magnetic signal from thin films\nis heavilypolluted by dia-orparamagneticsignalofthick\nsubstrates.This makes it difficult, if not impossible at\nall, to use integral magnetometric methods to study the\nSFTs. Neutron scattering, being a depth-selective mag-\nnetometric method is a widely used method for studying\nAFs and FEMs [33–35]. Similar to X-ray and light, neu-\ntrons diffract at periodic lattice with period Daccording\nto the well-known Bragg law nλ= 2Dsinθ. Hereλand\nθare the neutron wavelength and incident angle, and n\nis integer number corresponding to order of Bragg peak.\nPresence of spin one-half makes neutron scattering sen-2\nsitive to the magnetic lattice. In case of antiferromag-\nnetic lattice magnetic peak is doubled comparing to the\nstructural one, so that the magnetic Bragg peak appears\non the positions of n/2 of the structural Bragg peaks.\nApplying spin analysis, that is detecting neutron spin-\nstates before and after scattering, allows one to get ad-\nditional information about magnetic configuration. The\nnon-spin-flip (NSF) channels (++) and (- -) are sensi-\ntive to the sum and difference of nuclear potential and\ncollinear to the neutron polarization part of magnetiza-\ntion. Here first and second sign codes neutron polariza-\ntion along the external magnetic field Hbefore and after\nthe scattering process. Presence of non-collinear magne-\ntization causesspin-flip (SF) scattering(+-)and (-+). In\nBorn approximation the amplitude of the SF scattering\nis proportional to the spatial profile of the noncollinear\nmagnetization in reciprocal space. Thus the SF scatter-\ning is very sensitive channel to detect the SFTs.\nIn our prior work [36] we studied superlattice\n[Fe(3.5nm)/Pd(1.2nm)/Gd(5nm)/Pd(1.2nm)] 12. In the\nneutron experiment we measured intensity of SF scatter-\ning at the position of the first Bragg peak RSF\n1as a func-\ntion of external magnetic field at a temperature of 10K.\nAbove magnetic field of HSP=1.5kOe we detected a 20-\nfold increase of SF scattering which is the direct evidence\nfor the presence of SFT in our system. We note that\ntheHSPfield is much smaller than in spacer free Fe/Gd\nsystems. Subsequent structural studies by transmission\nelectron microscopy and synchrotron radiation [37] indi-\ncated presence of mutual diffusion at Gd/Pd interface.\nFor thin ( ∼1nm) Pd spacers this interdiffusion leads to\nalmost complete dissolution of Pd in Gd. As a result\nthe Curie temperature (and hence exchange energy) of\nthe (nominal) Gd layer decreases from 294K for bulk Gd\nto/lessorsimilar100K. Thus ability of Pd and Gd to form an alloy\nwith controllable suppression of exchange energy paves\nthe way for tuning of SFT by varying thickness of Pd\nspacer. To do this we prepared series of samples of nom-\ninal composition [Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)] 12\nvaryingtfrom 1.0 to 1.6 nm (details can be found in our\nprior works [36, 37]). Further we will code samples as\nPdYY, where YY is thickness of Pd layer in Angstroms.\nFig. 1a shows the X-ray low-angle diffraction\npatterns (reflectivities) measured at a wavelength of\nλ=1.54˚A from the samples under study. More than 10\norders of Bragg reflection are seen on the reflectivities,\nwhich indicates goodrepeatability ofthe Fe/Gd unit cell.\nFig. 1b shows the energy dispersive X-ray (EDX) mi-\ncroanalysis of scanning transmission electron microscopy\n(STEM) of Pd12 sample. The EDX analysis shows well-\ndefined Fe layers depicted by blue color and yellow layers\nofGdPd alloyinsteadofseparateredGd layersandgreen\nPd spacers. For the sake of simplicity, we will keep nam-\ning Gd layer, remembering however that in reality the\nlayer is a Gd xPd1−xalloy.\nPolarized neutron reflectometry (PNR) experiment/s50 /s52/s49/s48/s48/s49/s48/s50/s49/s48/s52/s49/s48/s54/s49/s48/s56\n/s110/s61/s57/s110/s61/s56/s110/s61/s54/s110/s61/s52/s110/s61/s51/s88/s45/s114/s97/s121/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s40/s176/s41/s32/s80/s100/s49/s48\n/s32/s80/s100/s49/s50\n/s32/s80/s100/s49/s52/s40/s97/s41\n/s110/s61/s49\n/s40/s98/s41\nFIG. 1. (a) X-ray low-angle diffraction (reflectivity) of sam -\nples under study. Vertical arrows show the position of sever al\nBragg peaks for sample Pd10. (b) The energy dispersive X-\nray (EDX) microanalysis of Pd12 sample.\nwas conducted on the monochromatic ( λ=4.3˚A) reflec-\ntometer NREX of the research reactor FRM-2 (Garch-\ning, Germany). Fig.2 shows the PNR data measured\non sample Pd10 at T=10 K in magnetic field H=1kOe\nand additional SF curve at T=10 K in magnetic field\nH=3kOe (solid line). In the neutron experiment 4 Bragg\npeaks were confidently measured. A large splitting of\n(++) and (--) NSF Bragg peaks indicates the presence\nof a collinear magnetic moment in the system. At the\nsame time we observed a much weaker (1-2 orders below\nNSF signal) SF scattering at Bragg peaks. The origin\nof this small, though not negligible SF signal can be as-\nsociated with noncollinear inhomogeneities at the Fe/Gd\ninterfaces. The data at H=1kOe can be quantitatively\ndescribed by a predominantly collinear AF state with\nmagneticmomentsofGd MGd≈5µBandFeMFe≈2µB\naligned parallel and antiparallel to H. By increasing the\nmagnetic field above HSP=2.3kOe (inset in Fig.2) we\nobserved a 20-fold increase of SF scattering at the first\nBragg peak RSF\n1. This SFT is similar to observed pre-\nviously spin-flop in Pd12 sample though taking place at\n1kOe higher magnetic field.\nBy measuring family of RSF\n1(H) scans at different tem-\nperatureswewereabletoconstructthenoncollinearmag-\nnetic phase diagram for the sample Pd10 in H-Tcoordi-\nnates (Fig. 3a). For this sample we observe a collinear\nAFstateinthetemperaturerangeupto30Kinmagnetic\nfields not exceeding 2 kOe. Above this field, the collinear\nAF state is replaced by a NC spin-flop state. Increasing\nthe temperature to 60K leads to a gradual shift of the\nSFT field towards lower values. Finally, above 60K, the\nspin-flip signal disappearsdue to the absenceof magnetic\nordering in Gd layer. Fig.3b and Fig.3c shows similar\nphase diagrams for Pd12 and Pd14 samples. One can see\nthat the transition field HSPdecreases with increase of\nt. For the samples with t=1.6nm (not shown) we did not\nobserve any detectable SF signal evidencing absence of\ncoupling of Fe and Gd layers.\nTo describe magnetic state of our systems we applied\nextended Stoner-Wohlfarth model widely used for de-3\n/s48 /s50 /s52 /s54/s49/s69/s45/s53/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s110/s101/s117/s116/s114/s111/s110/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s32/s40/s176/s41/s32/s40/s45/s32/s45/s41\n/s32/s40/s43/s32/s43/s41\n/s32/s40/s43/s32/s45/s41/s48 /s50 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s82/s83/s70\n/s49\n/s72/s32/s40/s107/s79/s101/s41/s72\n/s83/s80\nFIG. 2. Polarized neutron reflectivities of sample Pd10 mea-\nsured at T= 10 K at magnetic field H= 1 kOe (symbols)\nand SF curve at T=10 K,H=3kOe (solid line) Inset shows\nthe field dependence of intensity of SF scattering at the first\nBragg peak RSF\n1(H). Vertical arrow denotes the magnetic\nfield at which spin-flop transition takes place.\nscription of magnetic multilayers [8, 38]. Density of mag-\nnetic energy of one Fe/Gd unit cell can be written as\nE(αGd,αFe) =−H[mGdcos(αGd)+mFecos(αFe)]+\nJ1cos(αGd−αFe)+J2cos2(αGd−αFe).\n(1)\nIn Eq.1mX=MXdXis a product of magnetization and\nthickness (magnetic moment), αXis the angle between\nmagnetization and Hof a layer X(X=Fe,Gd). The first\nterm in (1) is Zeeman coupling which tends to align mag-\nnetic moments of the layers along the external field. The\nsecond term is bilinear antiferromagnetic exchange cou-\npling of Fe and Gd layers with strength parameter J1.\nThe third term describes biquadratic coupling tending\nto align the magnetic moments non-collinearly. As seen\nfrom (1) in case J2=0 the transition field can be esti-\nmated as HSP≈J1|mGd−mFe|/mGd·mFe.\nFor every magnetic field Hthe magnetic configuration\nof the system as a function of J1,2can be obtained by\nminimizing energy (1) varying angles αGdandαFe. The\nmagnetizationamplitudes MGd,Feandthicknesses dGd,Fe\nwere taken from PNR and SQUID data and fixed during\ncalculations. The angles α′\nGdandα′\nFecorresponding to\nthe minimum of energy for a given set of HandJ1,2is\nused to construct a theoretical SF reflectivity at the first\nBragg peak in Born approximation:\nRSF\n1,th=c[m2\nGd,⊥+m2\nFe,⊥+\n2mGd,⊥mFe,⊥cosdFe\ndFe+dGd]+Rbg,(2)\nwheremGd(Fe),⊥=mGd(Fe)sinα′\nGd(Fe)is the non-/s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s102/s41\n/s74\n/s49/s44/s32/s74\n/s50/s32/s40/s101/s114/s103/s47/s99/s109/s50\n/s41\n/s116/s32/s40/s110/s109/s41/s32/s74\n/s49\n/s32/s74\n/s50/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s101/s41\n/s74\n/s49/s44/s32/s74\n/s50/s32/s40/s101/s114/s103/s47/s99/s109/s50\n/s41\n/s84/s32/s40/s75/s41/s32/s74\n/s49\n/s32/s74\n/s50/s49/s50/s51/s52\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s32/s72/s40/s107/s79/s101/s41/s40/s100/s41/s32/s80/s100/s49/s48/s32/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s84/s32/s40/s75/s41/s50/s48/s46/s48/s48\n/s55/s54/s46/s56/s48\n/s49/s51/s51/s46/s54\n/s49/s57/s48/s46/s52\n/s50/s52/s55/s46/s50\n/s51/s48/s52/s46/s48\n/s51/s54/s48/s46/s56\n/s52/s49/s55/s46/s54\n/s52/s55/s52/s46/s52\n/s53/s51/s49/s46/s50\n/s53/s56/s56/s46/s48\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s40/s99/s41/s32/s80/s100/s49/s52/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s52/s46/s48/s48/s48\n/s54/s51/s46/s56/s48\n/s49/s50/s51/s46/s54\n/s49/s56/s51/s46/s52\n/s50/s52/s51/s46/s50\n/s51/s48/s51/s46/s48\n/s51/s54/s50/s46/s56\n/s52/s50/s50/s46/s54\n/s52/s56/s50/s46/s52\n/s53/s52/s50/s46/s50\n/s54/s48/s50/s46/s48/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s40/s98/s41/s32/s80/s100/s49/s50/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s48/s46/s48/s48/s48\n/s52/s54/s46/s56/s48\n/s57/s51/s46/s54/s48\n/s49/s52/s48/s46/s52\n/s49/s56/s55/s46/s50\n/s50/s51/s52/s46/s48\n/s50/s56/s48/s46/s56\n/s51/s50/s55/s46/s54\n/s51/s55/s52/s46/s52\n/s52/s50/s49/s46/s50\n/s52/s54/s56/s46/s48/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s83/s70/s80/s40/s97/s41/s32/s80/s100/s49/s48/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s84/s32/s40/s75/s41/s50/s46/s48/s48/s48\n/s54/s49/s46/s54/s48\n/s49/s50/s49/s46/s50\n/s49/s56/s48/s46/s56\n/s50/s52/s48/s46/s52\n/s51/s48/s48/s46/s48\n/s51/s53/s57/s46/s54\n/s52/s49/s57/s46/s50\n/s52/s55/s56/s46/s56\n/s53/s51/s56/s46/s52\n/s53/s57/s56/s46/s48/s65/s70\n/s78/s111/s32/s99/s111/s117/s112/s108/s105/s110/s103\nFIG. 3. (a)-(c) Experimental ( H,T) maps of RSF\n1for samples\nwith different Pd spacer. (d) Simulated map for Pd10 sample\n(e) Fit-resulted J1andJ2terms vs temperature for Pd10\nsample. (f) Thickness dependence of bilinear and biquadrat ic\nenergies J1andJ2obtained for T=10K.\ncollinearcomponentofmagneticmomentofGd(Fe)layer,\ncis scaling constant and Rbgis background intensity.\nThe latter two values were adjusted manually before the\nfit. We fitted then theoretical RSF\n1,thto the experimental\nH-dependencies RSF\n1by varying J1andJ2. The proce-\ndure was repeated for every Tso that for every sample\nwe obtained temperature dependencies of J1,2. Fig.3d\nshows results of such a fit for sample Pd10. It is rather\nnoticeable that despite of the simplicity of the Stoner-\nWohlfarth approach it allows to reproduce experimen-\ntal features quite well. Fig.3e shows the fit-resulted T-\ndependence of the exchange energies J1andJ2for Pd10\nsample. It can be seen that the bilinear term has a pre-\ndominant contribution, which gradually decreases with\ndecreasing temperature. Thus our analysis showed that\nfor a qualitative description of the SFT, a bilinear term\nis sufficient, but quantitatively the data are described\nbetter by including an additional biquadratic term.\nThe data for the other samples were fitted in a similar\nway. Fig.3fshowsthe dependency ofcouplingenergieson\nthickness of Pd spacer. As followsfrom the figure, the bi-\nlinear energy decreases almost linearly from 1.5 erg/cm2\natt=1nm to 0 at t=1.6nm. Biquadratic energy in turn4\nincreases with t. The obtained values are of the same or-\nders asJ1∼0.8 erg/cm2andJ2∼0.2 erg/cm2obtained\nin Ref.[39] for Gd/Pt/Co multilayers at T=10K.\nThe decrease in the bilinear component with the in-\ncrease in tcan obviously be correlated with a decrease in\nthe effective concentration of Gd in the GdPd layer. At\nthe same time, structural studies carried out earlier [37]\nindicate an increase in structural inhomogeneities with\nincreasing of t. It seems prudent to correlatethis growth\nwith an increase in the biquadratic component.\nIn conclusion, using PNR we performed a\nsystematic study of magnetic configuration of\n[Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)] 12heterostruc-\ntures with t=1.0-1.6nm. By measuring neutron spin-flip\nscattering we have detected presence of magnetically\nnon-collinear state at temperatures T/lessorsimilar50 K in mag-\nnetic fields of above H >500 Oe for the samples with\n1nm< t <1.4nm. By using of an extended Stoner-\nWohlfarth model we were able to describe the observed\ntransition as a competition of Zeeman energy, bilinear\ninteraction of order of 1 erg/cm2and biquadratic\naddition of order of 0.5 erg/cm2. The coupling energies\ncan be tuned by varying thickness of spacer between\n1nm and 1.4nm leading to the shift of the transition field\nbelow kilo-Oersted range. Our study opens perspectives\nfor a purposeful design of artificial FEMs with adjustable\nfield of spin-flop transition. Thus, the FEMs systems\nwith low Curie temperature components studied in this\nwork can be used in superconducting spintronics for\ngeneration of triplet superconductivitiy. An additional\nadvantage here is the good compatibility of gadolinium\nwith superconducting niobium [40, 41]. For the room\ntemperature applications one can use well-studied\nsynthetic AFs such as Fe/Cr [33–35], Fe/V [42, 43] or\nCo/Cu [44, 45] where subsequent adjustment can be\ncarried out by tuning of the coupling energy and the\nimbalance of the magnetic moments of the sub-lattices.\nWe would like to thank M.A. Milyaev for assistance\nin preparation of the samples, A.B. Drovosekov and\nD.I. Kholin for fruitful discussion of the results. 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B 72, 054437 (2005)."    },    {        "title": "1107.5225v1.Magnetic_Behaviour_of_Disordered_Ising_Ferrimagnet_in_High_Magnetic_Field.pdf",        "content": "arXiv:1107.5225v1  [cond-mat.mtrl-sci]  26 Jul 2011Magnetic Behaviour of Disordered Ising Ferrimagnet in High Magnetic Field\nSobhendu K.Ghatak\nDepartment of Physics\nRKMVivekananda University,Belur,Howrah-711202,India\nAbstract\nThe magnetic behaviour of a disordered ferrimagnetic syste mApB1−pwhere both AandBrepresent\nthe magnetic atoms with respective spin SA= 1/2 andSB= 1 in presence of high magnetic field is treated\ntheoretically.Assumingthemagneticinteraction canbede scribedthroughIsingHamiltonian theapproximate\nfree energy is obtained using the cluster-variational meth od.The field dependence of the magnetization\nis then obtained for different concentration pand exchange parameters ( JAA,JBBandJAB).Forp=\n0.5,the magnetization Min ferrimagnetic state and in absence of compensation tempe ratureTcmvanishes\natTC.Field induced reversal of Mis found at switching temperature TS(< TC) which is decreasing function\nof fieldH.A maximum in Mis found above TSand the maximum value of Mincreases with field.In\nferrimagnetic state Mincreases almost linearly at high Hregion. For system with large ferromagnetic\nJAA,the compensation temperature Tcmis increasing function of JBBandJAB.The decrease in compensation\ntemperature is linear at small field and tends to saturate at h igher field.The sharpness of the magnetization\nreversal is increased with H.For fully compensated state of the system with p= 2/3,the magnetization in\npresence of Halso exhibits switching behaviour at TS.Forp= 0.2 the field induced reversal of magnetization\noccurs more sharply.The orientational switching of the sub lattice magnetization MAandMBwith field\nincreases the Zeeman energy and is the origin of magnetizati on reversal at TS.\nPACS numbers: 05.50, +9;75.10Hk;75.10.-b\nKeywords :MixedSpinIsingmodel,Disorderedferrimagneticalloy,Rare-earth- transitionmetalalloy,Cluster-\nvariational method\n∗Corresponding author.\nE-mail address: skghatak@phy.iitkgp.ernet.in\n11 INTRODUCTION\nFerrimagnetic state in its simplest form is characterized by an oppos ing and unequal magnetization of two\nsublattices below a critical temperature Tc. The finite magnetization below Tcresults from unequal magnetic\nmoment of constituents metal ions of the material. In addition, diffe rences in rate of thermal demagnetization\nof sublattice magnetization can lead to complete cancelation at lower temperature -referred as compensation\ntemperature Tcmthat exists in number of ferrimagnetic system [1].The studies of ferr imagnetic materials are\nnormally centered around their importance in technical applications [2-4].In crystalline lattice the constituents\nmetal ions normally occupy respective sites.On the other hand, the site occupancy tends to be random in\ndisordered lattice. The relative composition of constituent metal io ns can also be varied over a wide range in\ndisordered (amorphous) state produced through rapidly quench ed method[6,7].The rare-earth -transition metal\nferrimagnetic alloys in amorphous state had been investigated for t heir potential in magneto-optical recording\n[8].The real alloy contains, apart from magnetic atom,glass former th at stabilizes the disordered state.The\namorphous alloy with two kinds of magnetic atom can be, to a first app roximation, considered as binary spin\nsystemwith twosublatticenetworks.Thefielddependenceofmagne tizationindisorderedferrimagneticmaterials\nhasbeenofrecentinterest,andthemagnetizationisfoundtobene arlylinearinfieldathighfieldregion[4,5]. The\ncompensation temperature is expected to be field dependent. In this context it is appropriate to examine\nthe field behaviour of the magnetization and the compensatio n temperature in ferrimagnetic\nstate, and is attempted here based on simple theoretical mod el.\nTheoreticalmodelfrequentlyutilizedtodescribethephasediagra mandthemagneticbehaviourofdisordered\nmagnetic alloy is disordered Ising model [9-13].The mixed spin system wit h two different spins whose interaction\nis Ising-like is considered as simple model for ferrimagnetic system.Diff erent theoretical methods like the mean-\nfield approximation [14],the effective field theories [15,16], the renorma lization-group calculations [17,18] and\nthe Monte-Carlo simulations [19,20] are used to get the phase diagra m and critical behaviour of Ising model\nwith spin S= 1/2 andS= 1 for two sublattices in ordered lattice. The model has also been st udied in different\ndecorated lattices [21,22] and it is predicted that the compensation temperature exists within a specific region\nofJAA−JBBplane where JAA(JBB) is intra-sublattice exchange interaction in A-(B) sublattice [22].\nIn this article the results of the field dependence of magnetization in ferrimagnetic state of mixed Ising spin\nsystemApB1−pwithSA= 1/2 andSB= 1 are presented.The approximate procedure as suggested by Og uchi\n[23] for pure Ising system and extended by Ghatak [13] for disorde red Ising system is utilized for evaluation\nof the free energy.The magnetization,compensation temperature and their field dependence are then obtained\nfrom the configuration averaged free energy.In sec.2 the model a nd the method of calculation are outlined and\nresults are presented in Sec.3.\n2 MODEL AND METHOD OF CALCULATION\nWe take a binary alloy ApBqof two magnetic atoms A and B with respective concentration pandq= 1−p. It\nis assumed that all magnetic interactions are localized and can be des cribed by Ising Hamiltonian\nH=−/summationdisplay\nijJijSizSjz−/summationdisplay\niHiSiz (1)\nWhereSizis the Ising spin at i−thsite and takes the value SA= 1/2 orSB= 1 depending upon the\noccupation of the site by AorBatom. These values are chosen to reduce the algebraic complexity .T he second\nterm is the Zeeman energy where the magnetic field Hi(expressed in dimension of energy) at i−thsite is in\n2z-direction. The nearest neighbour exchange interaction Jijtakes value JAA,JBBandJABfor the magnetic\nbond A-A, B-B and A-B respectively. For quenched disordered alloy the free energy F, given by [14]\nF=−kT[lnTrexp(−βH)]av (2)\nWhere [.....]avrepresents the average over all possible alloy configurations and β= 1/kT. The free energy\ncan be expressed as\nF=F0−kT[ln<exp(−βV)>]av (3)\nThe quantity F0=−kT[lnTrexp(−βH0)]avrefers to the configuration averaged free energy for non-\ninteracting system described by the Hamiltonian H0.The symbol < ... >in second term of Frepresents the\nensemble average over the states of H0and the operator V=H−H0. The non-interacing Hamiltonian H0is\ntaken as\nH0=/summationdisplay\niIiSiz (4)\nHereIiis parametric field to be determined.\nForevaluationofthesecondtermofEq.(3)theapproximateproce dureusedearlierisadopted.Theapproximation\nisbasedonassumptionthatthesystemisbuiltoutofsmallindepende nt”buildingblock”.Theinteractionamong\nspins within the block are treated exactly and the rest of the intera ction is represented by field Ii.The field Ii\nis determined from minimization of the approximate free energy Fvthat can be written as [23,13]\nFv=F0−kTL/summationdisplay\nl=1[ln<exp(−βVl)>]av (5)\nWhereVlis thelthdivision of Vwhich is divided into Lnumber of blocks. With decrease of number\n(L) of division, Fvtends to exact value of free energy. With the increase in size of the block the algebraic\ncomplexity grows at faster rate compared to improvement of the r esult related to transition temperature of\nan Ising model.The building block consisting of 4-spins for Ising model le ads to the results equivalent to the\nBethe approximation. Here the same block is taken for the disorder ed mixed spin system. The possible atomic\nconfigurations for 4-spin block with A or B distributed at lattice point s are shown in Fig. 1. The respective\nprobability of occurrence of the configuration is noted below the re spective figure. The number of spin states\nin a given configuration depends on number of A and B atoms. The num ber varies from maximum 34for block\nwith allS= 1 atoms to minimum 24for all A-atom block. In this approximation the configuration - aver aged\ntrial free energy Fvcan be expressed as [24]\nFv=F0−(z/8β)[p4lnZ0+q4lnZ4+4p3qlnZ1+4pq3lnZ3+2p2q22(lnZ2+2lnZ22)] (6)\nwhere\nF0=−((1−z/2)/β)[pln(cosh(βIA/2))+qln(1+2cosh( βIB))] (7)\nand\nZ0= 2[X4\nAcosh2α1+4coshα1+2+X−4\nA] (8)\nZ4= 2X4\nBcosh4α2+8X2\nBcosh3α2+4(3+2 XB)cosh2α2\n+ 8(3+ X2\nB)coshα2+8X−1\nBB+9+2X−4\nB(9)\nZ1= 2X2\nAXCcosh(α2+3α1/2)+2X2\nAX−1\nCcosh(−α2+3α1/2)+2[2+ X−2\nAXC]cosh(α2+α1/2)\n+ 2[2+ X−2\nAX−1\nC]cosh(−α2+α1/2)+2[2+ X−2\nA]cosh(α1/2)+2X2\nAcosh(3α1/2) (10)\n3AA\nB\nAB B\nB BA BB BA\nB\nA\nB4p3q\n4pq3 q44p2q2\n2p2q2B BAA\nAAA\nAA\nJAA JBB JABp4\nFigure 1: Schematic representation of ’building block’ of four atoms . Solid, Dashed and Dotted lines represent\nrespectively JAA,JABandJBB. The probabilities of different configurations are given below the diag rams.\nZ3= 2X2\nBXCcosh(3α2+α1/2)+2X2\nBX−1\nCcosh(3α2−α1/2)\n+ 2[3+ X−2\nBXC+2X1/2\nC]cosh(α2+α1/2)+2[3+ X−2\nBX−1\nC+2X1/2\nC]cosh(α2−α1/2)\n+ 2[2XBX1/2\nC+XC]cosh(2α2+α1/2)+2[2XBX−1/2\nC+X−1\nC] cosh(2α2−α1/2)\n+ [6+4( XC+X−C)X−1\nB]cosh(2α1/2) (11)\nZ2= 2X2\nCcosh(2α2+α1)+2X−2\nCcosh(2α2−α1)+4cosh(2 α2)+6cosh( α1)\n+ 8cosh( α2)+6+4 XCcosh(α2+α1)+4X−1\nCcosh(α2−α1) (12)\nZ22= 2XAXBXCcosh(2α2+α1)+2XAXBX−1\nCcosh(2α2−α1)+4XBX−1\nAcosh(2α2)\n+ 2XA[1+2X−1\nB]cosh(α1)+4XAX1/2\nCcosh(α2+α1)+4XAX−1/2\nCcosh(α2−α1))\n+ 2[2X−1\nA(X−1/2\nC+X1/2\nC]cosh(α2)+2X−1\nA+2X−1\nAX−1\nB(X−1\nC+XC) (13)\nα1= [IA(1−2/z)+2HA/z]β,α2= [IB(1−2/z)+2HB/z]β (14)\nwhere z =number of nearest neighbours, XA= exp(βJAA/4) ,XB= exp(βJBB) andXC= exp(βJAB). In\nabove expressions of α’s the symbols H’s andI’s represent respectively the magnetic field and the variational\nparameters at respective site.\nThe equations for the variational parameters IAandIBare obtained from the minimization of the trial free\nenergyFv\ndFv/dIA=dFv/dIB= 0 (15)\nThis leads to the coupled equations for IAandIBas\n2tanh(βIA/2) =p3(A0/Z0)+4p2q(A1/Z1)+4q3(A3/Z3)+2pq2(A2/Z2)+2(A22/Z22) (16)\n8sinh(βIB)\n1+2cosh( βIB)=q3(B4/Z4)+4p3(B1/Z1)+4pq2(B3/Z3)+2p2q((B2/Z2)+2(B22/Z22)) (17)\n4whereAi= (2β/z)∂Zi\n∂HAandBi= (2β/z)∂Zi\n∂HB.The non-trivial self-consistent solutions of the equs. (16-17 )th en\nprovide the approximate free energy.The sub-network magnetiza tion per atom then becomes\nMA= 0.5tanh(βIA0/2) (18)\nMB= 2sinh( βIB0)/[1+2cosh( βIB0)] (19)\nand the total magnetization Mper atom\nM=pMA+qMB. (20)\nHereIA0andIB0are the self-consistent solution of the coupled equations (16) and (17). The finite values of\nIA0andIB0lead to spontaneous sub-network magnetization which appears be low the transition temperature\nTc[24].The numerical results for magnetization are presented for p= 0.5,2/3 and 0.2 by varying magnetic\nfield.The model parameters e.g kT, magnetic field Hand the exchange interactions are scaled in terms of\nstrongest exchange parameter and number of nearest neighbou r is taken as z= 8.The direction of the magnetic\nfield is assumed to be parallel to the magnetization of A-sub-lattice.\n3 RESULTS AND DISCUSSIONS\n3.1 Magnetization\n(i) Concentration p= 0.5\nWe first examine the ferrimagnetic behaviour of the system with equ al concentration of AandBand the\nexchange interaction between AandBis anti-ferromagnetic.It is also assumed that the exchange integra lJAB\nis stronger than the ferromagnetic exchange between A−AandB−B. To represent this case, a typical values\nJAB=−1.0,JAA= 0.2 andJBB= 0.1 for the exchange interactions are taken.The spontaneous magn etization\nMA,MBofA- andB- sublattice and the net magnetization Mdecrease smoothly with Tfrom their maximum\nvalue at T= 0 and vanish at critical temperature Tc(Fig.-2). At low T,Mvaries little from its maximum\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s98/s32\n/s32/s77\n/s66\n/s84/s47/s84\n/s67/s77\n/s65\n/s84/s47/s84\n/s67/s97\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48\n/s84/s47/s84\n/s67/s77/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s48/s46/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s51\n/s32/s48/s46/s53\n/s32/s48/s46/s56\n/s32/s49/s46/s48/s77\nFigure 2: a) Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field H.\nb) that of sublattice magnetization MAandMBforH= 0 (dotted line) and 0 .5 (solid line). Value of exchange\nparameters are JAB=−1.0,JAA= 0.2 andJBB= 0.1.Inset Fig.: M/M0vsT/TCfor ordered AB(dashed\ncurve) and disordered (solid curve) case with same exchange para meters\nvalueM0=pMA+(1−p)MB=−0.25.There is no compensation point below TCfor this situation.In\npresence of a magnetic field along z-direction, the thermal d emagnetization becomes faster for\nT≤Tcand changes its sign at a particular temperature T=TS(< Tc)(Fig.2a) which depends\n5on the magnetic field.Above T > T S,the magnetization continues to increase until it reaches a\nmaximum at a temperature slightly higher than TS.With further increase in Ta slower decrease\nin the magnetization is found.As the field increases the reve rsal occurs at lower temperature\n,and so TS, termed as the switching temperature, decreases with the fie ld.The maximum value of\nthe magnetization ( ∆M) also becomes higher.This large change in behaviour of MnearTSis the\nresult of a different thermal behaviour of the sublattice mag netization in presence of a moderate\nfield.The results of MAandMBare presented in Fig. 2bforH= 0(dotted) and 0.5(solid) and\nshow that a simultaneous alteration of the orientation of su blattice magnetization with respect\nto the field direction occurs at TS.The switching of the orientation of the magnetization is th e\nresult of combined effects of the Zeeman and the exchange ener gies.As the magnetic moment of\nBis higher than that of A, the free energy in the switched state is lowered by increasing the m agnitude of\nthe Zeeman energy without any loss in the exchange energy.We also n ote that apart from the field,the inter-\nsublattice exchange parameters alter the switching temperature ,the sharpness of switching and the maximum\nvalue ofM.Inset Fig.2a shows thermal variation of the reduced magneti zation (dashed curve)of an\nordered ferrimagnet where two interpenetrating sublattic esAandBare antiferromagnetically\naligned.The corresponding variation of M/M0for disordered case is given by solid curve.The\nmagnetization at intermediate temperature interval in dis ordered system is reduced compared to\nthat of ordered state.Close to transition temperature the m agnetization are nearly equal in both\ncases.Thisis expected when the correlation length around TCis undisturbedby disorder.Similarity\nof the magnetization at low temperature is related to the abs ence of the spin wave excitation.The\nreversal of the magnetization in ordered system is found to o ccur at higher field.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s32\n/s72/s84\n/s83/s47/s32/s84\n/s67\n/s32/s77/s32/s47/s32/s77\n/s48\n/s97\n/s98/s77/s32/s47/s32/s73/s77\n/s48/s73\n/s72/s32/s32/s32/s32/s32/s32/s32/s84/s32/s47/s32/s84\n/s67\n/s32/s49/s46/s48\n/s32/s48/s46/s57\n/s32/s49/s46/s50\nFigure 3: a) Variation of ∆ M/|M0|andTS/TCwithHand b)Hdependence of magnetization M/M0at\nT/TC= 1,0.9 and 1.2 for same set of J’s as in Fig.2 and p= 0.5.\nIn Fig.3athe field variation of the reduced switching temperature TS/Tcand normalized ∆ M/|M0|are\ndepicted for range H= 0 to 1. In this range ∆ M/|M0|sharply increases at lower range of Hand tends to\nsaturate at higher region.On the other hand, the reduced switchin g temperature TS/Tcexhibits nearly linear\ndecrease at high Hregion.The field dependence of reduced M/|M0|is also shown in Fig.3 bforT/Tc= 1.2,1.0\nand 0.9.AtTc,Mgrows faster at low field ,and the growth rate slows down at higher fie ld. ForT < T c,M\nchanges sharply at certain field that depends on T. In high field region, a nearly linear dependence of MonH\nis found,and the slope becomes smaller at higher Tdue to more thermal fluctuation.\nNext we consider a system where the exchange parameters are su ch that the JAAis largest compared to\nothers. This is commonly the situation for the rare-earth transitio n metal alloy where the exchange integral\n6between the transition metal ions is much stronger than that betw een rare-earth and transition metal ions or\nbetween rare-earth ions.The moment of the rare-earth ( B) is larger than that of transition metal ion ( A).Again,\nthere are situations where JABis not very weak compared to JAA.In Fig.4 the magnetization M(4a) and the\nsublattice magnetization MAandMB(4b) are displayed for the exchange parameters JAA= 1.0,JAB=−0.5,\nandJBB= 0.05. The net magnetization Min the ferrimagnetic phase at H= 0 varies little for small\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s98\n/s84/s47/s84\n/s67/s77\n/s65\n/s77\n/s66\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s48/s46\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s52\n/s32/s48/s46/s53\n/s77\n/s32/s32\n/s97\nFigure 4: a) Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field\nH.b) that of sublattice magnetization MAandMBH= 0 (dotted line) and 0 .4 (solid line). Value of exchange\nparameters are JAA= 1.0,JAB=−0.5,andJBB= 0.05\nT/TC.However,a larger variation is found at higher temperature due to fa ster demagnetization of MB.This is\nassociated with smaller inter- and intra-exchange JBBinteractions. For this set of parameters there is no com-\npensation temperature and Mvanishes at TC.In presence of a magnetic field Mchanges sign at a temperature\nT=TS(field induced switching temperature) and attains a maximum at a tem perature TMless than TC. The\nreversal of MatTSis the effect of higher Zeeman energy gain when MAandMBswitches their orientation\nwith respect to the magnetic field (Fig.4b). The field variation of the m aximum value of the magnetization\nwith respect to its maximum magnitude at zero field MM/|M0|,, is given in Fig.5.A sharp increase is found at\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s77\n/s77/s47/s77\n/s48/s84\n/s77/s47/s84\n/s67\n/s72/s84\n/s83/s47/s84\n/s67\nFigure 5: Field variation of the maximum value of MM/|M0|,the normalized switching temperature TS/TCand\nthe temperature TM/TCwhereMis maximum.Other parameters are same as in Fig.4.\nlowHand tends to saturate at higher field region.With increase in HbothTMandTS(normalized by TC) are\nreduced.The variation is almost linear for TMwhereas nonlinearity is evident for TS.When the inter-sublattice\ninteraction is further reduced then the compensation temperatu re appears before the transition temperature\nTC.One such situation is described by the Fig.6 for a set of exchange JAA= 1.0,JAB=−0.1,andJBB= 0.05\nfordifferent field rangingfrom0 to0 .5.At lowtemperature Mis againdominated by Bsublattice magnetization,\n7/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s98/s32/s32/s77/s47/s73/s77\n/s48/s73\n/s72/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s47/s84\n/s67\n/s32/s32/s49/s46/s48\n/s32/s32/s49/s46/s50\n/s32/s32/s48/s46/s53/s49\n/s32/s32/s48/s46/s51/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s72\n/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s53\n/s97\nFigure 6: Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field H\n(a),and the reduced magnetization M/|M0|with field H(b) for different values of T/TC.Values of exchange\nparameters are JAA= 1,0,JAB=−0.1,andJBB= 0.05\nchanges its sign at the compensation temperature TCMand finally vanishes at TC. At compensation point the\nmagnetization of whole system becomes zero due to equal and oppo site magnetization of two sub-lattices. A\nfaster demagnetization of sublattice Bcompared to Asublattice occur as both JAB=−0.1,andJBB= 0.05\nare much weaker compared to JAA.In presence of Hthe compensation occurs at lower temperature and a well\ndefined maximum appears (Fig.6 a).The temperature dependence is also highly asymmetric about temp erature\nwhere the maximum appears. For high field the reversal of Mis sharp around TCM.In Fig.6bthe field\ndependence of Mis shown for temperature T/TC= 1.2,1,0.51and0.3.The curve labeled 0.51corre-\nsponds to the result at compensation point TCM= 0.51TCwhenH= 0.The curve labeled 0.3is the\nresult when temperature is selected as T < T CM.The magnetization at low temperature T < T CM\nchanges its direction at faster rate.At TCM,Mgrows with high slope for small field,however the\nvariation of Mbecomes flattened at a higher field. Nearly linear variation i s observed at high\nfield although the rate of variation depends on temperature. The compensation temperature\nTCMatH= 0depends on exchange parameters JBBandJAB[24]. The compensation point in\nferrimagnetic state appears when the sublattice Bwith higher moment is thermally demagne-\ntized at a faster rate compared to that of other sublattice A.This happens when intra-sublattice\ninteraction JBBis weak.For given JAB,it is expected that higher value of ferromagnetic JBBresults\nhigherTCM.The dependence of TCMonJBBfor two values of JAB=−0.1 and−0.2 is displayed in\nFig.7a,andTCMis linearly dependent on JBB.ForJAB=−0.2 the compensation point exists when JBBis less\nthan 0.1.The dependence of TCMonHis given in Fig.7 bforJAA= 1.0 and a) JAB=−0.2 ,JBB= 0.02, b)\nJAB=−0.1,JBB= 0.02,and c) JAB=−0.1,JBB= 0.05,JAB= 0.1. For low H, a linear decrease in TCMis\nfound for all cases,and at higher field TCMtends to saturate.\n(i) Concentration p= 2/3\nThe concentration p= 2/3 corresponds to the situation where total magnetization is comple tely compensated\natT= 0 due to choice of spin values. When the inter-sublattice exchange JAB=−1.0 is larger than the\nintra-sublattice exchange ( JAA=JBB= 0.4)interactions the state with M= 0 persists up to T/TC= 0.2\n(Fig.8a).Within intermediate temperature interval ( 0 .2≤T < T C) andH= 0,Mbecomes finite but\nsmall and dominated by B-sublattice magnetization as thermal demagnetization effect is sam e for both sublat-\ntices.However,the presence of field changes the behavior of M. There is a sharp variation of Mfrom negative to\n8/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s98/s99/s32/s84\n/s99/s109\n/s72/s97/s98\n/s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51 /s48/s46/s48/s52 /s48/s46/s48/s53/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s84\n/s99/s109\n/s74\n/s66/s66/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s65/s66\n/s32/s32/s32/s45/s48/s46/s50\n/s32/s32/s32/s45/s48/s46/s49/s97\nFigure 7: a)Dependence of the compensation temperature TCMwith exchange JBBfor two values of JAB\nand = 0.b) Field dependence of TCMfor three set of exchange parameters a) JAB=−0.2,JBB= 0.02,\nb)JAB=−0.1,JBB= 0.02,c)JAB=−0.1,JBB= 0.05 andp= 0.5\npositive value at switching temperature as Hincreases. With increases in field the switching temperature TSis\nlowered.The switching like behaviour of Mis due to simultaneous flipping of sublattice magnetization in order\nto maximize Zeeman energy and keeping antiferromagnetic alignment of the sublattice magnetizations.The field\ndependence at high field is also nearly linear as in earlier case of p= 0.5. We note that the magnitude of\nchange of the magnetization in this case much smaller compared to th e system p= 0.5. On the other hand\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s48/s46/s49/s50/s32\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s77/s97\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51/s98\nFigure 8: Variation of net magnetization Mforp= 2/3 with reduced temperature T/TCfor different field H.\nValue of exchange parameters are for a) JAA= 0.4,JAB=−1.0,andJBB= 0.4 and for b) JAA= 1.0,JAB=\n−0.5,andJBB= 0.2\nwhenJAA= 1.0 dominates overother exchangeinteractions ( JAB=−0.5,JBB−0.2) (Fig.8b) the magnetization\nalwaysremain parallelto H.This is due larger exchangeenergy in sublattice Acompared to that in Bsublattice,\nand the induce magnetization is dominated by change in MAfor all field.\n(i) Concentration p= 0.2\nThis refers to the situation where the concentration of ions with hig her values of magnetic moment is smaller\nand can simulate amorphous transition metal -rare-earth alloy with smaller concentration of rare-earth.Fig.9 a\ndisplays the net magnetization Mfor different field H= 0 to 0 .5 for exchange JAA= 1.0,JAB=−0.5,and\nJBB= 0.2.There is no compensation temperature for this set of parameter s.But the reversal occurs in presence\nof field.It is found that Mswitched very sharply from negative to positive value.Again the switc hing behaviour\nis associated with reversal of direction of MAandMBwith respect to the field direction at TS.\n9/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32\n/s72/s84\n/s83/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s65/s66\n/s32/s48/s46/s53\n/s32/s48/s46/s48/s53\n/s97/s98/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32\n/s32/s48/s46/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s53\nFigure 9: Variation of net magnetization Mforp= 0.2 with reduced temperature T/TCfor different field\nH(a).Value of the exchange parameters are JAA= 1.0,JAB=−0.5,andJBB= 0.2.b)Field dependence of\nTS/TCfor two values of JAB.\nIn this work it is assumed that the magnetization of the sub-l atticeAcarrying smaller moment\nper site is aligned parallel to +z-axis and antiferromagnet ically to the magnetization of sub-lattice\nBwith higher moment.This leads to net magnetization is in neg ative z-direction.In real system\nthe direction of the magnetization is determined by an aniso tropy energy which often is repre-\nsented by anisotropic field acting on sub-lattice. So by addi ng small anisotropic field along +z\ndirection in sub-lattice Athe presented ferrimagnetic state is realized.The switchi ng behaviour\nof the magnetization in the ferrimagnetic state with magnet ic field is expected to follow when\nthe field is in opposite direction of the magnetization.The s witching of magnetization had been\nobserved in multilayer of rare-earth and transition metal[ 25,26] or alloy[26].The magnetization of\nrare-earth layers is aligned opposite to that of transition metal layers due to antiferromagnetic in-\nteraction at the interface of two layers.The magnetization reverses its direction in presence of the\nmagnetic field and exhibits complex hystersis.Although the se systems are different compared to\nsystem considered here however, it is worth noting the simil arity of global field behaviour of the\nmagnetization.We envisage possible application of the fiel d induced magnetization reversal.The\nswitching device that relied on the sign of the magnetizatio n is a possible area of application.At\na fixed field the switching will be induced by changing tempera ture through TS.The ferrimag-\nnetic system with higher concentration of ion carrying high magnetic moment would be more\nappropriate due to sharp nature of reversal (Fig.9).\n4 CONCLUSIONS\nA disordered ferrimagnetic alloy ( ApB1−p) with Ising like interaction between the spins ( SA= 1/2andSB= 1)\nof sub-lattices is treated using a cluster-variational method in pre sence of magnetic field.In this method the\ninteractions within the cluster of different configurations are trea ted exactly and the rest of the interaction\nis described by variational field which is obtained from minimization of fr ee energy. The results on the mag-\nnetization and compensation temperature are presented for p= 0.5,2/3 and 0.2 for different values of field\nand exchange parameters.In absence of a compensation tempera ture at zero field,a field induced magnetization\nreversal in ferrimagnetic state is found at a temperature -terme d as switching temperature.With increase in the\nmagnetic field the switching occurs at lower temperature and the ma gnetization reversal becomes sharper.At\n10switching temperature the free energy is gained by interchanging o rientation of sublattice magnetization with\nrespect to applied field.The compensation temperature at zero field increases with increase in intra-sublattice\nexchange interaction of Bsublattice.In presence of magnetic field the the compensation point appears at smaller\ntemperature.The magnetization in paramagnetic state varies almos t in linear fashion with Hin high field re-\ngion.The nearlylinear increasein magnetizationhas been observedin r are-earth- transition metal alloy[4,5].For\nfully compensated composition ( p= 2/3) the magnetization reverses at a switching temperature when int er-\nsublattice exchangedominatesoverothers.Oncontrarythe magn etizationpassesthroughamaximum at T < T C\nwhen the intra-sublattice exchange is largest.The magnetization re versal is found to be much sharper for system\nwith higher concentration of Bwith weak JAB.The effect of field induced magnetization reversal can be utilized\nfor switching device.\n5 Acknowledgement\nThe author gratefully acknowledges assistance from authority of RKMVivekananda University.It is also my\ngreat pleasure to acknowledge Prof.D.Sherrington for his help and e ncouragement during work on disordered\nspin system.\nReferences\n[1] L.Neel Ann.Phys,(Paris) 3137 (1948)\n[2] T. Mallah, S. Tiebaut, M. Verdager and P. Veillet, Science 262, 1554 (1993); H. Okawa,N. Matsumoto,\nH. Tamaki and M. Obba, Mol. Cryst. Liq. Cryst. 233, 257 (1993); M. Turnbull, C. P. Landee, T. C.\nSoesbe and R. D. Willet,Mol. Cryst. Liq. Crys. 233, 269 (1993).\n[3] F. Tanaka, S. Tanaka, and N. Imanura, Jpn. J. Appl. Phys., Par t 126, 231 (1987); M. Alex, K. Shono,\nS. Kuroda, N. Koshino, and S. Ogawa, J. Appl. Phys. 67, 4432 (1990).\n[4] El-HagariM,Michor H,OzcanS,Giovamoni,MalarA,Heiba Z,KerschlP,Sch onhartM,BauerF,Grossinger\nR,Hilscher G,Freudenberg J,Rosner H,J.Phys.:Condens.Matter 184567 (2006)\n[5] R.Grossinger,M.Schonhart,P.Kerschl,S.OZcan,M.El-Hagary,J.Freude nberger and H.Michor\nJ.Phy.:Conference series 51139-142 (2006)\n[6] K.Moorjani and J.M.D.Coey Magnetic Glasses, Elsevier Publication Am sterdam (1984)\n[7] C.W.Chen Magnetism and Metallurgy of Soft Magnetic Materials,Dove r Publication,NewYork,(1986)\n[8] P.Choudhuri,J.J.Cuomo and R.J.Gambins Appl.Phys.Lett. 22,337 (1973)\n[9] M.Sajieddine,Oh.Baur,K.Cheriff,C.Dufour,G.Marchal and R.E.Camley Phy s.RevB49(1994) 8815,\nS.Honda and M.Nawate J.Magn.Magn.Mater. 136163 (1994)\n[10] D.Sherrington and S. Kirkpatrick Phys.Rev.Lett 341348 (1975)\n[11] M.F.Thorpe and A.R.McGurn Phys.Rev. B 202142 (1979)\n[12] T.Keneyoshi Phys.Rev. B 387688 (1986) and Solid State Comm. 93(1995) 691\n[13] S. K. Ghatak J.Phys C111401 (1978) and Pramana 22421 (1984)\n11[14] T.Kaneyoshi and J.C. Chen J.Magn.Magn.Mater. 98(1991) 210\n[15] A.F.Siqueira and ,I.P.Fittipaldi J.Magn.Magn.Mater. 54-57(1986) 678\n[16] G.J..A.Hunter,R.C.L.Jenkins and C.J.Tinley J.Phys. A23(1990) 4547\n[17] N.Benayad, Z.Phys.B: Condens.Matter 81(1990) 99.\n[18] J.Plascak, W. Figueiredo and B. C. S. Grandi, Braz. J.Phys. 29, (1999). 579\n[19] M.Zhang and C.M.Yang, Phys.Rev. B48(1993) 9452\n[20] G.M.Bunendia and M.A.Novotny ,J.Phys.: Condens.Matter 9(1997) 5951\n[21] M.Jascur and J. Strecka cond-mat/0507367 v2 31 Mar (2006)\n[22] J.Oitmaa, Phys.Rev. B72(2005) 224404\n[23] A.Oguchi, Prog.Theor.Phys. 561442 (1976)\n[24] S.K.Ghatak,Int.J.Mod.Phy.B 222421 (2008)\n[25] S.Demirtas,R.E.CamleyandA.R.Koymen,Appl.Phys.Lett. 8720211(2005),R.E.Camley,W.Lohstroh,G.P.Felcher,\nN.Hosoito and H.Hashizume,J.Magn.Magn.Mater 28665 (2005)\n[26] S.Demirtas,M.R.Hosu,R.E.Camley,H.C.Mireles and A.R.Koymen,Phys.Rev.B 87184433 (2005)\n12"    },    {        "title": "1203.6220v1.Geometric_and_disorder____type_magnetic_frustration_in_ferrimagnetic__114__Ferrites__Role_of_diamagnetic_Li__and_Zn2__cation_substitution.pdf",        "content": " 1 Geometric and disorder – type magnetic frustration in ferrimagnetic “114” \nFerrites: Role of diamagnetic Li+ and Zn2+ cation substitution.  \n \nTapati  Sarkar, V. Caignaert, V. Pralong and B. Raveau * \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal J uin, 14050 CAEN, France  \n \nDedicated to Professor Jacques Friedel on the Occasion of His 90th Birthday.  \n \nAbstract  \n \nThe comparative study of the substitution of zinc and lithium for iron in the “114” ferrites, \nYBaFe 4O7 and CaBaFe 4O7, shows that these diamagne tic cations play a major role in tuning \nthe competition between ferrimagnetism and magnetic frustration in these oxides. The \nsubstitution of Li or Zn for Fe in the cubic phase YBaFe 4O7 leads to a structural transition to a \nhexagonal phase YBaFe 4-xMxO7, for M = Li (0.30 \n  x \n 0.75) and for M = Zn (0.40 \n  x \n \n1.50). It is seen that for low doping values i.e. x = 0.30 (for Li) and x = 0.40 (for Zn), these \ndiamagnetic cations induce a strong ferrimagnetic component in the samples, in contrast to \nthe spin glass behaviour of the cubic phase. In all the hexagonal phases, YBaFe 4-xMxO7 and \nCaBaFe 4-xMxO7 with M = Li and Zn, it is seen  that in the low doping regime (x ~ 0.3 to 0.5), \nthe competition between ferrimagnetism and 2 D magnetic frustration is dominated by the  \naverage valency of iron. In contrast, in the high doping regime (x ~ 1.5), the emergence of a \nspin glass is controlled  by the high degree of cationic disorder, irrespective of the iron \nvalency.  \n \n \n \nKeywords : “114” Ferrites, ferrimagnetism and magnetic frus tration.    \n \n \n \n \n* Corresponding author: Prof. B. Raveau  \ne-mail:  bernard.raveau @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 1. Introduction   \n \n Strongly  correlated electron systems, involving transition metal oxides with a “square” \ncrystal lattice, namely perovskites, have been the subject of numerous investigations during the \nlast thirty years, for their superconducting properties in cuprates as well as their magnetic \nproperties in CMR mangani tes. Oxides with a triangular lattice have also been studied , as \nshown for the spinel family , [1 – 3] that exhibits strong ferrimagnetism and unique magnetic \ntransitions, as for example in Fe 3O4, and for the pyrochlore family and for spinels with a \npyrochl ore sublattice , [4 – 6] which have been investigated for magnetic frustration. \nNevertheless, the number of oxides with a triangular lattice is much more limited, and the \nrecent synthesis of the “114” cobaltites and ferrites [7 – 11] with original structure s closely \nrelated to the spinel offers a new playground for the investigation of the competition between \nmagnetic ordering and geometric frustration in this class of materials [12 – 14]. The \nconsideration of the structure of the “114” ferrites, cubic LnBaF e4O7 (Fig. 1 a), and hexagonal \nCaBaFe 4O7 (Fig. 1 b), shows that these oxides, both consist of similar layers of FeO 4 \ntetrahedra, called triangular (T) and kagom é  (K), and that their “Fe 4O7” frameworks can be \ndeduced from each other by a translation of one  triangular layer out of two. Therefore, the \ncorresponding iron sublattice is very different in the two structural families: it consists of a \npure tetrahedral framework of corner sharing “Fe 4” tetrahedra  in the cubic LnBaFe 4O7 (Fig. 2 \na), whereas it is bui lt up of rows of corner – sharing “Fe 5” bipyramids running along \n , \ninterconnected through “Fe3” triangles in the (001) plane for hexagonal CaBaFe 4O7 (Fig. 2 b ). \nIt is this geometry of the iron framework which is at the origin of two different kinds of \ngeometric frustration. The pure tetrahedral iron sublattice of LnBaFe 4O7 oxides is similar to \nthat of pyrochlore [4] and consequently generates a 3 D magnetic frustration. In contrast, the \nmixed “ bipyramidal – triangular” sublattice of CaBaFe 4O7, allows a com petition between a 1 D \nmagnetic ordering along \n  and a 2 D magnetic frustration in the (001) plane.  \n Recently, we have shown the possibility of stabilizing the iron “bipyramidal – \ntriangular” lattice at the cost of the tetrahedral lattice by substitution of zinc for iron in the \ncubic LnBaFe 4O7 oxides [15]. Surprisingly, it was observed that this doping with a \ndiamagnetic cation destroys the spin glass behaviour of LnBaFe 4O7 and induces \nferrimagnetism. However, with progressive increase of the Zn concentrat ion, the ferrimagnetic \ninteraction starts to weaken, and we get a spin glass for very high doping concentration. In \norder to understand the role of the different factors which govern the magnetic properties of  3 these ferrites, we have studied the substituti on of lithium and zinc , two diamagnetic cations \nwith different valencies,  for iron, in the ferrites YBaFe 4O7 and CaBaFe 4O7. We discuss, herein, \nthe relative influence of valence effects and cationic disordering upon the competition between \nmagnetic orderin g and frustration in these systems.  We will specifically attempt to decouple \nthe role of the Fe valency, and that of the disorder on the Fe sites in order to determine how the \ntwo separately affect the magnetic ground state. This will allow us the possibil ity to tune and \ncustomize the magnetic properties of these oxides by understanding the role of the two \ngoverning factors – the average Fe valency, and the degree of disorder on the Fe sites.  \n \n2. Experimental  \n \n All the samples used in this study were prepa red by standard solid state reaction \ntechnique. The details of the synthesis procedure can be found in our earlier publications [ 15, \n16]. The samples were chemically monophasic, and the phase purity was checked from X -ray \ndiffraction patterns registered wi th a Panalytical X’Pert Pro diffractometer. The d. c. \nmagnetization measurements were performed using a superconducting quantum interference \ndevice (SQUID) magnetometer with variable temperature cryostat (Quantum Design, San \nDiego, USA). The a.c. susceptib ility, \n ac(T) was measured with a PPMS from Quantum Design \nwith the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe and H ac = 10 Oe). All the \nmagnetic properties were registered on dense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \n3. Results and discus sion \n \n3.1. Zn substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In our previous work [15], we had synthesized the oxide series YBaFe 4-xZnxO7 with \nthe hexagonal symmetry , for x ranging between 0.4 – 1.5. For the sake of relevant \ncomparison, we have prepared CaBaFe 4-xZnxO7 samples with x = 0.5 and 1.5. We have \nspecifically chosen these two values of x so that we can investigate how the two factors, i.e. \nthe average Fe valency and the cationic disorder , affect the magnetic properties in two \nseparate regimes: the low do ping regime and the high doping regime.  The XRPD patterns of \nthese two samples clearly show that they are monophasic, keeping the hexagonal symmetry  of \nCaBaFe 4O7, and with cell parameters close to those of the virgin oxide i.e. a = 6.3527 (1)  Å \nand c = 10.3274 (2)  Å for x = 0.5, and a = 6.3668 (1)  Å and c = 10.29 75 (1)  Å for x = 1.5.   4  \n3.1.1.  Low doping regime: YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 \n \n YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 fall in the low doping regime. The degree of \ndisorder (measured in terms of the % of substituent cation) is the same and relatively small in \nthese two samples. We show the d. c. M vs T for the two samples in Fig. 3 . Both samples \nshow a ferrimagnetic transition (seen as a sharp rise in the M vs T curves) , similar to that \nobserve d for CaBaFe 4O7, [9] in accordance to the fact that the samples have been stabilized in \nthe hexagonal symmetry . However, their ordering temperatures , and importantly, their \nmagnetic moments are much smaller than those observed for CaBaFe 4O7 (TC = 270 K and  \nMFC(5K)  = 2.6 µ B/f.u.). Moreover,  the magnetic moments and the ordering temperatures  for the \ntwo samples  are very different , though they exhibit the same degree of disorder.  The \nCaBaFe 3.5Zn0.5O7 sample exhibits a much higher magnetic moment ( MFC(5K)  = 1.7 µB/f.u.) and \ntransition temperature (T C ~ 203.0 K) compared to the YBaFe 3.5Zn0.5O7 sample for which \nMFC(5K)  = 0.54 µ B/f.u. and T C ~ 119.5 K . Bearing in mind that both these oxides exhibit the \nsame “bipyramidal – triangular” iron sublattice ( Fig. 2 b ), the se results show that the \ncompetition between the 1 D ferrimagnetism that appears along \n  and the 2 D frustration in \nthe (001) plane of the hexagonal structure is strongly affected by the average valence of iron. \nIndeed, for the same degree of disorder (12. 5 % Zn), T C decreases and the magnetic \nfrustration increases significantly as the average value of Fe decreases from 2.57 for \nCaBaFe 3.5Zn0.5O7 to 2.29 for YBaFe 3.5Zn0.5O7. \nThe effect of the higher value of the average Fe valency of CaBaFe 3.5Zn0.5O7 is also  \nseen in  the M vs H loops of the two samples  (shown in the inset of Fig. 3 ). Not only is the \ncoercivity and remanence magnetization higher for CaBaFe 3.5Zn0.5O7, the shape of the M-H \nloops are also very different. CaBaFe 3.5Zn0.5O7 has a square loop, reminis cent of hard \nferrimagnets, while YBaFe 3.5Zn0.5O7 has a much softer M -H loop  signifying a weakening of \nthe ferrimagnetic interaction in YBaFe 3.5Zn0.5O7 compared to that in CaBaFe 3.5Zn0.5O7. Thus, \nin the low doping regime, the average Fe valency  of the ferri te is clearly the governing factor \nthat controls  the magnetic state and the strength of the magnetic interaction.  The degree of \ndisorder here plays a relatively minor role.  \n \n \n \n  5 3.1.2.  High  doping regime: YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 \n \nAn increase of  the doping concentration of the diamagnetic substituent subsequently \nleads to the appearance of spin glass behaviour in both YBaFe 4-xZnxO7 as well as CaBaFe 4-\nxZnxO7. This can be seen in Fig. 4 , where we have shown the d. c. M vs T  curves  for \nYBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7. In contrast to the sharp rise in the M(T) curves below \nthe ordering temperatures seen in the low doped samples (x = 0.5), the M(T) curves of the \nhigher doped samples (x = 1.5) show a more gradual rise in the magnetization with the \ndecrease in temperature terminating in a cusp -like behaviour at low temperature. The \ntemperature at which the ZFC M(T) curves of the two samples show cusps are T cusp = 35.5 K \nand 40.5 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively.  A. C. susceptib ility \nmeasurements \n '(T) of the two samples measured using different frequencies in the range 10 \nHz – 10 kHz  (Fig. 5 ) show that both samples show similar frequency dependent peaks with  Tg \n= 45 K and 50 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectivel y. The two samples \nalso have very similar narrow S – shaped loops with almost the same values of the coercivity \nand remanence magnetization  (see inset of Fig. 4). Moreover , the important point to note here \nis that the average Fe valency of these two compou nds is very different (Fe val = 2.40 and 2.80 \nfor YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively).  In spite of such a large difference in \nthe Fe valency, the two compounds behave strikingly similar to each other. We explain this \nresult as the dominating role of the cationic disorder in samples where the degree of disorder \nis large. Thus, for higher doped samples, the degree of disorder plays the deciding factor in \nstabilizing the magnetic ground state, and the Fe valency plays only a minor role.  \n Remarkab ly, the magnetic behaviour of these highly doped hexagonal phases is very \nsimilar to that of the virgin cubic sample YBaFe 4O7, which was shown to be a spin glass, with \na rather similar T g ≈ 50 K [10]. Thus, a high degree of cation disordering in the hexago nal \nphase has an effect similar to the pure geometric frustration of the tetrahedral iron sublattice \n(Fig. 2 a ) of the cubic phase i.e. it allows a complete magnetic frustration to be reached.  \n \n3.2. Li substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In the previo us sections, we have shown that the magnetic properties of hexagonal Zn \nsubstituted “114” ferrites  YBaFe 4O7 and CaBaFe 4O7 can be tuned on the basis of two doping \nregions – the low doping regime and the high doping regime. While the average Fe valency \nplays  a crucial factor in determining the magnetic state of the oxide in the low doping regime,  6 the degree of cationic disorder becomes the dominant factor controlling the magnetic state in \nthe regime of high doping.  In order to confirm that this effect is not specific to Zn, but is, in \nfact, a rather general phenomenon, we carry out similar studies on YBaFe 4O7 with a second \ndiamagnetic substituent, Li+, and we compare the magnetic behaviour of YBaFe 4-xLixO7 with \nthat previously observed for CaBaFe 4-xLixO7 [16].  \n Due to its univalent character, lithium has the advantage of inducing an average iron \nvalency which is different from that of the zinc compounds for the same substitution rate, \nthereby allowing the relative effects of  the Fe valence and the cationic dis order to be \ncompared further. The size of Li+, which is similar to that of Zn2+, and its ability to adopt the \ntetrahedral coordination , are favourable to such a substitution. In contrast to the case of zinc \nsubstitution, the maximum amount of lithium that was substituted was limited by the \nexperimental conditions of synthesis. Indeed, working in sealed tubes, and using the \nprecursors Y 2O3, BaFe 2O4, LiFeO 2, Fe 2O3 and Fe in order to avoid any reaction with the \nsupport and any Li 2O volatization, only the compo sitions YBaFe 4-xLixO7 with x \n  0.75 could  \nbe prepared , keeping the oxygen and lithium stoichiometry intact.  \n The first important point deals with the fact that lithium substitution, like zinc, \nstabilizes the hexagonal symmetry at the cost of the cubic phas e. Quite remarkably, a smaller \nlithium content, x = 0.30 only, is sufficient to stabilize the hexagonal form, instead of x = 0.40 \nfor Zn . In any case, the cell parameters of the Li substituted yttrium phase vary only slightly \nwith composition from a = 6.30 74 (1)  Å, c = 10.35 86 (2)  Å for x = 0.30 to a = 6.293 2 (1)  Å, c \n= 10.3 104 (1)  Å for x = 0.75.  \n The magnetic study of the compounds YBaFe 4-xLixO7 clearly show s that for the \nmaximum substitution rate i.e. x = 0.75, the complete spin glass behaviour cannot be  reached. \nThus, for the sake of comparison with other substituted phases, we discuss, in this section, the \nresults obtained in the low doping regime.  \n  \n3.2.1.  YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7 \n \n Li+ substitution in hexagonal CaBaFe 4O7 has been studied b y us before [16]. For the \npurpose of comparison with Li+ in YBaFe 4O7, we choose the samples with x = 0.3 from the \ntwo series. This is because, as we have stated before, x = 0.3 is the minimum amount of Li+ \nrequired to stabilize monophasic YBaFe 4-xLixO7 with the hexagonal symmetry.  \n The d. c. magnetization results  of YBaFe 3.7Li0.3O7 have been shown in Fig. 6 . In \naccordance with its hexagonal symmetry, YBaFe 3.7Li0.3O7 is ferrimagnetic. However,  7 considering the fact that the average Fe valency in YBaFe 3.7Li0.3O7 (Fe val = 2.35) is much less \nthan that in Ca BaFe 3.7Li0.3O7 (Fe val = 2.62), the ferrimagnetic interaction in YBaFe 3.7Li0.3O7 \nshould be weaker than that in Ca BaFe 3.7Li0.3O7. This is indeed the case as can be seen from  \nTable 1 , where we have compared the va lues of T C, M FC(T=5K)  and H C(T=5K)  for the two \nsamples.  The values for Ca BaFe 3.7Li0.3O7 in Table 1  have been quoted from reference 16.  \nCaBaFe 3.7Li0.3O7 has higher values of T C, M FC(T=5K)  and H C(T=5K)  than those of \nYBaFe 3.7Li0.3O7 , showing that the ferrim agnetic interaction is stronger and the magnetic \nfrustration is weaker in CaBaFe 3.7Li0.3O7.  \n \n3.2.2.  CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 \n \n In all the above cases, we have compared samples with different average Fe valenc ies, \nand shown that in the regime of low doping, the Fe valency controls the magnetic properties \nof the oxide, while in the regime of high doping, the degree of cationic diso rder is the \ndeciding factor.  We have seen that in the low doping regime, samples with the same degree of \ndisorder, b ut with different Fe valency behave differently vis – à – vis their magnetic \nproperties. In our final section, we investigate what happens in the reverse case i.e. for \nsamples with different degree of disorder (but well within the low doping regime), but  having  \nthe same average Fe valency. For this purpose , we compare the  two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7 which fall in the low doping regime, and have almost the same average \nFe valency (Fe val = 2.57 and 2.58 for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 respectively).   \nThe d. c. magnetization results of CaBaFe 3.5Zn0.5O7 have been shown in Fig. 7. In accordance \nwith its hexagonal symmetry, CaBaFe 3.5Zn0.5O7 shows a sharp ferrimagnetic transition, and a \nlarge square hysteresis loop. However, what is more striking is the almost exact one – to – one \ncorrespondence of the magnetic parameters obtained for the two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7. These are listed in Table 2 .  The values for CaBaFe 3.8Li0.2O7 have \nbeen quoted from reference 16.  This remarkable correspondence in the magnetic properties of \nthe two samples with the same average Fe valency proves unambiguously that in the low \ndoping regime the Fe valency is indeed the main factor controlling the magnetic state of the \noxide.   \n \n \n \n  8 4. Conclusion  \n \n This study shows the great impact of the substitution of diamagnetic cations such as \nzinc or lithium for iron upon the competition between ferrimagnetism and magnetic frustration \nin “114’ ferrites. The first effect is structural – it is seen t hat the substitution of these cations \nfor iron in the cubic phase, YBaFe 4O7, leads to a hexagonal symmetry, and consequently \ndestroys the 3 D geometric frustration at the benefit of a competition between a 2 D geometric \nfrustra tion and 1 D magnetic orderin g, thereby inducing ferrimagnetism. The second effect, \nobserved in the hexagonal phases such as CaBaFe 4O7, modifies the competition between the 2 \nD frustration and the 1 D magnetic ordering in two different ways, depending on the \nsubstituent concentration.  For low doping values, the modification of the average iron valency \nthat is induced by this substitution dominates the magnetism of these compounds leading to a \ndecrease of ferrimagnetism at the benefit of 2 D magnetic frustration, whereas in the high \ndoping regime, the disordering of the cations dominates, inducing a complete magnetic \nfrustration, irrespective of the iron valency. The nature of the ferrimagnetism of these ferrites, \ntill date, has not been completely elucidated, and in particular, it is st ill not known whether the \niron spins lie in plane or out of the triangular planes, so that a vast field is still open for the \ninvestigation and understanding of this new type of magnetic frustration.  \n \n5. Acknowledgements  \n \nWe acknowledge the CNRS and the Co nseil Regional of Basse Normandie for financial \nsupport in the frame of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \n \n \n \n \n \n \n \n \n \n  9 6. References  \n  \n      [1]    Walz , F.: J. Phys. Condens. Matter 14, R285  (2002)   \n      [2]    Verwey , E. J. W.,  Haayman, P. W.: Physica  8, 979  (1941 ) \n      [3]    Goodenough,  J. B. Magnetism and the Chemical Bond, Interscience Monographs on  \n               Chemistry , Vol. 1 5 Huntington, New York , 1976 )  \n      [4]    Greedan,  J. E.:  J. Alloys and Compounds  408 – 412, 444  (2006 ) \n      [5]    Muraoka,  Y., Tabata , H., Kawai,  T.: J. Appl. Phys. 88, 7223  (2000)  \n      [6]    Delgado,  G. E.,  Sagredo , V., Bolzoni,  F.: Cryst. Res. Technol . 43, 141  (2008)  \n      [7]    Valldor , M., Andersson,  M.: Solid State Sciences   4, 923  (2002 ) \n      [8]    Valldor,  M.: J. Phys.: Condens. Matter . 16, 9209  (2004)  \n      [9]    Raveau, B., Caignaert,  V., Pralong,  V., Pelloquin  D., Maignan,  A. : Chem. Mater. 20,  \n              6295  (2008)  \n      [10]   Caignaert,  V., Abakumov,  A. M., Pelloquin, D., Pralong,  V., Maignan,  A., Van   \n                Tendeloo , G., Raveau,  B.: Chem. Mater . 21, 1116  (2009)  \n      [11]   Pralong, V., Caignaert,  V.,  Maignan , A., Raveau,  B. : J. Mater. Chem . 19, 8335  \n                (2009)   \n      [12]   Chapon, L. C., Radaelli, P. G., Zheng , H., Mitchell, J. F.: Phys. Rev. B  74, 17240  \n               (2006)  \n      [13]   Manuel, P., Chapon,  L. C.,  Radaelli, P. G., Zheng , H., Mitchell, J. F.:  Phys. Rev.  \n                Lett. 103, 037202  (2009 ) \n      [14]   Caignaert, V., Pralong,  V., Hardy,  V., Ritter C., Raveau,  B.: Phys. Rev. B  81, 094417  \n                (2010 ) \n      [15]    Sarkar, T., Pralong,  V., Caignaert , V., Raveau,  B. : Chem. Mater . 22, 2885 (2010)  \n      [16]    Vijayanand hini, K., Simon, Ch., Pralong,  V., Caignaert V., Raveau,  B.: Phys. Rev. B  \n                79, 224407  (2009 ) \n \n \n \n \n \n \n \n  10 Figure Captions  \n \nFigure 1 : Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two \nsorts of layers of FeO 4 tetrahedra  called triangular (T) and kagomé (K).  \nFigure 2 : Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \nFigure 3 : MZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at \nH = 0.3 T. The ins et shows the magnetization as a function of magnetic field at T = 5 K for \nthe two samples.  \nFigure 4 : MZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at \nH = 0.3 T. The inset shows the magnetization as a function of magnetic field  at T = 5 K for \nthe two samples.  \nFigure 5 : Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \nFigure 6 : MZFC(T) and M FC(T) curves  for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets \n(a) show the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation of T C.  \nFigure 7 : MZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0. 3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C. \n \nTabl e Captions  \n \nTable 1 : TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \nTable 2 : T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \n \n \n \n \n  11 \n \n \nFig. 1. Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two sorts \nof layers of FeO 4 tetrahedra called triangular (T) and kagomé (K).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  12 \n  \nFig. 2 . Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \n \n \n \n \n  13 \n \n \nFig. 3. M ZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at H \n= 0.3 T. The inset shows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  14 \n \n \nFig. 4. M ZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at H \n= 0.3 T. The inset s hows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 \n \n \nFig. 5. Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 6. M ZFC(T) and M FC(T) curves for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets (a) \nshow the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation  of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 7. M ZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0.3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a  function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n  18 Table 1. TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \n \n \nSample  \n  \nTC (K) \n  \nMFC(T=5K)   (µB/f.u.)  \n HC (at T = 5 K) (T)  \nYBaFe 3.7Li0.3O7  115.7  0.34 0.87 \nCaBaFe 3.7Li0.3O7 150.7  0.50 0.98 \n \nTable 2. T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \nSample  \n TC (K) \n θCW (K) \n µeff (µB/f.u.)  \n HC (at T = 5 K) \n(T) \nCaBaFe 3.8Li0.2O7 191.6  211.9  4.37 1.17 \nCaBaFe 3.5Zn0.5O7 203.0  218.8  4.05 1.12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "1909.11381v1.Stability_and_Absence_of_a_Tower_of_States_in_Ferrimagnets.pdf",        "content": "Stability and Absence of a Tower of States in Ferrimagnets\nLouk Rademaker,1Aron Beekman,2and Jasper van Wezel3\n1Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland\n2Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan\n3Institute for Theoretical Physics Amsterdam, University of Amsterdam,\nScience Park 904, 1098 XH Amsterdam, The Netherlands\n(Dated: September 26, 2019)\nAntiferromagnets and ferromagnets are archetypes of the two distinct (type-A and type-B) ways\nof spontaneously breaking a continuous symmetry. Although type-B Nambu{Goldstone modes arise\nin various systems, the ferromagnet was considered pathological due to the stability and symmetry-\nbreaking nature of its exact ground state. However, here we show that symmetry-breaking in\nferrimagnets closely resembles the ferromagnet. In particular, there is an extensive ground state\ndegeneracy, there is no Anderson tower of states, and the maximally polarized ground state is\nthermodynamically stable. Our results are derived analytically for the Lieb{Mattis ferrimagnet and\nnumerically for the Heisenberg ferrimagnet. We argue that these properties are generic for type-B\nsymmetry-broken systems, where the order parameter operator is a symmetry generator.\nI. INTRODUCTION\nSpontaneous symmetry breaking (SSB) is the phe-\nnomenon that the thermal equilibrium state of a many-\nbody system has lower symmetry than the Hamilto-\nnian that governs it. For a continuous symmetry, there\nis a multitude of degenerate symmetry-breaking ground\nstates in the thermodynamic limit where the number of\nconstituents Ngoes to in\fnity. Conversely when Nis \f-\nnite, most such systems possess a unique and symmetric\nground state. This was explicitly shown for the Heisen-\nberg antiferromagnet by Marshall, and Lieb and Mat-\ntis [1, 2], and is now understood to be quite general [3, 4].\nHowever, these unique ground states are not stable, in the\nsense that adding even a small symmetry-breaking per-\nturbation\u000fwill lead to a symmetry-broken ground state;\nin the thermodynamic limit an in\fnitesimal perturbation\nsu\u000eces to break the symmetry. Examples of this type of\nsymmetry breaking include the breaking of Z2(up{down)\nsymmetry in Ising models, of U(1) (phase rotation) sym-\nmetry inXY-models, of SU(2) (spin-rotational) symme-\ntry in Heisenberg antiferromagnets, and of translational\nsymmetry breaking in crystals [5].\nThe instability of the symmetric ground state of \fnite-\nsized systems may be intuitively understood by realizing\nit is actually a type of `Schr odinger cat state', namely\na superposition of macroscopically distinct states which\neach break the symmetry di\u000berently [5, 6]. There must\nthen be some observable Awith an extensive expectation\nvaluehAi\u0018O (N) whose variance scales as Var A\u0018N2,\nviolating the cluster decomposition property [7]. In other\nwords, the symmetric ground state contains macroscopic\nuncertainty of an extensive observable, making it exceed-\ningly susceptible to local perturbations. The symmetry-\nbroken state, on the other hand, does not contain macro-\nscopic uncertainties and is thermodynamically stable. All\nthe while it is not an energy eigenstate; instead, it is a\nsuperposition of the ground state and zero-wavenumber\nlow-energy states. If the broken symmetry is continuous,\nthe gap of these states is of the order O(1=N). The ex-istence of this low-energy tower of states was observed\nfor quantum antiferromagnets by Anderson [8], and sub-\nsequently shown to be generic in SSB systems with a\nsymmetric ground state [3, 4, 9{17]. To understand the\nphysics of \fnite-size systems with SSB, stability is at least\nas important as the energy spectrum [18]. The existence\nof the tower of states can be an important numerical di-\nagnostic to show the propensity to SSB even in very small\nsystems [19, 20].\nThis behavior is however not completely general. In-\ndeed it has long been known that the Heisenberg fer-\nromagnet has degenerate, symmetry-breaking ground\nstates for systems of any size. Furthermore its order\nparameter is a symmetry generator itself and hence a\nconserved quantity [21], there is only a single Nambu{\nGoldstone modes while two symmetry generators are bro-\nken, and this Goldstone mode has a quadratic dispersion.\nIt has recently been cleared up that these two features\ngo hand-in-hand: whenever the commutator of two bro-\nken symmetries has a non-vanishing expectation value,\ntwo Goldstone \felds conspire to form a single, quadrati-\ncally dispersing gapless mode accompanied by a gapped\npartner mode.[22{26] Such Goldstone modes have been\ndubbed type-B, while ordinary, linearly dispersing Gold-\nstone modes are called type-A.\nIt is now the question whether the other ferromagnet\nphenomenology|degenerate, thermodynamically stable\n\fnite-size ground states and no tower of states|also car-\nries over to any type-B SSB system. A natural starting\npoint is the ferrimagnet , a state with antiferromagnetic\ncorrelations between two unequal-size spin species, which\nimplies in addition to antiferromagnetic order alsoferro-\nmagnetic order. Earlier, one of the authors suggested\nthat ferrimagnets would feature a tower of states, since\ntheir classical ground states are not eigenstates of the\nquantum Hamiltonian [27]. On the other hand, it has\nlong been known that spin systems with any non-zero\nmagnetization have macroscopically degenerate ground\nstates [2, 28].\nHere we show that the Heisenberg ferrimagnet is fararXiv:1909.11381v1  [cond-mat.str-el]  25 Sep 20192\nmore akin to a ferromagnet than to an antiferromagnet:\nwe demonstrate explicitly that there exists a thermody-\nnamically stable \fnite-size ground state, and that there\nis no tower of states separated from the ground state by\nan excitation gap of order O(1=N). This stable ground\nstate can be understood to be a classical (product) state\nsupplemented by quantum corrections, in the same way\nthat the SSB states of type-A systems are [8, 21]. We pro-\nvide an analytic derivation of the stability of this state in\nthe simpli\fed case of the Lieb{Mattis model, and provide\nnumerical evidence for the stability in the full Heisenberg\nHamiltonian.\nWe furthermore argue that this behavior is general\nfor any system with exclusively type-B SSB. This paints\na comprehensive picture of SSB: if the ground state is\nunique, it must be accompanied by a tower of states in\norder for thermodynamically stable SSB states, as a su-\nperposition of very-closely spaced energy eigenstates, to\nexist. In type-B systems such a tower of states is absent,\nbut SSB is possible because thermodynamically stable,\nsymmetry-breaking exact ground states exist even for\n\fnite-size systems.\nThis article is organized as follows. In Section II we\nbrie\ry outline the Lieb{Mattis argument which leads to\nthe tower of states in antiferromagnets, the archetype\nfor type-A SSB. In Section III we show that a tower of\nstates in absent in Heisenberg ferrimagnets, while there\nis a ground state degeneracy. These ground states all\nbreak theSU(2) spin-rotation symmetry as is shown in\nSection IV. In Section V we calculate the overlap of the\nSSB ground state with the classical N\u0013 eel state and com-\npare with the situation in the antiferromagnet. The cen-\ntral part of this work is the demonstration that the two\nground states which have maximal positive or negative\nmagnetization are thermodynamically stable. In Sec-\ntion VI this is shown analytically for the Lieb{Mattis\nmodel. Numerical evidence of the stabily of small 1D\nferrimagnets is provided in Section VII by exact diago-\nnalization. We conclude with a comprehensive picture of\nSSB and directions for further research in Section VIII.\nII. ANTIFERROMAGNETS\nTo set the stage, we shall \frst recall some well-known\nfacts about Heisenberg antiferromagnets on bipartite lat-\ntices. The Heisenberg Hamiltonian is\n^HH=JX\nhiji^~Si\u0001^~Sj: (1)\nHere^~Siis a spin-soperator[29] on site i, the sum is over\nnearest-neighbor lattice sites, and J > 0 is a coupling\nconstant. This Hamiltonian is invariant under global\nSU(2)-spin rotations. If the lattice is bipartite it can\nbe divided in A- andB-sublattices such that each site\nhas neighbors only on the other sublattice. The classi-\ncal ground states are N\u0013 eel states with spins anti-alignedon the two sublattices, breaking the SU(2) symmetry\ntoU(1); the direction of N\u0013 eel ordering is spontaneously\nchosen. If furthermore the number of sites of each sub-\nlattice is the same (for instance in square and hexagonal\nlattices), there is no net magnetization hP\ni^~Sii= 0.\nThe N\u0013 eel states are not eigenstates of Eq. (1), and will\nbe a\u000bected by quantum corrections. But even stronger,\na \fnite-size system governed by this Hamiltonian has a\nunique ground state with total spin value S= 0, which\ntherefore does not break any symmetry. This was shown\nby Marshall [1], and can be understood due to an elegant\nargument by Lieb and Mattis [2]: consider the following\nHamiltonian (\\Lieb{Mattis model\")\n^HLM=2J\nN^~SA\u0001^~SB=J\nN(^S2\u0000^S2\nA\u0000^S2\nB); (2)\nwhere^~S=P\ni^~Siis the total spin of the system,^~SA;B=\nP\ni2A;B^~Sithe total sublattice spin, and ^S2=^~S2etc.\nNote that^~S=^~SA+^~SB. In this model each spin on the A-\nsublattice interacts with all spins on the B-sublattice and\nvice versa. This Hamiltonian simultaneously commutes\nwith ^S2\nA,^S2\nB,^S2and^Sz, and eigenstates can therefore be\ndesignated by the quantum numbers jSASBSMzi, with\nenergiesE=J\nN\u0000\nS(S+ 1)\u0000SA(SA+ 1)\u0000SB(SB+ 1)\u0001\n.\nClearly the energy is minimal when Sis minimal and\nbothSAandSBare maximal. The minimal value is\nS= 0, and therefore the ground state is a total spin\nsinglet, is unique, and does not break any symmetry.\nIt can be easily seen that the Lieb{Mattis model is\nequal to only the ~k=~0 and~k=~Q= (\u0019;\u0019;:::;\u0019 ) con-\ntributions of the Fourier-transformed Heisenberg Hamil-\ntonian Eq. (1). Lieb and Mattis have shown that for\nany \fniteNthe overlap between the ground state of\nEq. (2) and the ground state of Eq. (1) is non-vanishing.\nTherefore, these two states must have the same quantum\nnumbers, and also the ground state of the Heisenberg\nantiferromagnet is a total spin singlet.\nThe correspondence between the two Hamiltonians\ngoes further. Excitations that keep SAandSB\fxed\nwhile increasing Scost an energyO(J=N). For large\nN, these energy levels are almost degenerate with the\nground state. There is therefore a tower of extremely\nlow-lying states with SAandSBmaximal,Mz= 0, and\ndi\u000beringS, and this carries over to the Heisenberg an-\ntiferromagnet by the same argument. Note that excita-\ntions in the Lieb-Mattis model that change SAorSBcost\nenergy of at least O(J), just as local excitations such as\nspin \rips, while Nambu{Goldstone modes (spin waves) in\nthe Heisenberg model have lowest energy O(J=L) withL\nthe linear size of the system.\nThe variance of the local N\u0013 eel order parameter ^Nz\ni=\n(\u00001)i^Sz\niin the symmetric ground states is of order\none [4], so that variance of the total N\u0013 eel order parameter\nscales asN2, indicating that this state is not thermody-\nnamically stable. This is easy to see, when one realizes\nthe ground state of the Lieb{Mattis model is equal to3\nthe equal-weight superposition of classical N\u0013 eel states in\nall magnetization directions [6]. This is a Schr odinger\ncat state, which is extraordinarily sensitive to external\nperturbations.\nIII. THE ABSENCE OF A TOWER OF STATES\nWe will now begin demonstrating the di\u000berences with\nthe picture painted in Section II, for the case where the\nmagnetization is \fnite. For concreteness, we study the\nHeisenberg ferrimagnet governed by Hamiltonian Eq. (1),\nbut now the spins on A- andB-sublattices are sAandsB\nrespectively, with sA6=sB. Without loss of generality\nwe choosesA> sB. On a bipartite lattice with equal\nnumber of sublattice sites, the classical N\u0013 eel state has\na \fnite magnetization h^Szi=N(sA\u0000sB)=2, and stag-\ngered magnetization h^Nzi=N(sA+sB)=2. Everything\nwe say here also holds for antiferromagnets where the\nnumber ofA-sublattice sites is di\u000berent from number the\nB-sublattice sites; the imporant feature is that the total\nspin isS=jSA\u0000SBj>0 whereSA;B=P\ni2A;BsA;B.\nIt is known that the number of ground states of this\nmodel is equal to 2 S+ 1 = 2jSA\u0000SBj+ 1 =NjsA\u0000\nsBj+ 1 [2, 28], which can again be inferred from the\noverlap of these states with the ground states of the\ncorresponding Lieb{Mattis model. This number is ex-\ntensive (proportional to N) sinceSis extensive. More-\nover, the lowest excitation according to Eq. (2) has spin\nS=jSA\u0000SBj\u00061 whileSAandSBare the same, with\nenergy gap \u0001 E=J\u0000\n(sA\u0000sB) +2\nN\u0001\n. Crucially, this en-\nergy gap isO(J) instead of order O(J=N) sincesA6=sB.\nThere is therefore no tower of states with energy gap\nO(1=N) that would vanish in the thermodynamic limit.\nIndeed, the exchange energy Jis typically of order 1{10\nmeV, which is certainly not neglible, possibly even mea-\nsurable.\nRecall that the Lieb{Mattis Hamiltonian is the ~k=\n~0;~Qpart of the full Heisenberg Hamiltonian. The tower\nof states consists precisely of the zero-wavenumber exci-\ntations, and therefore these states and their energies are\nidentical for both models. We can therefore conclude that\nthe lowest excitations in Heisenberg ferrimagnets will be\nlow (non-zero) wavenumber collective excitations: spin\nwaves with quadratic dispersion whose energy scales as\nO(J=L2) withLthe linear system size (so N=Ld).\nWhile this energy can get arbitrarily low as L!1 , it\nwill not be as low as the gap in a putative tower of states\nind>2 dimensions. We will con\frm this numerically in\nSection VII.\nIn Ref. [4], Tasaki provides a proof of the existence of\na tower of states with gaps of order O(1=N), based on\nseveral assumptions. One of these assumptions is that\nthe ground state be unique andbe an eigenstate of a\nsymmetry generator with eigenvalue M. In the present\ncase, although the ground states are eigenstates of ^Sz,\nthey are degenerate and not unique. Below we will argue\nthat whenM > 0, the ground state is always degenerate.Tasaki's derivation applies therefore only to the case M=\n0 and the results of this section are not in contradiction\nwith the proof. The alternative is when the symmetry\ngenerator is broken itself in the type-A way, which we\nshortly discuss in Section VIII.\nIV. SPONTANEOUS SYMMETRY BREAKING\nFrom the standard viewpoint of SSB, the absence of\na tower of states naively poses a conundrum. In type-\nA systems, all ~k=~0 energy eigenstates, including the\nsymmetric ground state, are thermodynamically unsta-\nble, and a tiny perturbation will be able to break the\nsymmetry. The existence of a tower of states is nec-\nessary to be able to construct the dynamically stable,\nsymmetry-breaking superpositions of energy eigenstates,\nas the energy \ructuations of these superpositions fall o\u000b\nasO(J=N). We have just seen the smallest energy gap\ntowards total spin excitations in ferrimagnets is instead\nO(J). What does this imply for the symmetry breaking?\nThe answer is in fact quite simple: the exact ground\nstates already break the symmetry themselves. This is\neasy to see from the Lieb{Mattis model. We have shown\nthat its spectrum can be assigned de\fnite quantum num-\nbersSandMz(where the z-axis is chosen arbitrarily).\nThe only such state which has full SU(2) spin-rotation\nsymmetry is the one with S=Mz= 0. (It is not su\u000e-\ncient to have only Mz= 0. Recall for instance that a two-\nspin-1\n2system in the triplet state s= 1;mz= 0 breaks\nSx- andSy-rotation symmetry.) The ground states of\nthe ferrimagnet instead have S=jSA\u0000SBj>0. In fact,\nthe symmetric state with S= 0 has quite a high energy\ncompared to this state.\nWe can also reach this result more formally, by con-\nsidering the usual SSB procedure of adding an external\nstaggered magnetic \feld Bcoupling to the order param-\neter, to the Lieb{Mattis model:\n^HLM=J\nN\u0010\n^S2\u0000^S2\nA\u0000^S2\nB\u0011\n\u0000B\u0010\n^Sz\nA\u0000^Sz\nB\u0011\n: (3)\nHere ^Sz\nA;B=P\ni2A;B^Sz\nirepresent the z-component of\nthe total spin on sublattices AandB. The matrix ele-\nments of the symmetry-breaking \feld in the basis of Lieb{\nMattis eigenstates, hSASBSMzj(^Sz\nA\u0000^Sz\nB)jS0\nAS0\nBS0M0\nzi,\nare known exactly (reproduced in Appendix A for com-\npleteness) [30]. For zero \feld B= 0, the ground states\nhave maximal SA, andSB, minimal S=jSA\u0000SBj,\nand are degenerate for any value of Mz. For non-zero\n\feld,B > 0, the degeneracy is lifted, and the state\nwith the lowest expectation value of the energy has mag-\nnetizationMz=S. This state must be a superpo-\nsition of states with di\u000berent values of the total spin,\njSA\u0000SBj<S < (SA+SB), because the staggered mag-\nnetization operator Sz\nA\u0000Sz\nBcouples state with total spin\nStoS\u00061 states.\nIn the limit of large Nthe matrix elements of the\nHamiltonian can be conveniently expressed in terms of4\nthe shifted total spin ~S=S\u0000jSA\u0000SBj. The Hamilto-\nnian is then, up to order O(1\nN):\n^H\u0019E0+X\n~S;~S0jSASB~SMzi\u0010\nf~S+1\u000e~S;~S0\u00001\n+a~S\u000e~S;~S0+f~S\u000e~S;~S0+1\u0011\nhSASB~S0Mzj;\nwithE0=J\n4(sA\u0000sB)2N+J\n2(sA\u0000sB)\n\u0000B\n2(sA+sB)N\u00002BsB\nsA\u0000sB;\na~S=~S\u0012\nJ(sA\u0000sB) + 2BsA+sB\nsA\u0000sB\u0013\n;\nf~S=\u00002BpsAsB\nsA\u0000sB~S: (4)\nThis expression for the Hamiltonian is a special case of\na tridiagonal matrix discussed in Appendix B. As shown\nthere, the ground state in the large Nlimit is a superpo-\nsition of states with di\u000berent ~S=S\u0000jSA\u0000SBj,\nj 0(B;N )i=X\n~S (~S)j~Si; (5)\nwith weights given by an exponentially decaying function\n (~S) =ce\u0000~S=\u0015. The decay `length' \u0015is given by\n\u0015= 1=log \n1\n2\u0000\n1 +p\n1\u00004\u000f2\u0001\nj\u000fj!\n; (6)\nwith\u000f=\u00002BpsAsB\nJ(sA\u0000sB)2+ 2B(sA+sB): (7)\nIn the limit of zero staggered \feld B!0, the decay\nlength vanishes, and the Lieb{Mattis ground state j 0iis\nthe total-spin eigenstate with S=jSA\u0000SBjandMz=S.\nContrary to the case of type-A SSB systems, the ther-\nmodynamic limit and the limit of vanishing \feld com-\nmute for the ferrimagnet:\nlim\nB#0lim\nN!1j 0(B;N )i= lim\nN!1lim\nB#0j 0(B;N )i:(8)\nJust as for the ferromagnet, a particular orientation (i.e.\nthe choice of the z-axis) for the ferrimagnetic ground\nstate can be singled out from the ground state manifold\nby an in\fnitesimal \feld even for \fnite system size. There\nis thus a discontinuity in the ground state as a function\nof applied \feld for any system size:\nlim\nB#0j 0(B;N )i6= lim\nB\"0j 0(B;N )i: (9)\nNumerical evaluation of the ground state of the ferri-\nmagnetic Lieb{Mattis Hamiltonian for large but \fnite N\nandB > 0 con\frms that it is an exponentially decaying\nfunction in ~S, as shown in Fig. 1. The decay length\nfollows the analytical result of Eqns. (6), (7).\nWe conclude that although the ferrimagnet has 2 S+ 1\ndegenerate ground states, any small external staggered\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★●N=20\n◆N=200\n★N=2000\nEqs.(6)-(7)0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.00.10.20.30.40.50.60.7\nB/JλFIG. 1. The ground state of the ferrimagnetic Lieb{Mattis\nmodel with staggered magnetic \feld Bis a superposition of\nstates with di\u000berent ~S=S\u0000jSA\u0000SBj, with weights given\nby e\u0000~S=\u0015. Here, a numerically determined decay length \u0015is\nplotted as a function of B=J for increasing system size up to\nN= 2000. The dashed black line is the exact result in the\nthermodynamic limit following Eqs. (6)-(7). It is important\nto observe that in the thermodynamic limit N!1 the decay\nlength\u0015remains smooth near B= 0.\n\feld is su\u000ecient to lift the degeneracy, upon which the\nground state will be remain dominated by the spin state\nwithS=jSA\u0000SBj, and the magnetization direction\nparallel to the external \feld.\nV. THE CLASSICAL N \u0013EEL STATE\nBesides the total (ferromagnetic) magnetization Mz,\nferrimagnets also have a nonzero staggered (antiferro-\nmagnetic) magnetization n=h^Sz\nA\u0000^Sz\nBi 6= 0. Using\nthe matrix elements in Appendix A, the staggered mag-\nnetization expectation value for the maximally polar-\nized ground state of the Lieb{Mattis ferrimagnet, with\nMz=SandS= (sA\u0000sB)N=2, and no applied \felds, is\nfound to be:\nn=NsA+sB\n2\u00002sB\njsA\u0000sBj+O(N\u00001) (10)\nTo leading order, this agrees with the expectation from\nthe classical limit for a ferrimagnet, which is the N\u0013 eel\nproduct statej N\u0013 eeliwith all spins on the Asublattice\npointing maximally up and on the Bsublattice maxi-\nmally down. That state has maximal staggered magne-\ntizationn=NsA+sB\n2.\nThe N\u0013 eel state itself can be written as a superpo-\nsition of states with di\u000berent total spin Sbut \fxed\nMz=SA\u0000SB, andSA;B=sA;BN=2. Becausej N\u0013 eeli\nis the ground state of the Hamiltonian in Eqn. (3) with\nJ= 0 andB > 0, the N\u0013 eel state wavefunction has the\nsame exponentially decreasing form  (~S) =ce\u0000~S=\u0015as\nthe ferrimagnetic ground state. For J= 0, however,\nthe decay length \u0015in Eqn. (6) is independent of Nand\nB. This means that the N\u0013 eel state has nonzero overlap5\nwith the ground state of the Lieb{Mattis Hamiltonian for\nany system size, and even in the thermodynamic limit\nN!1 . Its value can be found by normalising the N\u0013 eel\nstate wavefunction:\nh N\u0013 eeljS=jSA\u0000SBji=p\n1\u0000e\u00002=\u0015(J=0) (11)\nFor the special case of sA= 1 andsB= 1=2, we have\n\u000f(J= 0) =\u0000p\n2\n3and the overlap is found to be1p\n2.\nBecause the N\u0013 eel state is a superposition of total spin\nstates, which do not form a tower of states in the ferri-\nmagnet, the N\u0013 eel state cannot be become energetically\ndegenerate with the ground state of the Lieb{Mattis\nHamiltonian for J > 0 andB= 0. The energy di\u000ber-\nence follows directly from\nh N\u0013 eelj^HLMj N\u0013 eeli=J\n4NjsA\u0000sBj2+J(s2\nA+s2\nB):(12)\nThis should be compared with the Lieb{Mattis ground\nstate energy E0=J\nN(SA\u0000SB)(SA\u0000SB+1) =J\n4NjsA\u0000\nsBj2+J\n2(sA\u0000sB). The energy di\u000berence between j N\u0013 eeli\nand the actual ground state is thus:\n\u0001EN=J\u0012\ns2\nA+s2\nB\u00001\n2sA+1\n2sB\u0013\n: (13)\nFor the speci\fc case with sA= 1 andsB= 1=2, the\nenergy di\u000berence is exactly \u0001 EN=J. In all cases, it is\nof orderO(J), and does not vanish in the thermodynamic\nlimit. This is to be contrasted with the antiferromagnet,\nwhere the N\u0013 eel state becomes exactly degenerate with\nthe ground state in the thermodynamic limit of the Lieb{\nMattis model.\nThe fact that the staggered magnetization in Eqn. (10)\ndi\u000bers from its classically expected value at sub-leading\norder, may be interpreted as indicating that the Lieb{\nMattis ground state involves zero-wavenumber quantum\ncorrections on top of the classical N\u0013 eel state. These quan-\ntum corrections correspond precisely to the suppression\nof total-spin components outside the ground state man-\nifold. Going towards more realistic models, one should\nnote that the ground state of the Heisenberg ferrimagnet\nis not the same as that of the Lieb{Mattis Hamiltonian.\nBecause the Heisenberg model includes only local inter-\nactions, its ground state will have quantum corrections\nat all wave numbers as compared to the N\u0013 eel state, and\ntheir overlap vanishes in the thermodynamic limit (con-\n\frmed numerically below), even though both states will\nagree to leading order on the expectation value of stag-\ngered magnetization. The vanishing overlap is in fact\na generic property for ground states of distinct models\nin an exponentially large Hilbert space, and is expected\nalso for example for the overlap between the ground state\nof the Heisenberg antiferromagnet and the classical N\u0013 eel\nstate.VI. STABILITY\nIn Section IV we addressed one part of the conundrum\nthat the absence of a tower of states poses, namely how\nthe ground state breaks the symmetry. However, zero-\nwavenumber energy eigenstates are global|they are not\ntensor products of local states|and typically unstable as\nwe have shown for instance for the ground state of the\nantiferromagnet in Section II. The question is whether\nthe symmetry-breaking exact ground states of the \fnite\nsize ferrimagnet are stable.\nIn this section we show that the exact ground states of\nthe Lieb-Mattis model with maximal polarization, Mz=\n\u0006S=\u0006jSA\u0000SBjare thermodynamically stable. In\nthe next section we provide numerical evidence that the\nmaximally polarized ground states in the full Heisenberg\nmodel are also stable.\nRecall that a state is de\fned to be unstable if there\nexists an extensive observable ^Awhose variance Var ^A=\nh^A2i\u0000h ^Ai2scales asN2[4, 5, 18]. This de\fnition is\nequivalent to saying there exists a connected correlation\nfunction that does not satisfy the cluster decomposition\nproperty [5, 7].\nTo prove that a state is stable one thus needs to com-\npute the variance of all possible extensive observables. In\nthe case of the Lieb{Mattis Hamiltonian in Eqn. (3), all\nstates are global, and we can su\u000ece by computing the\nvariance of the transverse total spin, Var ^Sx, and of the\ntotal sublattice magnetization, Var( ^Sz\nA\u0000^Sz\nB).\nThe variance of the transverse total spin ^Sxis inde-\npendent of SAandSB. BecausehSMzj^SxjSMzi= 0, we\n\fnd\nVarh\n^Sxi\n=hSMzj(^Sx)2jSMzi\n=1\n4hSMzj(^S++^S\u0000)2jSMzi\n=1\n4hSMzj(^S+^S\u0000+^S\u0000^S+)jSMzi: (14)\nThe maximally polarized states, with Mz=\u0006S, are an-\nnihilated by ^S\u0006, and we then \fnd the variance to be\nS=2\u0018O(N). On the other hand, for jMzj<Swe \fnd:\nVarh\n^Sxi\n=1\n2S(S+ 1)\u00001\n2M2\nz: (15)\nAs long asjMzjdoes not scale with system size in exactly\nthe same way as S, the variance of the total transver-\nsal spin scales as O(N2), implying instability. Therefore\nonly the states with jMzj=S!1 are thermodynamically\nstable with respect to the total transversal spin.\nThe variance of the sublattice magnetization ^Sz\nA\u0000^Sz\nB\ncan be computed directly using the matrix elements in\nAppendix A. For the maximally polarized ground state\nwithMz=SandsA>sB, it is to leading order:\nVarh\n^Sz\nA\u0000^Sz\nBi\n= 4sAsB\n(sA\u0000sB)2+:::\u0018O(1):(16)6\n◆◆◆◆◆◆★★★★★★(L2)\n(L)Mz=0 or 1/2\nMz=S\n4 6 8 10 12140.5125\nLVarS x\n◆◆◆◆◆◆★★★★★★(L)\n(L)Mz=0 or 1/2\nMz=S\n4 6 8 10 12142468\nLVar(Sz)A-(Sz)B\nFIG. 2. A numerical comparison of the stabilities of the max-\nimally (Mz=S=L=2, green stars) and minimally ( Mz= 0\nor 1=2, red diamonds) polarized ground states of the 1D fer-\nrimagnetic sA= 1,sB=1\n2Heisenberg model. The left panel\nshows Var ^Sx, and the results exactly follow Eq. (15) (shown\nas dashed lines). The variance in the maximally polarized\nstate scales asO(L) and thus signal stability, whereas the\nO(L2) scaling in the minimally polarized state implies in-\nstability. The right panel shows the variance of ^Sz\nA\u0000^Sz\nB.\nHere, the results do not match the variance found for the\nground states of the Lieb{Mattis model, because SAandSB\nare not symmetries of the ferrimagnetic Heisenberg Hamil-\ntonian. Nevertheless, the variances still scales as O(L) (the\ndashed lines represent a \ft) which implies that all ground\nstates are stable with respect to the uncertainty of the anti-\nferromagnetic correlations.\nThis shows that the maximally polarized ground state\nis thermodynamically stable with respect to all exten-\nsive observables even in \fnite-sized ferrimagnets. For\nstates with non-maximal polarization, M\u0018 O (1), we\nhave Varh\n^Sz\nA\u0000^Sz\nBi\n=sAsB\nsA\u0000sBN+:::, which also suggests\nstability. However, since we have already seen that these\nstates with non-maximal polarization are unstable with\nrespect to total transverse spin ^Sx, the two maximally\npolarized ground states are the only thermodynamically\nstable ground states.\nVII. NUMERICAL RESULTS\nThe Mermin{Wagner{Hohenberg{Coleman theorem\ndoes not prohibit type-B SSB from occurring in one-\ndimensional systems at zero temperature [5, 31, 32]. We\ncan therefore con\frm the generality of the analytic re-\nsults of the Lieb{Mattis model in Section VI using numer-\nical results for a one-dimensional ferrimagnetic Heisen-\nberg chain. The Hamiltonian is given by:\n^H=JX\ni^~Si\u0001^~Si+1: (17)\nFor concreteness, we take all the even sites to have sA= 1\nspins and the odd sites sB= 1=2 spins. We consider\nchains up to lengths L= 14 using exact diagonaliza-\ntion, and evaluate degeneracies and the stability of the\n● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●\n● ● ● ● ● ●●● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ●\n● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ●● ●● ●\n● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●●●●●●●●●\n●●●●●\n●●●●●●●●●●●●●●●\n● ●● ●● ●● ●● ●● ●● ●● ●● ●\n● ●● ●\n● ●● ●\n● ●● ●\n● ●● ●\n● ●● ● ● ●\n● ●● ●● ●\n● ●● ●● ●\n● ● ● ●● ●● ●● ●\n●● ●●●●●● ●●●\n● ●\n● ●● ●● ●\n● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●\n● ●● ●● ●● ●● ●● ●\n● ●● ●\n● ●● ●● ●● ●● ●● ●● ●● ●\n● ●● ●\n● ●●●\n● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●★ ★ ★ ★★ ★★ ★ ★ ★\n★★ ★ ★★ ★ ★ ★ ★★★★ ★ ★ ★ ★★ ★ ★★ ★ ★\n★ ★★ ★ ★ ★★ ★ ★ ★ ★ ★★ ★★ ★★ ★ ★ ★ ★ ★★ ★ ★ ★★ ★★ ★ ★ ★\n★ ★ ★★ ★ ★ ★ ★★ ★ ★ ★ ★ ★ ★★ ★ ★★★★ ★ ★★★★★★★★★★★★★★★★★★★★★★★★\n★★★★★★★★★★★★★★★★★★\n★ ★ ★ ★◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆\nk≠0 states\nk=0 states, S>|S A-SB|\nk=0 states, S<|S A-SB|●\n◆\n★(1)\n(1/L2)\n(1/L2)6 8 10 12 140.20.512\nLΔE/JFIG. 3. An overview of all low-energy excited eigenstates\nwith their energy gap in units of J, in the 1D ferrimagnetic\nsA= 1,sB=1\n2Heisenberg chain, as a function of system\nsizeL. Black dots represent states with \fnite wave number,\nk6= 0, while red stars indicate states with zero wave number\nk= 0 andS <jSA\u0000SBj, and green diamonds have k= 0 and\nS >jSA\u0000SBj. HerejSA\u0000SBj=L=4 is the ground state total\nspin. In one dimension, the excitation gap towards Goldstone\nmodes with nonzero wave number (black dots) is O(1=L2)\n(indicated by the black dashed line), because \u000fk\u0018k2and\nthe smallest momentum scales as k\u0018O(1=L). States with\nk= 0 andS <jSA\u0000SBjare two-Goldstone mode states (red\nstars), which follows from the fact that their magnetization\neigenvalueMzis smaller than the ground state value. Finally,\nthe states that would constitute the tower of states in an\nantiferromagnet, namely those with k= 0 andS >jSA\u0000SBj\n(green diamonds), have a gap that is independent of system\nsize (green dashed line).\nground states. Also note that because our system is one-\ndimensional, the linear size Land total system size N\nare equal.\nWe con\frm that the sA= 1,sB=1\n2Heisenberg ferri-\nmagnet has a L=4+1-dimensional ground state manifold\nwith total spin S=L=4, just like the Lieb{Mattis ground\nstates.\nNext we analyze the stability of the states in the\nground state manifold. Following Sec. VI, we compute\nthe variance of the transverse total spin and the stag-\ngered magnetization, shown in Fig. 2. The variance of\nstaggered magnetization scales with the system size, in-\ndependent of the magnetic number Mz. All states in the\nground state manifold are therefore stable with respect\nto staggered magnetization. However, the variance of\ntransverse spin scales as O(L2) for the state with mini-\nmalMzbut it scales asO(L) for the maximally polarized7\nNéel/Lieb-Mattis\nHeisenberg/Lieb-Mattis\nNéel/Heisenberg\n4 6 8 10 12 140.20.40.60.81.0\nLOverlaps|<ψ|ϕ>2\nFIG. 4. The overlap-squared between ground states of vari-\nous models shown on a log-log scale. The overlap between the\nmaximally polarized ground state of the Lieb{Mattis model\nstate and the classical N\u0013 eel state follows Eqn. (11) and stays\nnonzero even in the thermodynamic limit. On the other hand,\nthe presence of spin \rips or quantum corrections in the maxi-\nmally polarized ground state of the ferrimagnetic Heisenberg\nmodel causes its overlaps with both the N\u0013 eel state and the\nLieb{Mattis state to vanish with increasing system size.\nstateMz=S. In fact, the results are exactly equal to\nEq. (15). We therefore con\frm that the only thermody-\nnamically stable ground state is the maximally polarized\nstate withMz=S.\nThe absence of a tower of states is con\frmed by an\nanalysis of the low-lying eigenstates, shown in Fig. 3.\nFor each low-energy eigenstate we computed its energy,\nmomentum k, total spin Sand polarization Mz. If a\ntower of states would be present, the energy gap towards\nstates with k= 0,Mz=jSA\u0000SBjandS >jSA\u0000SBj\nwould vanish as O(1=L). This is not the case, as we\nargued in Sec. III: in Fig. 3 we see that these states (green\ndiamonds) have a gap of order O(1) (green dashed line).\nNonetheless, there are states whose gap vanishes as\nO(1=L2). We identify here the excitation of a Goldstone\nmode, with dispersion \u000f\u0018k2and the smallest momen-\ntum scaling as k\u0018O(1=L). Goldstone modes necessarily\nhave nonzero momentum, shown in Fig. 3 as black cir-\ncles. The excitation of two Goldstone modes with oppo-\nsite momentum now leads to states with zero momentum\nbut still a gap that scales as O(1=L2), shown as red stars.\nWe thus have analyzed the full eigenvalue spectrum and\nexcluded the possibility of a tower of states.\nFinally, Fig. 4 shows the overlap of the Lieb{Mattis\nand Heisenberg ground states with the classical N\u0013 eel\nstate. While the former have a nonzero overlap in the\nthermodynamic limit, the overlap of the latter decreases\nwith system size. This can be understood as a conse-\nquence of the extensive number of single and few-spin\n\rips contained in the Heisenberg ground state relative to\nthe N\u0013 eel state. These can be seen as quantum corrections\nto the classical state at all wave numbers, and although\nthey hardly a\u000bect the macroscopic staggered magneti-zation, and do not a\u000bect the magnetization at all, they\ndo cause the overlap with the N\u0013 eel state to vanish in\nthe thermodynamic limit. The latter also occurs in the\n(type-A) antiferromagnet.\nVIII. OUTLOOK\nWe found the ferrimagnet to have an extensive ground\nstate degeneracy and no tower of states. Furthermore,\nwe found that within the ground state manifold, the\nmaximally polarized states are thermodynamically sta-\nble. The ferrimagnet shares these features with the fer-\nromagnet, which turns out to be less unique than often\nassumed [24, 27, 31].\nAlthough we found these results for the speci\fc case\nof the ferrimagnet, we hypothesize that these conclusions\napply to type-B systems in general. The de\fning prop-\nerty of such systems is that the the expectation value of\nthe commutator of two broken symmetry generators does\nnot vanish. Apart from some pathological cases [22], such\na commutator is a linear combination of symmetry gen-\nerators itself, implying that the order parameter operator\ncommutes with the Hamiltonian , i.e. the order parameter\noperator is a symmetry generator.\nLet us consider the Lie algebra structure of the Hamil-\ntonian and its eigenstates. The symmetry generators ^Qa,\nwhich by de\fnition are Hermitian and commute with the\nHamiltonian, can be expressed in the Cartan{Weyl basis,\nin which the Cartan subalgebra is spanned by a maximal\nset ofrmutually commuting generators ^Fi, whereris\ncalled the rank of the Lie algebra. The remaining gen-\nerators can be expressed in pairs of Hermitian-conjugate\nroot generators ^E\u000b;^E\u0000\u000b, with\u000bcalled the root vector\nand commutation relations [ ^E\u000b;^E\u0000\u000b] =\u000bi^Fi. Watan-\nabe and Brauner have shown that the Cartan subalgebra\ncan be chosen in such a way that only the Cartan gener-\nators ^Fican obtain an expectation value, and hence lead\nto type-B SSB [23]. We can simultaneously diagonalize\nthe Hamiltonian and the Cartan generators. Eigenstates\nof the Cartan generators ^Fiareweight statesj\u0016iwith\neigenvalues \u0016i, collected in a weight vector \u0016.\nNow we recall several important theorems from Lie\ngroup theory [33, 34]. First, irreducible representations\nof a semisimple Lie algebra are completely classi\fed by\nspecifying the highest weight \u0016 \u0016. Second, a Lie algebra\nof rankrcontainsrCasimir operators, which commute\nwith the entire Lie algebra. Third, by Schur's lemma,\nany operator that commutes with all generators of the\nLie algebra is proportional to the unit matrix in any ir-\nreducible representation.\nFor discussing the ground states and the tower of\nstates, we only need to consider the k= 0-part of the\nHamiltonian, which therefore only depends on internal\ndegrees of freedom which transform under the Lie group\nof symmetry transformations. Consequently we are go-\ning to assume that the k= 0 Hamiltonian can be com-\npletely speci\fed in terms of Lie algebra generators, in8\nother words it is a spectrum-generating algebra [34]. It\nfollows that the k= 0 Hamiltonian, which commutes\nwith the entire algebra, consists only of Casimir opera-\ntors.\nFor type-B SSB, a ground state is an eigenstate of sym-\nmetry generators in the Cartan subalgebra, i.e. a weight\nstate, where at least one weight component is non-zero.\nSince a symmetry generator is an extensive operator, this\nweight component is extensive. Because the irreducible\nrepresentation of this generator is speci\fed by its high-\nest weight, the highest weight must also be extensive,\nwhich implies that the representation space has exten-\nsive dimensions. And since the Hamiltonian consists of\nCasimir operators, which are proportional to the iden-\ntity matrix in any irreducible representation, this implies\nthat there is a ground state degeneracy of the extensive\ndimension of this representation. This proves that un-\nder mild assumptions type-B SSB involves an extensive\nground state degeneracy.\nAs for the tower of states, we shall discuss the\ncase where the k= 0 Hamiltonian is proportional to\nthe quadratic Casimir operator ^C2, which consists of\nquadratic combinations of generators ^Qa. This com-\nprises most models of interest, including the Heisen-\nberg ferrimagnet and (anti)ferromagnet. Because the\n^Qaare extensive, the Hamiltonian must be of the form\n^Hk=0=1\nN^C2, up to factors of order O(1), so that the en-\nergy is extensive. In the Cartan-Weyl basis, the quadratic\nCasimir operator can be expressed as [35]\n^C2=X\ni^Fi^Fi+X\n\u000b2\u0001+(^E\u000b^E\u0000\u000b+^E\u0000\u000bE\u000b); (18)\nwhere \u0001 +is the set of positive root vectors. Since\nCasimir operators are proportional to the identity ma-\ntrix, we can \fnd the proportionality constant by acting\non the highest weight state for which ^E\u000bj\u0016\u0016i= 0. Then,\nacting on this state, the term in brackets is equal to\n[^E\u000b;^E\u0000\u000b] =\u000bi^Fi. We therefore \fnd the energy of all\nstates in the irreducible representation \u0016 \u0016to be\nE=1\nNX\ni\u0016\u0016i\u0000\n\u0016\u0016i+X\n\u000b2\u0001+\u000bi\u0001\n: (19)\nWith all this, we can analyze the putative tower of\nstates. Denoting the representation to which the ground\nstate belongs by \u0016 \u00160, it consists of k= 0 states in dif-\nferent irreducible representations \u0016 \u00160, but with the same\neigenvalues \u0016for the Cartan generators, which implies\nthat \u0016\u00160>\u0016\u00160\u0015\u0016. From Eq. (19) we see that, since \u0016 \u00160\nis extensively non-zero, the energies of any excited state\ncontain at least a factor of1\nN\u0016\u00160\nj, and therefore are at least\nO(1). There is no tower of states with gaps O(1=N).\nThere is one caveat: it could be that the weight state\nhas eigenvalue \u0016j= 0 for one (or more, but not all) Car-\ntan generators. Then Eq. (19) would allow for a tower of\nstates with energy gaps O(1=N). But if a Cartan genera-\ntor has eigenvalue 0, there are two possibilities: \frst, theSSB# ground\nstatestower of\nstates# NG\nmodesNG\ndispersionlower critical\ndimension\ntype-A 1 yes n linear 1\ntype-BO(N) non\n2quadratic 0\nTABLE I. Comparison between type-A and type-B SSB phe-\nnomenology. They di\u000ber in the number of ground states;\nthe existence of a tower of states; the number of Nambu{\nGoldstone modes in terms of the number of broken symmetry\ngenerators n; their dispersion relation; and the lower criti-\ncal dimension, which is the lowest dimension at which zero-\ntemperature SSB is possible (the lower critical dimension at\n\fnite temperature is 2 for both type-A and type-B SSB).\nstate could be invariant under one or more SU(2) sub-\ngroups, in which case the root generators which would\nconstruct the tower of states also annihilate the state.\nSecond, the symmetry is broken in the type-A way, in\nwhich case a tower of states is expected. We leave de-\ntailed investigation of systems with both type-A and\ntype-B breaking for future work.\nWe therefore conclude that states which feature type-B\nSSB exclusively, have an extensive ground state degener-\nacy and no tower of states with energy gaps O(1=N).\nThe distinction between symmetry breaking where the\nexpectation value of the Casimir operator is minimal or\nmaximal has been discussed before [36], although these\nauthors do not consider the case where \u0016 < \u0016\u0016as is the\ncase for almost all type-B SSB [27].\nThe nonzero overlap between the ground state of the\nferrimagnetic Lieb{Mattis Hamiltonian and the classical\nN\u0013 eel state suggests it may be possible to \fnd a simple ex-\nact representation of this ground state, which we leave for\nfuture work. It also opens up the question of the entan-\nglement structure of the ferrimagnet, which unlike that of\nthe product state Heisenberg ferromagnet, is quite subtle.\nIt would be interesting to see whether the entanglement\nin type-B SSB systems exhibits the same Goldstone mode\ncounting as type-A SSB systems.[16, 17]\nConcluding, the distinction between type-A and type-\nB SSB seems to go much further than the counting of\nGoldstone modes [24, 25], as is summarized in Table I.\nType-A, ordinary, SSB has a unique symmetric ground\nstate and a tower of low-lying states with energy gap\n\u0018 O (1=N). There is a linearly dispersing Goldstone\nmode for each broken symmetry. Furthermore it has\nquantum corrections to the classical SSB state, and due\nto the Coleman theorem type-A SSB in one dimension\nat zero temperature is forbidden in the thermodynamic\nlimit. Conversely, here we have found that type-B SSB\nis accompanied by an extensive ground state degeneracy\nand has no tower of low-lying states. Instead, at least one\nof the ground states is thermodynamically stable. Two\nbroken symmetry generators lead to one quadratically\ndispersing Goldstone mode and a gapped partner mode.\nFinally, type-B systems do not su\u000ber from the Cole-\nman theorem and are stable in one dimension [5, 31, 32]9\n(although both type-A and type-B systems are subject\nto the Mermin{Wagner{Hohenberg theorem that for-\nbids SSB in two or lower dimensions at \fnite tempera-\nture [5, 31]). There seems to be only one essential di\u000ber-\nence between general (\\ferri\") type-B SSB and the pecu-\nliar case of the ferromagnet: the latter is the same as the\nclassical ferromagnet, whereas the ferrimagnet is a clas-\nsical N\u0013 eel state dressed with quantum corrections [27].\nAnother di\u000berence is that the ferromagnet does not con-\ntain a gapped mode partnered with the gapless Goldstone\nmode; this can be interpreted as the gap being pushed\nto in\fnity [5, 37]. Nevertheless, here we have seen that\neven if quantum corrections are present in type-B sys-\ntems, the lowest energy gap of zero-wavenumber states is\nnotO(1=N) butO(1) and they do not constitute a tower\nof states in the sense of Refs. [3, 4, 8{11].\nACKNOWLEDGMENTS\nWe thank Hal Tasaki for inspiring discussions and for\nsharing a draft version of Ref. [28]. This work is sup-\nported by the Swiss National Science Foundation via an\nAmbizione grant (L. R.); by the MEXT-Supported Pro-\ngram for the Strategic Research Foundation at Private\nUniversities \\Topological Science\" (Grant No. S1511006)\nand by JSPS Grant-in-Aid for Early-Career Scientists\n(Grant No. 18K13502) (A. J. B.);\nAppendix A: Matrix elements\nFor completeness, we reproduce here the matrix ele-\nments of the staggered magnetic \feld in the basis of Lieb{\nMattis eigenstates jSASBSMzi, as described in Ref. [30]:\nhSASBSMzj(Sz\nA\u0000Sz\nB)jS0\nAS0\nBS0M0\nzi=\u000eSA;S0\nA\u000eSB;S0\nB\n\u0002\u000eMz;M0z[fS+1\u000eS;S0\u00001+gS\u000eS;S0+fS\u000eS;S0+1];(A1)\nHere, we de\fned the functions:\nfS\u0011r\n(S2\u0000(SA\u0000SB)2)((SA+SB+1)2\u0000S2)(S2\u0000M2z)\n(2S+1)(2 S\u00001)S2 ;\ngS\u0011(SA\u0000SB)(SA+SB+ 1)Mz\nS(S+ 1):(A2)\nFor the speci\fc case of SA=B=sA=BN=2,Mz= 0, and\nlargeN, the matrix elements can be conveniently ex-\npressed in terms of the shifted total spin ~S=S\u0000jSA\u0000\nSBj, up to orderO(1=N):\nf~S\u00192psAsB\nsA\u0000sB~S\ng~S\u00191\n2(sA+sB)N\u00002sB+ (sA+sB)~S\nsA\u0000sB\nS(S+ 1)\u00191\n4(sA\u0000sB)2N2\n+1\n2(sA\u0000sB)(2~S+ 1)N (A3)Appendix B: Ground state of a tridiagonal matrix\nConsider a symmetric tridiagonal matrix, where the\nonly nonzero elements are on the diagonal and just below\nand above it:\nM=0\nBBB@a1b1\nb1a2b2\nb2a3b3\n.........1\nCCCA: (B1)\nWe are interested in the ground state eigenvector and\neigenvalue for the special case where both axandbxin-\ncrease linearly with x > 0. By rescaling the matrix we\nde\fne:\nax=x; bx=\u000fx: (B2)\nAs an ansatz for the ground state eigenvector, with\neigenvaluev, we choose:\n x= (\u0000sgn[\u000f])xe\u0000x=\u0015: (B3)\nThe eigenvalue equation now reads, for each row x:\nj\u000fj\u0010\n(x\u00001)e1=\u0015+xe\u00001=\u0015\u0011\n=x\u0000v: (B4)\nBy looking at the \frst row, with x= 1, we can relate \u000f,\n\u0015andv:\nj\u000fje\u00001=\u0015= 1\u0000v: (B5)\nInserting this back into the equation for general x > 1,\nwe \fnd:\nj\u000fj2x\u00001\n1\u0000v+x(1\u0000v) =x\u0000v; (B6)\nwhich can be simpli\fed to:\n(x\u00001)(j\u000fj2\u0000v(1\u0000v)) = 0: (B7)\nBecause the second factor must equal zero for all x, we\nfound an expression for vin terms of\u000fthat is independent\nofx. This proves that  xis indeed an eigenvector of M\nwith eigenvalue:\nv=1\n2\u0010\n1 +p\n1\u00004j\u000fj2\u0011\n: (B8)\nThe exponential decay length is given by:\n\u0015= 1=log0\n@1\n2\u0010\n1 +p\n1\u00004j\u000fj2\u0011\nj\u000fj1\nA: (B9)10\n[1] W. Marshall, Antiferromagnetism, Proc. Roy. Soc. Lon-\ndon A , 48 (1955).\n[2] E. H. Lieb and D. Mattis, Ordering Energy Levels of\nInteracting Spin Systems, J. Math. Phys. 3, 749 (1962).\n[3] T. Koma and H. Tasaki, Symmetry breaking and \fnite-\nsize e\u000bects in quantum many-body systems, J. Stat.\nPhys. 76, 745 (1994).\n[4] H. Tasaki, Long-range order, \\tower\" of states, and sym-\nmetry breaking in lattice quantum systems, J. Stat. Phys.\n174, 735 (2019).\n[5] A. Beekman, L. Rademaker, and J. van Wezel,\nAn Introduction to Spontaneous Symmetry Breaking,\narXiv:1909.01820 (2019).\n[6] L. Rademaker, Exact Ground State of Lieb-Mattis\nHamiltonian as a Superposition of N\u0013 eel states,\narXiv:1909.09663 (2019).\n[7] A. Shimizu and T. 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D 92, 045028 (2015)."    },    {        "title": "1808.04326v1.Application_of_two_sublattice_bilinearly_coupled_Heisenberg_model_to_the_description_of_certain_ferrimagnetic_materials.pdf",        "content": "arXiv:1808.04326v1  [cond-mat.other]  13 Aug 2018Application of two-sublattice bilinearly coupled Heisenb erg model to\nthe description of certain ferrimagnetic materials\nHassan Chamati chamati@issp.bas.bg\nand Diana V. Shopova sho@issp.bas.bg Corresponding author\nAddress:Institute of Solid State Physics, Bulgarian Academy of Scie nces, 1784 Sofia,\nBulgaria\nKeywords; ferrimagnetism, Landau theory, Heisenberg model, ph ase diagram.\nPACS: 75.10.Dg, 71.70.Gm, 75.50.Gg\nAbstract\nWestudyphenomenologically onthebasisoftwobilinearlyc oupledHeisen-\nbergmodelsthephasediagramofsomeferrimagneticsubstan ces. Calculations\nare performed with the help of Landau energy obtained throug h applying the\nHubbard-Stratonovich transformation to the initial micro scopic Heisenberg\nHamiltonian. Thephase transitions within the model are of s econd order with\nthe emergence of a compensation point at lower temperatures for some values\nof parameters of the system. The main phase is a two-sublatti ce collinear\nferrimagnet but also a metastable non-collinear phase is pr esent within the\nexchange approximation presented here. The numerical resu lts give a detailed\ndescription of temperaturedependenceof magnetization on the strength of in-\ntersublattice interaction and the difference between the effec tive exchanges of\ntwo ferromagnetically ordered sublattices.\n1 Introduction\nFerrimagnets are substances made of various components having different magnetic\nproperties. Thedifferences inmagneticmoments leadtoageometric frustrationthat\nmay arise because either different elements occupy the lattice sites or the same ele-\nment occupies nonequivalent crystallographic sites surrounded by adifferent number\nor type of non-magnetic ions, which effectively results in different ma gnetic prop-\nerties. For complex alloys a combination of both may take place (For a n extensive\nreview see Ref. [1] and references therein). Within mean-field appr oach it is gen-\nerally accepted that ferrimagnets can be modeled with the help of se veral interpen-\netrating sublattices each ordered ferromagnetically with effective antiferromagnetic\ncoupling between them. In the pioneering works of N´ eel on ferrima gnetism within\n1the molecular field approach (see e.g. [2]), a two sublattice model is u sed to compute\nthe thermal magnetization behaviour of ferrimagnets, and six pos sible magnetiza-\ntion curves were derived. Special attention there is paid to iron gar nets, where the\nspontaneous magnetization in comparison with experiment can be int erpreted by\napplying a three-sublattice model. In view of experimental study of magnetocaloric\neffect of rare-earth based ferrimagnets [3] which has great pote ntial for technologi-\ncal applications in environmentally-friendly refrigeration, the theo retical mean-field\ndescription of such alloys with three sublattices, is further elabora ted [4]. Such stud-\nies are based on considering microscopic classical Heisenberg models with different\nexchange and spin-orbit interactions depending on the crystal st ructure, chemical\ncomposition of the particular alloy under study.\nThere is another theoretical mean field approach based on conside ring mixed spin\nIsing model fordescription of ferrimagnets, see for example [5, 6, 7], where a very de-\ntailed review of the literature on this approach is presented. In the present paper we\nwill consider ferrimagnets which can be described by different magne tic ions sitting\non two interpenetrating sublattices in a body centred cubic struct ure. The interac-\ntion of ions on each sublattice is supposed ferromagnetic, while ions o n the different\nsublattices are coupled antiferromagnetically. The magnetic prope rties will be inves-\ntigated on the basis of bilinearly coupled Heisenberg classical model in a mean-field\napproximation which is treated using the Hubbard-Stratonovich tr ansformation for\nobtaining the respective Landau free energy and its analysis.\nThe rest of the paper is organized as follows: in Section 2 we describe in detail how\nwe calculate the Landaufreeenergy fromclassical Heisenberg mod el with competing\ninteractions on the basis of previously applied approach [8] for deriv ation of mean\nfield approximation. In Section 3 the solutions of equations of state obtained by\nthe minimization of Landau energy derived in Section 1 are discussed. Section 4\nsummarizes the results both in strong and weak-coupling limit for fer rimagnetic\nsubstances under consideration. Section 5 generalizes the conclu sions and possible\nfurther development of our study.\n2 The model and derivation of Landau free en-\nergy\nThe microscopic Heisenberg Hamiltonian which describes two coupled s ubsystems\nconsisting of classical spins with different magnetic properties which antiferromag-\n2netically between them through a bilinear term can be written in the fo llowing form\n:\nH=−1\n22N/summationdisplay\nij/bracketleftBig\nJ(1)\nijS(1)\ni·S(1)\nj+J(2)\nijS(2)\ni·S(2)\nj+2KijS(1)\ni·S(2)\nj/bracketrightBig\n.(1)\nHereS(1,2)\nj, aren-component classical Heisenberg spins whose magnitude is nor-\nmalized on the unit sphere in spin space through the condition |S(1,2)\nj|= 1. The\nexchange parameters J(1,2)\nij,Kijin the general case are N×Nsymmetric matrices\nwithN- the number of lattice sites considered equal for both subsystem s. This\ncondition simplifies the consideration as makes the system symmetric with respect\nto the interchange of the subsystems. The exchange matrices J(1,2)\nijdenote the inter-\naction between magnetic atoms of the same sort and Kij- between magnetic atoms\nof different sorts.\nThe above Hamiltonian may be applied to the description of magnetic sy stems\nwhich consist of two different magnetic materials and no matter what is the mi-\ncroscopic origin of this difference, it is effectively described by differe nt exchange\ninteractions within the two subsystems. There may be other situat ion when the\nsubstance is made only of one type of magnetic ions but they occupy two different\ncrystallographic positions in the Bravais lattice and are separated b y a number of\nnon-magnetic atoms. Such substance may also be considered as bu ilt of two mag-\nnetic subsystems with different exchange interactions within them.\nIn order to analyse the behaviour of magnetization and the phase t ransitions in sys-\ntems that can be described by the above microscopic Hamiltonian we h ave to find\nthe mean-field energy for the Hamiltonian (1) by calculating the part ition function\nwhich in this case is represented by functional integral in n-dimensio nal spin space,\nwheren- is the number of spin components. To do this we apply the Hubabrd\n-Stratonovich transformation; see for example [10] and the pap ers cited therein. We\nhave used this approach in [8] for ferromagnetic coupling between the two magnetic\nsubsystems where a detailed description of procedure is given. Her e we will just\noutline the important steps in the derivation of Landau free energy , especially in\nrelation of antiferromagnetic coupling between the subsystems.\nWe present the two interacting different magnetic subsystems on a body-centered\ncrystal lattice, for which the corners of elementary cube are occ upied by one sort of\nmagnetic atoms, and at the center of the cube atoms of different m agnetic sort are\nlocated. So the nearest neighbours belong to different magnetic su bsystems and the\nnext-nearest neighbour tosubsystems 1 and2, respectively. Th us thesystem maybe\ndescribed as two interpenetrating sublattices,consisting of differe nt magnetic atoms\nand we assume that the interaction within the sublattices J1,2\nijis ferromagnetic and\n3between them, Kij, it is antiferromagnetic. The Hubbard-Stratonovich transfor-\nmation renders the initial microscopic Hamiltonian in new n-component variables\nΨ(1)\ni,Ψ(2)\nidefined in real space, directly connected with the initial spins (see [8 ]),\nnamely:\nH=1\n22N/summationdisplay\nij/parenleftBig\nJ(1)\nijΨ(1)\ni·Ψ(1)\nj+J(2)\nijΨ(2)\ni·Ψ(2)\nj+2KijΨ(1)\ni·Ψ(2)\nj/parenrightBig\n(2)\n−ln/bracketleftBigg2N/summationdisplay\niIn/2−1(x(1)\ni)(x(1)\ni)\n2)−n/2Γ(n\n2)/bracketrightBigg\n−ln/bracketleftBigg2N/summationdisplay\niIn/2−1(x(2)\ni)(x2\ni\n2)−n/2Γ(n\n2)/bracketrightBigg\n.\nHereIn/2−1(x(1,2)\ni) is the modified Bessel function, and Γ(n\n2) is the Gamma function.\nIn the above expression the exchange parameters J(1,2)\nijandKijare connected to\nthose in the initial Hamiltonian (1) by the relations:\nJ(1,2)\nij=J(1,2)\nij\nT\nKij=Kij\nT(3)\nwithT- the temperature.\nWe denote by x(1)\niandx(2)\niin (2) the following expressions:\nx(1)\ni=/vextendsingle/vextendsingle/vextendsingleJ(1)\nijΨ(1)\nj+KijΨ(2)\nj/vextendsingle/vextendsingle/vextendsingle;x(2)\ni=/vextendsingle/vextendsingle/vextendsingleJ(2)\nijΨ(2)\nj+KijΨ(1)\nj/vextendsingle/vextendsingle/vextendsingle\nThe terms containing Bessel functions in (2) will be further used on ly in the form\nof expansion with respect to x(1,2)\niup to forth order by using the relation:\nΓ(n\n2)(2\nx)(n/2−1)In/2−1(x) = 1+∞/summationdisplay\nk=1(x2/4)k\nk!(k+n/2+1)(k+n/2−2)...n/2\nThe next step is to perform Fourier transformation to k-space , and pass to con-\ntinuum limit in kas the finite size effects will not be considered at this stage. The\nquadratic part of the obtained Hamiltonian again contains a bilinear te rm with re-\nspect toΨ(1,2)(k) andwe have todiagonaliseit. This isdonewiththehelp ofunitary\nmatrixˆS:\nˆS=/parenleftBigg\nS0(k)−S1(k)\nS∗\n1(k)S0(k)/parenrightBigg\n. (4)\nThe eigenvalues of the matrix ˆSread:\nλ1,2(k) =1\n2/bracketleftBig\nJ1(k)+J2(k)±/radicalbig\n(J1(k)−J2(k))2+4K(k)2/bracketrightBig\n, (5)\n4whereJ1,2(k) andK(k) are the Fourier transforms of J(1,2)\nijandKij, respectively. In\norder to compute the integral we use the steepest-descent met hod,i.e.the integra-\ntion contour is taken around the maxima of the eigenfunctions (5). The calculation\nfor bcc structure show that if we take the nearest neigbour inter action between\natoms of the same sort and the nearest neighbour interaction bet ween the atoms of\ndifferent sort, λ1,2(k) has a maximum in the centre of the Brillouin zone that gives\nferromagnetic ordering for the sublattices with antiferromagnet icK <0 interaction\nbetween them which is the focus of our consideration below. We shou ld note that\nλ1,2(k) has a maximum also at the border of the Brillouin zone k=π/a(ais the\nlattice constant) which supposes antiferromagnetically ordered s ublattices. There\nmay be also some local maximum inside the Brillouin zone, which may give so me\nincommensurate ordering within the sublattices, but this case is bey ond the scope\nof the present study.\nAfter performing the reverse Fourier transform to real space w e obtain the dimen-\nsionless Landau energy in the following form:\nF\nT=t1\n2− →ψ2\n1+t2\n2− →ψ2\n2+g\n4/bracketleftBig\n(− →ψ2\n1)2+(− →ψ2\n2)2/bracketrightBig\n+b\n2− →ψ2\n1− →ψ2\n2+b(− →ψ1·− →ψ2)2+w(− →ψ2\n2−− →ψ2\n1)(− →ψ1·− →ψ2).(6)\nThe coefficients of the Landau energy are expressed by the compo nents of the ma-\ntrix (4) and its eigenvalues (5) for k= 0:\nS0=1\nD/parenleftBig\nJ1−J2+/radicalbig\n(J1−J2)2+4K2/parenrightBig\n,\nS1=2K\nD(7)\nwhereDis introduced to satisfy the condition /bardblˆS/bardbl= 1, namely S2\n0+S2\n1= 1:\nD=√\n2/bracketleftbig\n(J1−J2)2+4K2/bracketrightbig1/4/bracketleftBig\nJ1−J2+/radicalbig\n(J1−J2)2+4K2/bracketrightBig1/2\n.(8)\nWe will write here the explicit expressions for the coefficients of landa u energy as we\nwill need them further in solving the mean field equations and discussio n of obtained\nresults;\nt1,2=1\nλ1,2−1\nn(9a)\ng=u\n2(S4\n0+S4\n1) (9b)\nb=uS2\n1S2\n0 (9c)\nw=u\n2S0S1(S2\n0−S2\n1); (9d)\n5hereu=n2(n+2) withn- the number of order parameter components. The real\nvector fields− →ψ1and− →ψ2in the Landau free energy (6) play the role of two coupled\norder parameters, and the averaged sublattice magnetizations a re related to them\nby the equations:\n− →m1=S0\nλ1− →ψ1−S1\nλ2− →ψ2 (10a)\n− →m2=S1\nλ1− →ψ1+S0\nλ2− →ψ2 (10b)\n3 Solving mean-field equations\nThe initial microscopic Hamiltonian is symmetric with respect to the rot ation of\nall spins through the same angle. The application of Hubbard-Strat onovich trans-\nformation for derivation of landau free energy , given in previous se ction, preserves\nthe symmetry of initial hamiltonian also with respect to field variables− →ψ1,2which\nmeans that we can find the magnitude and the mutual orientation be tween order pa-\nrameters− →ψ1,2but not their orientation with respect to crystallographic axes. Th is\nmaybedoneforparticularmagneticsubstancebyincluding intheinitia lmicroscopic\nHamiltonian terms accounting for the magnetic anisotropy. For pur e exchange inter-\nactions we can introduce the following notations [8]:− →ψ1i=|− →ψ1|βiand− →ψ2i=|− →ψ2|δi,\nwhere|− →ψ1|=ψ1,|− →ψ2|=ψ2are the magnitudes of the vector fields, and βi, δiare\nthe respective direction cosines, which fulfil the condition:\n3/summationdisplay\ni=1β2\ni= 1 and3/summationdisplay\ni=1δ2\ni= 1. (11)\nThe equations of state then will be:\n∂f\n∂Xi= 0,whereXi={ψ1,ψ2,βi,δi} (12)\nSolvingtheaboveequationswithrespect todirectioncosines βi, δigives twopossible\norientations between the vector fields− →ψ1,− →ψ2forK <0:\n1. The collinear phase with/summationtext\niβiδi=−1, that is,− →ψ1and− →ψ2are antiparallel, and\n2. Thenon-collinearphasewith/summationtext\niβiδi= 0, thatis,− →ψ1and− →ψ2areperpendicular.\nBelow we will discuss in detail the non-collinear phase 2. The angle betw een the\norder parameters− →ψ1and− →ψ2,i.e.,is;\n3/summationdisplay\niβ1δi=−w(ψ2\n2−ψ2\n1)\n2bψ1ψ2\n6and is defined only when ψ1/ne}ationslash= 0 andψ2/ne}ationslash= 0. ForK <0, the analysis shows\nthat the non-collinear phase exists only when the order parameter s− →ψ1and− →ψ2are\nof equal magnitudes, meaning that the order parameters− →ψ1and− →ψ2are mutually\nperpendicular. The magnitude ψ=ψ1=ψ2for the non-collinear phase in analytical\nform reads:\nψ2=−t1+t2\nu. (13)\nThen the sublattice magnetization magnitudes calculated using the a bove expres-\nsions for the non-collinear phase will be:\n|− →m1|=ψ/radicalBigg\nS2\n0\nλ2\n1+S2\n1\nλ2\n2, (14a)\n|− →m2|=ψ/radicalBigg\nS2\n1\nλ2\n1+S2\n0\nλ2\n2. (14b)\nNote that the sublattice magnetizations are not perpendicular but form an angle\n∠(− →m1,− →m2) =γwith each other, expressed by\ncos(γ) =S0S1(λ2\n1−λ2\n2)/radicalbig\n(S2\n0λ2\n2+S2\n1λ2\n1)(S2\n1λ2\n2+S2\n0λ2\n1).\nThe calculations show that this non-collinear phase for K <0 within the exchange\napproximation has no domain of stability. We should mention here that the free\nenergy (6) is very sensitive to the sign of interaction Kbetween the sublattices.\nWhenK >0,i.e., the interaction between the sublattices is ferromagnetic there is\nsmall domain in which the respective non-collinear phase is stable [9].\nFor antiparallel− →ψ1and− →ψ2it is obvious that the sublattice magnetizations (10) will\nbe also antiparallel. We may write the resulting equations for the magn itudes of the\norder parameters ψ1andψ2of the collinear phase and K <0 in the following form:\nt1ψ1+gψ3\n1+3bψ1ψ2\n2−wψ2(ψ2\n2−3ψ2\n1) = 0, (15a)\nt2ψ2+gψ3\n2+3bψ2\n1ψ2−wψ1(3ψ2\n2−ψ2\n1) = 0, (15b)\nwith the stability conditions given by:\nt1+3gψ2\n1+3bψ2\n2+3wψ1ψ2>0 (16)\n(t1+3gψ2\n1+3bψ2\n2+6wψ1ψ2)(t2+3gψ2\n2+3bψ2\n1−6wψ1ψ2)\n−9[w(ψ2\n1−ψ2\n2)+2bψ1ψ2]2≥0 (17)\nWe will make some remarks on the dependence of solutions of above s ystem on\nthe magnitude of exchange parameters J1, J2andK. WhenJ1<|K|,J2<|K|,\n7the leading interaction is determined by the antiferromagnetic coup ling between the\ntwo sublattices. This may be called a strong coupling limit for which the e igenvalue\nλ2(k= 0)\nλ2=1\n2/bracketleftBig\nJ1+J2−/radicalbig\n(J1−J2)2+4K2/bracketrightBig\n, (18)\nbecomes negative. This isequivalent totheinequality K2−J1J2>0. The coefficient\nt2in front ofψ2\n2becomes positive; see (9), and the field− →ψ2becomes redundant. The\nLandau free energy (6) will be:\n(F\nT)s=fs=t1\n2− →ψ12+g\n4(− →ψ12)2(19)\nThe minimization of above equation gives for− →ψ1the solution:\n(− →ψ1)2=−t1\ng(20)\nwhich exists and is stable for t1<0.\nThe sublattice magnetizations:\n− →m1=S0\nλ1− →ψ1 (21)\n− →m2=S1\nλ1− →ψ1\nwill be antiparallel as S1∼K/DandK <0. The phase described by the above\nequations will be presented by two antiparallel sublattices with differ ent magnitudes\nof sublattice magnetizations.\nIn the weak coupling limit for antiparallel configuration, i.e., whenJ1>|K|,J2>\n|K|, or equivalently J1J2> K2, the system of equations (15), together with the\nstability conditions (18), (19) should be solved. This can be done num erically and\nthe results will be presented in the next section.\n4 Results and discussion\nThe analytical result for sublattice magnetizations in the limiting case of strong\ncoupling (21) gives for the magnitude of total magnetization |− →M|=|− →m1+− →m2|the\nfollowing expression:\n|− →M|=S0|ψ1|\nλ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+S1\nS0/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nwith|ψ1|2, given by (20). The phase transition is obviously of second order an d\nthe total magnetization behaviour with temperature is smooth res embling the one\n8of Weiss ferromagnet with the exception that no saturation is reac hed forT= 0.\nAccording to the Neel’s classification of ferrimagnets, see [2], the c hange of magneti-\nzation with temperature in the strong coupling limit falls within R-type c urve. For\nexample, similar curve is obtained theoretically and compared with the respective\nexperiment for Y 3Fe5O12[4] where two sublattice model with strong antiferromag-\nnetic coupling is considered.\nIn the limiting case when J1=J2=Jthe relation will hold S0=−S1= 1/√\n2 and\nan antiferromagnetic structure with− →m1=−− →m2will appear, only if− →ψ2≡0 and\n|− →ψ1|2=−2t1/u. The transition temperature for antiferromagnetic ordering will b e\ngiven by:ta\nc= (J+|K|)/n.\nFurther we will present the numerical results for the temperatur e dependence of\nsublattice magnetizations and the total magnetization of the syst em in the weak-\ncoupling limit which we define here in the following way: J1>|K|andJ2>|K|.\nSuch a situation is present, for example in some ferrimagnetic compo unds like\nGdCo 12B6[11]. It is experimentally found that the exchange constants within s ub-\nlattices are ferromagnetic and larger than the antiferromagnetic coupling between\nthe sublattices; moreover there the magnetic anisotropy is small.\nExperiments for some R-T compounds where R is a rare earth elemen t and T is\na transition element, show that the exchange in the transition meta l sublattice is\nleading in magnitude, while the exchange in the rare-earth ion sublatt ice can be\nsafely ignored and considered as negligible. The intersublattice inter action is also\nsmall see, for example, DyFe 5Al7[12], ErFe 11TiH [13], RCo 2(R = Tb and Gd and R\n= Er, Ho, and Dy) [14]. In our notations the relation between the exc hange integrals\nin this case will be J1>|K| ≫ J 2, so this does not fall into our assumption of weak\ncoupling and will not be considered here.\nIn order to solve numerically the equations of state (15) for weak c oupling between\nsublattices we introduce the following dimensionless parameters.\nt=T\nJ1+J2, (22a)\nα=J1−J2\nJ1+J2, (22b)\nβ=K\nJ1+J2, (22c)\nwitht– the dimensionless temperature. In the above expression we have supposed\nthatJ1>J2which in view of symmetry in interchanging the sublattices does not\nlimit the consideration; then α>0 andβ <0 asK<0.\n9The weak coupling between the sublattices, namely J1>|K|andJ1>|K|may be\nexpressed by the parameters from (22) by the relation:\nα2+β2<1.\nThe parameter αis a measure for the difference in exchange parameters of the two\nsublattices and by its definition 0 <α<1.\nThe total magnetization of the system is the sum of sublattice magn etizations (10):\n− →M=− →m1+− →m2=S0+S1\nλ1− →ψ1+S0−S1\nλ2− →ψ2. (23)\nHereafter we will use the following notations for magnitudes of subla ttice magneti-\nzations and total magnetization both in the text and in figures:\nm1=|− →m1|;m2=|− →m2|;andM=|− →M|\nThecalculationsshowthatthephasetransitiontoorderedferrima gneticstateoccurs\nat temperature :\ntc=1\n6(1+/radicalbig\nα2+β2)\nwhich grows when either the difference between the exchange inter actions in sublat-\ntices grows, or when the antiferromagnetic coupling is bigger, or bo th. The phase\ntransition from disordered to ordered phase is of second order.\nWe want to note that within the exchange approximation used here f or the regime of\nweak coupling defined above with the decrease of temperature a co mpensation point\nappears no matter how small is the difference between the exchang e interactions of\nsublattices. At the compensation temperature tcomp, the sublattice magnetizations\n− →m1and− →m2areequalinmagnitudeandantiparallel, so M= 0. Therelationbetween\nthe order parameters magnitudes there is defined by:\nψ1=λ1\nλ2(S0−S1)\n(S0+S1)ψ2. (24)\nAs the calculations show ψ2< ψ1for all values of αandβ, butψ1grows with the\ndecrease of temperature in a monotonic way, while ψ2grows more rapidly. The\nquantity\nλ1\nλ2(S0−S1)\n(S0+S1)=(6tc)2\n1−α2−β2/parenleftBigg/radicalbig\nα2+β2−β\nα/parenrightBigg\nis always>1 asβ <0∼Kandα >0 so at some temperature tcomp< tc,\nand values of ψ1, ψ2the condition (24) is fulfilled. In the following figure we\nshow the change of net magnetization magnitude M(t) with the temperature for\n100.000.020.040.06\n0.00 0.05 0.10 0.15 0.20α = 0.1M\ntβ = - 0.08\nβ = - 0.11\nβ = - 0.13\nβ = - 0.20\nFigure 1: The dependence of net magnetization Mon reduced temperature tfor for\nfixedαand the different values of antiferromagnetic coupling β\n.\nα= 0.1,i.e.,J2= 0.83J1and different values of β. It is seen from Fig.1 that\nthe increase of antiferromagnetic coupling between the sublattice s slightly shifts the\ncompensation temperature to higher values, and M(t) grows more rapidly below\nthe compensation temperature and reaches higher values as t−→0. We suppose\nthat within the exchange approximation and in weak coupling limit the ke y factor\nfor the compensation point to appear is the weakness of antiferro magnetic exchange\nbetween the sublattices compared to the ferromagnetic exchang e of sublattices 1\nand 2 , respectively.\nWe will discuss in more detail the influence of difference between the m agnitudes\nof exchange interaction in the sublattices, represented by the pa rameterαonM(t)\nand sublattice magnetizations m1,andm2. Forα= 0.08,i.e.,J2= 0.85J1,M(t),\nm1,andm2are shown in Fig. 2. At tcthe transition is of second order and when\nlowering the temperature a compensation point tcompappears, which is located close\ntotc. Sublattice magnetizations change with temperature in monotonic w ay, and in\nthe temperature interval tcomp<t<t c, the relation between sublattice magnetiza-\ntions ism1>m2, as expected as in sublattice 1 the exchange interaction J1>J2.\nBelowtcompthe magnetization of weaker sublattice m2becomes bigger than m1.\nFor intermediate values of α= 0.4,orJ2= 0.43J1, see Fig.3, with decrease of\n110.000.050.100.150.200.25\n0.00 0.05 0.10 0.15 0.20α = 0.08, β = -0.3\ntc tcomp.\ntm1m2M\nFigure 2: The dependence of net magnetization Mand sublattice magnetizations\nm1andm2on reduced temperature tfor for small difference between sublattice\nexchange parameters\n.\ntemperature below the compensation point the total magnetizatio n rapidly grows in\nnon-monotonic way as t→0, similar to V-curve according to Neel’s classification.\nThe behaviour of sublattice magnetizations with temperature is quit e different; m1\n, which is the sublattice magnetization with stronger exchange inter actionJ1grows\nin smooth way with decrease of temperature, while m2for weaker sublattice inter-\nactionJ2, below compensation point grows drastically in non-monotonic way an d\nin the temperature interval 0 <t<t comp,m2≫m1.\nSuch behaviour is described in detail in [15] for ferrimagnets with com pensation\npoint for many experimentally observed substances. There explan ation ofM(t)\nbehaviour below compensation point is done by introducing the notion of weak sub-\nlatticewheredepending ontheparticular substance considered diff erent mechanisms\nfor explanation of this effect are pointed out. Within our exchange m odel the effect\nof weak sublattice is readily seen and mainly depends on the difference J1−J2∼α.\nWhenαfurther grows the compensation point is shifted to lower temperat ures and\nthe magnetization m1of the stronger sublattice decreases for t→0. This is illus-\ntrated in Fig.4 for J2= 0.19J1.\n120.00.10.20.30.4\n0.00 0.05 0.10 0.15 0.20 0.25α = 0.4, β = -0.095\ntc tcomp.\ntm1m2M\nFigure 3: The dependence of net magnetization Mand sublattice magnetizations\nm1andm2on reduced temperature tfor intermediate difference between sublattice\nexchange parameters\n.\nA qualitatively similar behaviour of the total magnetization can be see n in the\nexperiments with ErFe 2[16] and GdCo 12B6[11] althoughdirect comparison with the\nexperimental curves can hardly be made, as these substances ha ve crystallographic\nand magnetic structure that differs from the one assumed within ou r model.\nThe influence of parameter αis summarized in the next figure, see Fig. 5, where\nthe net magnetization is displayed for small values of β= 0.08 and different values\nof parameter α. As the difference between the magnitudes of sublattice exchange\ninteractions grow the transition temperature is shifted to higher v alues as expected\nand the compensation temperature is lowered. Another effect is th e change of net\nmagnetization behaviour below tcompfrom monotonic to nearly exponential when α\nincreases.\n5 Conclusions\nOur mean-field analysis of this relatively simple model with competing int eractions\non a bcc lattice shows that the behaviour of net magnetization of th e two-sublattice\nferrimagnet depends essentially on the difference in magnetic intera ctions between\n130.00.20.40.60.8\n0.000.050.100.150.200.250.30α = 0.7, β = -0.1\ntc\ntm1m2M\ntcomp.\nFigure 4: Illustration of the dependence of net magnetization Mand sublattice\nmagnetizations m1andm2onreducedtemperature twhentheexchangeinsublattice\n2 is very small compared to sublattice 1\n.\nthe sublattices within the weak coupling limit presented here. The influ ence of anti-\nferromagnetic coupling is more prominent when its magnitude is of the same order,\nor larger than the difference between the exchange parameters o f the sublattices.\nThe model may be generalized to include the expansion of the effectiv e Hamiltonian\nup to sixth-order terms in xi, i= 1,2 in order to consider the possibility of first\norder phase transition to ferrimagnetic state due to the influence of external param-\neters as pressure and change of concentration.\nWithin our approach the parameters of Landau energy can be direc tly related to\nthe averaged microscopic exchange interactions as described in se ction 2 , and this\nrelation is quite simple in the case of high symmetry crystal structur e as considered\nhere. It will be of interest also to include the influence of an externa l magnetic\nfield and to perform calculations for ferrimagnets, for which the re lation between\nexchange interactions fulfills the condition J1>|K| ≫ J 2, that is, when one of the\nsublattices is very weak.\n140.00.20.40.6\n0.000.050.100.150.200.250.30β = - 0.08M\ntα = 0.1\nα = 0.4\nα = 0.6\nα = 0.7\nFigure 5: The dependence of net magnetization Mfor fixed antiferromagnetic ex-\nchangeβand growing difference αbetween the ferromagnetic exchanges of the two\nsublattices\n.\nAcknowledgments\nThis work was supported by the Bulgarian National Science Fund und er contract\nDN08/18 (14.12.2017).\nReferences\n[1] R. Skomski, Simple models of magnetism, Oxford\nGraduate Texts, Oxford University Press, NY, 2008.\ndoi:10.1093/acprof:oso/9780198570752.001.0001 .\n[2] L. N´ eel, Magnetism and Local Molecular Field, Science 174 (1971) 985.\ndoi:10.1126/science.174.4013.985 .\n[3] A. S. Andreenko, K. P. Belov, S. A. Nikitin, A. M. Tishin, Magnetoc aloric\neffects in rare-earth magnetic materials, Sov. Phys. Usp. 32 (8) ( 1989) 649.\ndoi:10.1070/PU1989v032n08ABEH002745 .\n15[4] P. J. von Ranke, B. P. Alho, E. J. R. Plaza, A. M. G. Carvalho, V. S . R.\nde Sousa, N. A. de Oliveira, Theoretical investigation on the magnet ocaloric\neffect in garnets R 3Fe5O12where (R = Y and Dy), J. Appl. Phys. 106 (2009)\n053914.doi:10.1063/1.3213383 .\n[5] T. Kaneyoshi, J. Chen, Mean-field analysis of a ferrimagnetic mixe d spin\nsystem, Journal of Magnetism and Magnetic Materials 98 (1991) 20 1.\ndoi:10.1016/0304-8853(91)90444-F .\n[6] J.W. Tucker, The ferrimagneticmixed spin-1\n2andspin-1 Ising system, J. Magn.\nMagn. Mater. 195 (1999) 733. doi:10.1016/S0304-8853(99)00302-9 .\n[7] F. Abubrig, Magnetic properties of a mixed-Spin-3/2 and spin-2 I sing ferri-\nmagnetic system in an applied longitudinal magnetic field, World J. Cond ens.\nMatter Phys. 03 (2013) 111. doi:10.4236/wjcmp.2013.32018 .\n[8] D. V. Shopova, T. L. Boyadjiev, Mean field analysis of two coupled Heisenberg\nmodels, J. Phys. Stud. 5 (2001) 341\n[9] D.V. ShopovaandT. L.Boyadjiev, Theexistence ofastablenon collinear phase\nin Heisenberg model with complex structure, Phys. Lett. A 311 (20 03) 438\n[10] R. Brout, Phys. Rep.10 , 1 (1974)\n[11] O. Isnard, Y. Skourski, L. V. B. Diop, Z. Arnold, A. V. Andreev , J. Wosnitza,\nA. Iwasa, A. Kondo, A. Matsuo, K. Kindo, High magnetic field study o f the\nGd-Co exchange interactions in GdCo 12B6, J. Appl. Phys. 111 (2012) 093916.\ndoi:10.1063/1.4710995 .\n[12] D. Gorbunov, A. Andreev, N. Mushnikov, Magnetic properties\nof a DyFe 5Al7single crystal, J. Alloys Comp. 514 (2012) 120.\ndoi:10.1016/j.jallcom.2011.11.020 .\n[13] N. V. Kostyuchenko, A. K. Zvezdin, E. A. Tereshina, Y. Skour ski, M. Do-\nerr, H. Drulis, I. A. Pelevin, I. S. Tereshina, High-field magnetic beh avior and\nforced-ferromagnetic state in an ErFe 11TiH single crystal, Phys. Rev. B 92,\n104423 (2015) doi:10.1103/PhysRevB.92.104423 .\n[14] E. Z. Valiev, A. E. Teplykh, Magnetic properties of RCo 2compounds in the\nexchange-striction model of ferrimagnets, Phys. Metals Metallog r. 118 (2017)\n21.doi:10.1134/S0031918X16120164 .\n16[15] Belov K P, Ferrimagnets with a ’weak’ magnetic sublattice, Phys. Usp. 39\n(1996)623634\n[16] I. Chaaba, S. Othmani, S. Haj-Khlifa, P. de Rango, D. Fruchar t,\nW. Cheikhrouhou-Koubaa, A. Cheikhrouhou, Magnetic and magnet ocaloric\nproperties of Er(Co 1−xFex)2intermetallic compounds, J. Magn. Magn. Mater.\n439 (2017) 269. doi:10.1016/j.jmmm.2017.05.033 .\n17"    },    {        "title": "1211.0123v1.Spin_Seebeck_effect_in_antiferromagnets_and_compensated_ferrimagnets.pdf",        "content": "arXiv:1211.0123v1  [cond-mat.mtrl-sci]  1 Nov 2012Spin Seebeck effect in antiferromagnets and compensated fer rimagnets\nYuichi Ohnuma,1,∗Hiroto Adachi,2,3Eiji Saitoh,1,2,3,4and Sadamichi Maekawa2,3\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n3CREST, Japan Science and Technology Agency, Sanbancho, Tok yo 102-0075, Japan\n4WPI Research Center, Advanced Institute for Material Resea rch, Tohoku University, Sendai 980-8577, Japan\n(Dated: August 22, 2018)\nWe theoretically investigate the spin Seebeck effect (SSE) i n antiferromagnets and ferrimagnets,\nand show that the SSE vanishes in antiferromagnets but survi ves in ferrimagnets even at the magne-\ntization compensation point despite the absence of its satu ration magnetization. The non-vanishing\nSSE in ferrimagnets stems from two non-degenerate magnons. We demonstrate that the magnitude\nof the SSE in ferrimagnets is unchanged across the magnetiza tion compensation point.\nPACS numbers: 85.75.-d, 72.25.Mk, 75.30.Ds\nI. INTRODUCTION\nMuch attention is now focused on the thermal effects\nin spintronics, and the emergent research field of spin\ncaloritronics is rapidly developing.1,2One of the most\nimportant issues in spin caloritronics is the spin Seebeck\neffect (SSE).3The SSE is the mechanism by which a\nspin voltage is generated from a temperature gradient\nin a magnetic material over a macroscopic scale of sev-\neral millimeters.4Because the spin voltage is a potential\nfor electron spins to drive spin currents, this spin voltage\ninjects a pure spin current, i.e., a spin polarized current\nwhich is unaccompanied by a charge current, from the\nferromagnet into an attached nonmagnetic metal. The\ninverse spin Hall effect (ISHE)5,6converts the injected\nspin current into a transverse electric voltage and hence\nthe SSE is electrically detectable. Since its discovery in\n2008,this phenomenonhasdrawnmuchinterestasasim-\nple way of generating pure spin currents that are needed\nfor future spin-based technology,7,8and the recent obser-\nvation of the giant SSE in InSb9has attracted a consid-\nerable attention.\nThe SSE has been observed in various ferro-\nmagnetic materials ranging from metallic ferromag-\nnets, Ni 81Fe193and Co 2MnSi,10to semiconduct-\ning ferromagnet (Ga,Mn)As,11,12to insulating mag-\nnets LaY 2Fe5O1213and (Mn,Zn)Fe 2O4.14Although\nLaY2Fe5O12and (Mn,Zn)Fe 2O4are classified into fer-\nrimagnets in a rigorous terminology, the current under-\nstanding of the SSE in these systems relies on a model-\ningasferromagnets15,16becausethe low-energymagnetic\npropertiesrelevant to the SSE arewell described by a fer-\nromagnet modeling owing to the large gap between the\nacoustic and optical magnons. These observations have\nestablishedtheSSEasauniversalaspectofferromagnets.\nBesides ferromagnets, ferrimagnets and antiferromag-\nnets are known as prototypes of magnetic materials.17A\nferrimagnet is an ordered spin system in which two sub-\nlattice magnetizations point in the opposite directions,\nand an antiferromagnet is classified as a special case of\na ferrimagnet for which both sublattices have equal sat-uration magnetizations. Recently, there has been an on-\ngoing attempt to develop antiferromagnetic metal spin-\ntronics, and several experimental18and theoretical19–21\nwork are already in progress. Regarding ferrimagnets,\nthe intriguing characteristics of ferrimagnetic ordering\nare now drawing considerable attention22,23in develop-\ning a ultrafast magnetization manipulation technique.\nTherefore, it is quite natural to ask whether the SSE\ncan be observed in antiferromagnets and ferrimagnets.\nIn this paper, we address the issue of observing the\nSSE in antiferromagnets and ferrimagnets. Especially,\nwe focus on the SSE in ferrimagnets with magnetization\ncompensation. A certain class of ferrimagnets are known\nto possess a magnetization compensation temperature\nTM(angular-momentum compensation temperature TA),\nat which the two sublattice magnetizations (spins) have\nthe same magnitudes but opposite directions, leading to\nnet-zero saturation magnetization (spin angular momen-\ntum).24–28We show that two non-degenerate magnons\ngive rise to the non-vanishing SSE at TMorTAdespite\nthe absence ofnet saturationmagnetizationortotal spin.\nAlso, we show that for a uniaxial antiferromagnet the\nSSE vanishes because the thermal spin injection by the\ntwodegeneratemagnonsisperfectlycompensated. More-\nover, the SSE in an easy-plane antiferromagnet is shown\ntodisappearbecauseinthis instanceneithermagnoncar-\nries spins.\nThis paper is organized as follows. In Sec. II, we in-\nvestigate the SSE in uniaxial antiferromagnets as well as\nferrimagnets with magnetization compensation. Next, in\nSec. III we discuss the SSE in easy-plane antiferromag-\nnets. Finally, in Sec. IV we summarize and discuss our\nresults.\nII. SPIN SEEBECK EFFECT IN UNIAXIAL\nANTIFERROMAGNETS AND FERRIMAGNETS\nAs a general model of ferrimagnets and antiferromag-\nnets, we consider the following Hamiltonian defined on a2\nFIG. 1: (Color online) Schematic view of a hybrid structure\ncomposed of a nonmagnetic metal ( N) and a ferrimagnet ( F)\nwith two sublattices AandB.\nlattice composed of two sublattices AandB,29\nHF=−JA/summationdisplay\n/angbracketlefti,i′/angbracketright∈ASA,i·SA,i′−JB/summationdisplay\n/angbracketleftj,j′/angbracketright∈BSB,j·SB,j′\n+JAB/summationdisplay\n/angbracketlefti∈A,j∈B/angbracketrightSA,i·SB,j+δHA+δHB,(1)\nwhereJAandJB(JAB) are the nearest-neighbor intra-\nsublattice (inter-sublattice) exchange integrals, and /angbracketleft,/angbracketright\nspecifies nearest neighbor bonding (see Fig. 1). The last\ntwo terms in Eq. (1) for sublattice L=A,Bare given\nbyδHL=/summationtext\ni∈L[gLµ0H0·SL,i−DL\n2(/hatwidez·SL,i)2], where\nµ0is the Bohr magneton, H0=−H0/hatwidezis the external\nmagnetic field, gLandDLare the effective g-factor and\nthe anisotropy constant for sublattice L.\nFirst, we use the spin-waveapproximationto diagonal-\nize Eq. (1). Following standard procedures30using the\nlinear Holstein-Primakoff transformation for spin opera-\ntorsS±\nL,i=Sx\nL,i±iSy\nL,i(L=A,B), the Hamiltonian (1)\nis diagonalized to be\nHF=/planckover2pi1/summationdisplay\nq/parenleftBig\nω+\nqα†\nqαq+ω−\nqβ†\nqβq/parenrightBig\n, (2)\nwhereω±\nq=1\n2/radicalBig\n(ǫAq+ǫBq)2−4η2q±(ǫA\nq−ǫB\nq), and the\nprecise forms of ǫA\nq,ǫB\nq, andηqare given by ǫA\nq=\n2z0JASA[1−γq] +z0JABSB+ (gAµ0H0+DASA) and\nǫB\nq= 2z0JBSB[1−γq]+z0JABSA+(−gBµ0H0+DBSB).\nHere,γq=z−1\n0/summationtext\nδeiq·δis defined by the sum over\nz0nearest neighbors of the original lattice, and ηq=\nJAB√SASB/summationtext\nδ′eiq·δ′is defined by the sum over z0near-\nest neighbors of the sublattice AorB. In this paper,\nwe assume a cubic lattice for simplicity. In Eq. (2),\nthe magnon operators αqandβqare defined by the\nBogoliubov transformation31aq=u+\nqαq+u−\nqβ†\nqand\nbq=u−\nqα†\nq+u+\nqβq, where and aqandbqare the\nFourier transforms of operators ai= (2SA)−1\n2S+\nA,iand\nbi= (2SB)−1\n2S−\nB,iwithSA=|SA|andSB=|SB|, and\nu+\nq2−u−\nq2= 1.\nFIG. 2: (Color online) Spin-wave spectra ( H0= 0) with q\nalong the [111] direction calculated from Eq. (2) using pa-\nrameters for (a) a uniaxial antiferromagnet NiO, and (b) a\ncompensated ferrimagnet Er 3Fe5O12. The wavevector qis\nmeasured in units of the inverse of the nearest-neighbor dis -\ntance.\nIn Fig. 2, the spin-wave spectra ( H0= 0) calculated\nfrom Eq. (2) for a uniaxial antiferromagnet NiO and a\ncompensated ferrimagnet Er 3Fe5O12are plotted. For\nNiO, we use JAB= 6.3 meV ( JA=JB= 0),D=\n0.1 meV,SA=SB= 0.92,28,32whereas for Er 3Fe5O12,\nwe assign the net spin of the rare-earth ions (the fer-\nric ions) to SA(SB) on a model cubic lattice, and we\nsetJA= 0 meV, JB= 0.68 meV, JAB= 0.19 meV,\nSA= 4.2,SB= 2.5,gA= 1.4,gB= 2.0DA= 3.5×\n10−3meV, and DB= 3.0×10−4meV to reproduce the\nN´ eel temperature TN´ eel= 556 K and the magnetization-\ncompensation temperature TM= 83 K.28,33As is well\nknown, the two antiferromagnetic magnons are degener-\nate ifH0= 0, whereas the two ferrimagnetic magnons\nare non-degenerate because of the inequivalence of the\ntwo sublattices.\nWe discuss now the SSE in uniaxial antiferromagnets\nand ferrimagnets modeled by Eq. (1). Note that a uniax-\nial antiferromagnet can be modeled as a special case of\na ferrimagnet. We consider a model shown in Fig. 1, in\nwhich a ferrimagnet ( F) and a nonmagnetic metal ( N)\nare interacting weakly through the s-dexchange interac-\ntion at the interface. We assume that the ferrimagnet F\nhas a local temperature TF, and the nonmagnetic metal\nNhas a local temperature TN. We analyze the SSE in\nthe longitudinal configuration34by employing the linear-\nresponse formulation of the SSE in a ferromagnet de-\nveloped in Refs. 15 and 35. The s-dinteraction at the\ninterface is modeled by\nHsd=/summationdisplay\ni,j∈F/N-interface/parenleftBig\nJA\nsdσi·SA,i+JB\nsdσj·SB,j/parenrightBig\n,(3)\nwhere, for sublattice L=A,B,JL\nsdis thes-dexchange\ninteraction at the F/Ninterface, σiis the itinerant spin\ndensity operator in N. The total Hamiltonian of the\nsystem,H, is then given by\nH=HF+HN+Hsd, (4)3\nFIG. 3: (Color online) Feynman diagram representing two\nprocesses relevanttotheSSEinuniaxialantiferromagnets and\nferrimagnets. (a) Spin current injected by αqmagnons ( I+\ns).\n(b) Spin current injected by βqmagnons ( I−\ns). The signs of\nI+\nsandI−\nsare opposite. The solid and wavy lines represent\nmagnon and itinerant spin-density propagators, respectiv ely.\nwhereHNis the single-particle Hamiltonian of the con-\nduction electrons in N(see, e.g., Eq. (31) in Ref. 36).\nThe central quantity that characterizes the SSE is the\nspin current Isinjected into N, because it is proportional\nto the experimentally detectable electric field EISHEvia\nISHE:5,6\nEISHE=θSHρJs×σ, (5)\nwhereθSHandρare respectively the spin-Hall angle and\nthe resistivity of N, andJs= (Is/Aint)/hatwidexis the spin-\ncurrent density across the F/Ninterface having a con-\ntact area Aint. Following Refs. 15 and 35, we calculate\nIsas the rate of change of the spin accumulation in N,\ni.e.,Is=/summationtext\ni∈N/angbracketleft∂tσz\ni/angbracketrightwhere/angbracketleft···/angbracketrightdenotes the statisti-\ncal average. What is special in the present calculation\nis that we need to express the s-dinteraction [Eq. (3)]\nin terms of the αqandβqoperators [“ ±” branches in\nEq. (2)], because these are the magnon operators in F.\nFollowing procedures presented in Appendix A, the spin\ncurrent injected in Nis expressed as\nIs=−2√\n2√NNNF/planckover2pi1Re/summationdisplay\nk,q/integraldisplay\nω/bracketleftBig\nJ+\nsd(k,q)AK\nk,q(ω)\n+J−\nsd(k,q)BK\nk,q(ω)/bracketrightBig\n, (6)\nwhereAK(BK) is the Keldysh component of the inter-\nface correlation function between magnon operator αq\n(β†\nq) and the itinerant spin-density operator σ−\nk(see Ap-\npendix A), and we have introduced the shorthand no-\ntation/integraltext\nω=/integraltext∞\n−∞dω\n2π. HereJ±\nsd(k,q) is the effective s-d\ninteraction written in terms of magnon operators, and\nthe precise definition is given in the Appendix A.\nWe perform the perturbative approach in term of the\ns-dinteraction at the interfaceto evaluate Eq. (6). Then,\nthe spin current Isis given by the two diagrams shown\nin Fig. 3, and accordingly, Ishas two terms:\nIs=I+\ns+I−\ns, (7)\nFIG. 4: (Color online) Temperature dependence of the SSE\nsignalIs[red, Eq. (7)], saturation magnetization Ms[blue,\nEq. (9)], and total angular momentum Stot[green, Eq. (10)],\ncalculated for a compensated ferrimagnet Er 3Fe5O12using\nthesameparametersasinFig.2(b). Thecase fora Mspinned\nby the anisotropy field is shown; the data is normalized by its\nvalue at T/TN´ eel= 0.1. Inset: The case for a Mspinned by\nthe external magnetic field is shown.\nwhereI±\ns, representing the contribution from the ±\nbranch, is expressed by\nI±\ns=±/summationdisplay\nk,q8Nint[[|J±\nsd(k,q)|2]]\nNNNF/planckover2pi12/integraldisplay\nωImχR(k,ω)\n×ImGR\n±(q,ω)/bracketleftBig\ncoth(/planckover2pi1ω\n2kBTN)−coth(/planckover2pi1ω\n2kBTF)/bracketrightBig\n.(8)\nHereNintis the number of localized spins at the F/N\ninterface, NN(NF) is the number of lattice sites in N\n(sublattice sites in F), [[|J±\nsd(k,q)|2]] =SA(JA\nsdu±\nq)2+\nSB(JB\nsdu∓\nq)2. In Eq. (8), χR(k,ω) =χN/(1 +λ2\nNk2−\niωτsf) where χN,λN,τsfare respectively the param-\nagnetic susceptibility, the spin-diffusion length, and the\nspin-flip relaxation time in N, andGR\n±(q,ω) = 1/(ω−\nω±\nq+ iα±ω) where α±is the damping parameter in F.\nNote that the signs of the spin current injected by the\nαqmagnons ( I+\ns) and that by the βqmagnons ( I−\ns) are\nopposite.\nWe first consider the SSE in a uniaxial antiferromag-\nnet. As is depicted in Fig. 2 (a), the two magnons in a\nuniaxialantiferromagnetaredegenerateif H0= 0. More-\nover,owingtothe equivalenceofsublattices AandB, the\ns-dexchange interactions at the interface for these two\nsublattices are the same ( JA\nsd=JB\nsd). From these condi-\ntions we obtain |I+\ns|=|I−\ns|resulting in a null SSE due\nto Eq. (7), i.e., Is= 0. Thus, the SSE vanishes in a\nuniaxial antiferromagnet under a negligibly small exter-\nnal magnetic field because of the perfect compensation\nof the spin injection by the two degenerate magnons.\nWe next consider the SSE in a ferrimagnet close to\nthe magnetization compensation point, in which the two4\nmagnons are no longer degenerate. Figure 4 shows the\ntemperature dependence of the SSE signal Is(T) calcu-\nlated from Eqs. (7) and (8) for a compensated ferrimag-\nnet Er 3Fe5O12by using the same parameters as in Fig. 2\n(b). In Fig. 4 we also plot the saturation magnetization\nMs=µ0/parenleftBiggA\nNF/summationdisplay\ni∈A/angbracketleftSz\nA,i/angbracketright+gB\nNF/summationdisplay\nj∈B/angbracketleftSz\nB,j/angbracketright/parenrightBig\n(9)\nto determine the magnetization compensation point de-\nfined by Ms(TM) = 0. In addition, we plot the total\nangular momentum\nStot=/angbracketleftSz/angbracketright (10)\ntodeterminetheangular-momentumcompensationpoint\ndefined by Stot(TA) = 0. Here, Szis thez-component of\nthe total spin S=SA+SB, i.e.,\nSz=1\nNF/summationdisplay\ni∈ASz\nA,i+1\nNF/summationdisplay\nj∈BSz\nB,j.(11)\nClearly we see that the SSE signal is unchanged across\nboth compensation points, either TM≈0.15TN´ eelor\nTA≈0.32TN´ eel. We performed the same calculation\nfor several different choices of parameters, and confirmed\nthat the SSE is unchanged across TMandTA.\nIII. SPIN SEEBECK EFFECT IN EASY-PLANE\nANTIFERROMAGNETS\nIn this section, we show that the SSE in easy-plane an-\ntiferromagnets vanishes under a zero magnetic field be-\ncause neither of magnons carries spins in easy-plane an-\ntiferromagnets. We consider the following Hamiltonian\nfor easy-plane antiferromagnets:37\nHeAF=J/summationdisplay\n/angbracketlefti∈A,j∈B/angbracketrightSA\ni·SB\nj\n+/summationdisplay\nL=A,B/summationdisplay\ni∈L[gµ0H0·SL\ni−D\n2(/hatwidez·SL\ni)2] (12)\nwhereJisthenearest-neighborexchangeintegrals, H0=\nH0/hatwidexis the external magnetic field, and gis the g-factor\nandD <0 is the anisotropyconstant which selects the x-\nyplaneasaneasyplane. Notethattheexternalmagnetic\nfield is applied along the xaxis, and we assume SA/bardbl/hatwidez\nandSB/bardbl −/hatwidezwhenH0= 0. Following Ref. 37, we in-\ntroduce the linear Holstein-Primakoff transformation by\nchoosing the direction of each canted sublattice spin in\nthe ground state as a spin quantizing axis. Performing a\nπ/4 rotation to the operators to separate the mixing of\nthe two spin operators and using the Bogoliubov trans-\nformation, Eq. (12) is diagonalized to be\nHeAF=/planckover2pi1/summationdisplay\nq/parenleftBig\nε+\nqξ†\nqξq+ε−\nqζ†\nqζq/parenrightBig\n,(13)whereε±\nq=/radicalBig\n(A±q+2B±q)(A±q−2B±q),A±\nq=\n2z0JScos2θ+gµ0H0sinθ+|D|S∓z0JS(cos2θ−1)γq,\nB±\nq=∓z0JS(cos2θ−1)γq− |D|S/2, and γq=\nz−1\n0/summationtext\nδeiq·δis defined by the sum over z0nearest neigh-\nbors. In the above equation, θis the canted angle of\nthe sublattice magnetization, and the magnon operators\nξqandζqare defined by the Bogoliubov transformation\n1√\n2/parenleftbig\naq+b−q/parenrightbig\n=uqξq+vqξ†\n−qand1√\n2/parenleftbig\naq−b−q/parenrightbig\n=\nxqζq+yqζ†\n−q, whereu2\nq−v2\nq= 1 and x2\nq−y2\nq= 1 are\nrealcoefficients, and aqandbqarethe Fouriertransforms\nof Holstein-Primakoff operators aiandbi. As is seen in\nFig. 3 of Ref. 37, the two magnons in the easy-plane an-\ntiferromagnet are not degenerate even when H0= 0.\nNow we discuss the SSE in an easy-plane antiferro-\nmagnet modeled by Eq. (12) by using the same proce-\ndure as in the previous section. As before, we consider\na system in which an easy-plane antiferromagnet ( eAF)\nhaving local temperature TeAFand a nonmagnetic metal\n(N) having local temperature TNare interacting weakly\nthrough the s-dexchange interaction at the interface. In\nthe absence of an external magnetic field, a direct calcu-\nlation shows that the spin current Isinjected into Nis\nidentically zero, i.e.,\nIs= 0. (14)\nThis is understood by investigating the z-component of\nthe total spin S=SA+SB[Eq. (11)]. In the case\nof a uniaxial antiferromagnet discussed in the previous\nsection, the expectation value of Szis given by\n/angbracketleftSz/angbracketright=/parenleftBig\nSA−1\nNF/summationdisplay\nq/angbracketleftα†\nqαq/angbracketright/parenrightBig\n−/parenleftBig\nSB−1\nNF/summationdisplay\nq/angbracketleftβ†\nqβq/angbracketright/parenrightBig\n,\n(15)\nwhereαqandβqare the magnon operators defined in\nSec. II. Equation (15) means that the αqmagnons car-\nries spin one while βqmagnon carries spin minus one in\na uniaxial antiferromagnet. On the other hand, the ex-\npectation value of Szin an easy-plane antiferromagnet\nunder discussion is calculated to be identically zero, i.e.,\n/angbracketleftSz/angbracketright= 0. (16)\nEquation (16) means that magnons in an easy-plane an-\ntiferromagnet are similar to a linearly-polarized photon\nand hence neither of magnons carries spins if H0= 0.\nIn the presence of a finite external magnetic field\nH0=H0/hatwidexwith a nonzero canted angle θ, however, the\nx-component of Sbecomes nonzero. In this situation,\nthe spin current Isinjected into Nis shown to have the\npolarization along the xaxis, and its magnitude is given\nby\nIs=gµ0H0\nJz0/parenleftbig\nI+\ns+I−\ns/parenrightbig\n, (17)5\nwhere\nI±\ns=4J2\nsdNint\nNNNF/planckover2pi12/summationdisplay\nk,q/integraldisplay\nωImχR(k,ω)ImFR\n±(q,ω)\n×/bracketleftBig\ncoth(/planckover2pi1ω\n2kBTN)−coth(/planckover2pi1ω\n2kBTeAF)/bracketrightBig\n.(18)\nHere,FR\n±(q,ω) = 1/(ω−ε±\nq+iα±ω) is the retarded com-\nponent of the magnon propagator with α±is the damp-\ning parameter. Note that the signs of I+\nsandI−\nsare the\nsame, in contrast to the case of a uniaxial antiferromag-\nnet.\nFrom Eqs. (14) and (17), we conclude that the SSE\nvanishes in an easy-plane antiferromagnet if H0= 0.\nIV. DISCUSSION AND CONCLUSION\nThe main result of this paper is that the SSE in an-\ntiferromagnets vanishes, whereas the SSE in ferrimag-\nnets persists and is insensitive to either magnetization or\nangular-momentum compensation effects. The interpre-\ntationis asfollows. Forthe SSEtooccur, the existenceof\nthe transverse fluctuations of the total spin, i.e., Sx\ntotand\nSy\ntot, is needed. For a ferrimagnet at TMorTA, fluctua-\ntionsofSx,ydonot vanishevenwhen Stot= 0orMs= 0.\nTherefore, ferrimagnetic magnons can always generate\nthe SSE. Only for a uniaxial antiferromagnet, where the\ntwo magnons are degenerate, the SSE from the two de-\ngenerate magnons with the opposite sense compensates\nperfectly. Note that neither magnon in an easy-plane\nantiferromagnet carries spins.\nOur conclusion is not modified by considering the\nphonon-drag contribution to the SSE38because, as dis-\ncussed in Refs. 39 and 40, phonon drag can be taken\ninto account by replacing TFandTNin Eq. (8) with an\neffective magnon temperature T∗\nFand effective spin accu-\nmulationtemperature T∗\nN. We alsonotethat themagnon\nexcitations are well defined even at TA. The presumed\ndivergence of the magnon damping parameter at TA41\ndoes not exist when we recall the condition justifying the\nexpansion used in Ref. 42, where such an effect mani-\nfests itself as an enhancement of the damping parameter\nwithout any divergence (see Appendix B). Note that the\nmagnitude of magnon damping has less effect on the lon-\ngitudinal SSE, although it has a large influence on the\ntransverse SSE.\nTo conclude, we have theoretically investigated the\nSSE in antiferromagnets and ferrimagnets, and shown\nthat the SSE vanishes in antiferromagnets whereas it\npersists at either the magnetization or the angular-\nmomentum compensation points of ferrimagnets, despite\nthe absence of its saturation magnetization or total spin.\nBecause a fringing field by saturation magnetization is\nsuppressed at the magnetization compensation point,\nthis phenomenon can be useful for constructing a pure\nspin current device which is free from crosstalk of the\nfringing field.Acknowledgments\nWe are grateful to K. Uchida. This work is was finan-\ncially supported by a Grant-in-Aid from MEXT, Japan,\nand a Fundamental Research Grants from CREST-JST,\nPRESTO-JST, Japan.\nAppendix A: Linear-response expression of\nmagnon-driven spin injection in ferrimagnets\nIn this Appendix, we derive Eq. (6) in the main text.\nWe consider a system described by the Hamiltonian (4),\nand calculate the spin current Is=/summationtext\ni∈N/angbracketleft∂tσz\ni/angbracketright. We\nuse the momentum representation of σz\niand calculate\nthe quantity Is=√NN/angbracketleft∂tσz\nk0/angbracketrightk0→0. The Heisenberg\nequation of motion for σz\nk0gives\n∂tσz\nk0=i\n/planckover2pi1/summationdisplay\nk,q/bracketleftBig√8SAJA\nsd(k,q)√NFNN(u+\nqα−\nq+u−\nqβ+\nq)σ−\nk\n+√8SBJB\nsd(k,q)√NFNN(u+\nqβ+\nq+u−\nqα−\nq)σ−\nk/bracketrightBig\n+h.c.,(A1)\nwhere σ±\nk=1\n2(σx\nk±iσy\nk),JL\nsd(k,q) =/summationtext\ni∈N/F(L)JL\nsdei(k−q)·rifor sublattice L=A,B,\nand we have used the relation [ σz\nk,σ±\nk′] =±2√NNσ±\nk+k′.\nTaking the statistical average of the above quantity, the\nspin current is calculated to be\nIs(t) =−4√\n2√NNNF/planckover2pi1Re/summationdisplay\nk,q/bracketleftBig\nJ+\nsd(k,q)A<\nk,q(t,t)\n+J−\nsd(k,q)B<\nk,q(t,t)/bracketrightBig\n, (A2)\nwhereJ±\nsd(k,q) =JA\nsd(k,q)√SAu±\nq+JB\nsd(k,q)√SBu±\nq.\nHere,A<\nk,q(t,t′) =−i/angbracketleftαq(t′)σ−\nk(t)/angbracketrightandB<\nk,q(t,t′) =\n−i/angbracketleftβ†\nq(t′)σ−\nk(t)/angbracketrightmeasure the interface correlation func-\ntions between the magnon operators ( αqandβq) and\nspin-density operator σ−\nk. In the steady state the inter-\nface correlations A<\nk,q(t,t′) andB<\nk,q(t,t′) depends only\non the time difference t−t′. Introducing the frequency\nrepresentation A<\nk,q(t,t′) =/integraltext∞\n−∞dω\n2πA<\nk,q(ω)e−iω(t−t′)as\nwell as using the relationship A<=1\n2[AK−AR+AA],\nwe finally obtain Eq. (6) in the main text.\nAppendix B: Magnon damping near compensation\npoints\nIn this Appendix, we calculate temperature depen-\ndence of magnon damping parameter close to the com-\npensation points and show that the magnon excitation\nis well defined even at compensation points without any\ndivergences. We begin with two Landau Lifshitz Gilbert\nequations for sublattice L=A,B:41\ndML\ndt=−γLML×HL+αL\nMs,LML×dML\ndt,(B1)6\nFIG. 5: (Color online) Temperature dependence of the\neffective magnon damping parameter αeff[red, Eqs. (B8)\nand (B9)], saturation magnetization Ms[blue, Eq. (9)], and\ntotal angular momentum Stot[green, Eq. (10)], calculated for\na compensated ferrimagnet Gd 23Fe74.6Co3.4. The data is nor-\nmalized by its value at T/TN´ eel= 0.1.\nwhereMLis the sublattice magnetization with its mag-\nnitude given by ML,HLis the effective magnetic field,\nγL=gLµ0//planckover2pi1is the gyromagnetic ratio, and αLis the\nGilbertdampingparameter. Theeffectivefieldsaregiven\nbyHA=H0+Han\nA−λMBandHB=H0+Han\nB−λMA,\nwhereH0=H0/hatwidezisexternalmagneticfield, Han\nA=Han\nA/hatwidez\nandHan\nB=−Han\nB/hatwidezare the anisotropy fields, and λMA\nandλMBare the inter-sublattice exchange fields with\nλ=z0JAB/(gAgBµ2\n0). Because we here focus on the\nuniform mode, the intra-sublattice exchange couplings\nλA=z0JA/(gAµ0)2andλB=z0JB/(gBµ0)2are dis-\ncarded in Eq. (B1).\nBelow the magnetization compensation point we set\nMA=MA/hatwidez+mAandMB=−MB/hatwidez+mB, such that\nthe effective fields can be written as\nHA= (H0+Han\nA+λMB)/hatwidez−λmB,(B2)\nHB= (H0−Han\nB−λMA)/hatwidez−λmA.(B3)\nIntroducing the representation Eeff\nA=−(H0+Han\nA+\nλMB) andEeff\nB=−(H0−Han\nB−λMA), and linearizing\nwithrespectto mAandmB, theLandau-Lifshitz-Gilbert\nequations are transformed to be\ndmA\ndt=/hatwidez×/bracketleftBig\nγA(λMAmB−Eeff\nAmA)+αAdmA\ndt/bracketrightBig\n,\n(B4)\ndmB\ndt=−/hatwidez×/bracketleftBig\nγB(λMBmA+Eeff\nBmB)+αBdmB\ndt/bracketrightBig\n.\n(B5)We introduce m±=mx±imyand substitute m+\nL(t) =\nm+\nLe−iωtinto Eqs. (B4) and (B4). Then we obtain\n(−iω−αAω+iγAEeff\nA)m+\nA−iλγAMAm+\nB= 0,\n(B6)\n(−iω+αBω+iγBEeff\nB)m+\nB+iλγBMBm+\nA= 0.\n(B7)\nThe eigenfrequency ωis determined by the equation:\n(ω−iαAω−γAEeff\nA)(ω+iαBω−γBEeff\nB)\n+λ2γAγBMAMB= 0. (B8)\nAbove the magnetization compensation temperature,\nwe setMA=−MA/hatwidez+mAandMB=MB/hatwidez+mB\nbecause we consider a situation in which the satura-\ntion magnetization is pinned by an external magnetic\nfield. This situation can be analyzed by rewriting Eeff\nA=\n−(H0−Han\nA−λMB) andEeff\nB=−(H0+Han\nA+λMA) as\nwell as reversingthe signs of αAandαB. We numerically\nsolve Eq. (B8) by setting\nω=ω0+iαeffω0. (B9)\nFigure 5 shows the temperature dependence of the\neffective Gilbert damping parameter αeffthe lower fre-\nquency mode, calculated for a compensated ferrimagnet\nGd22Fe70Co8.41,43,44We assign Asublattice for Gd ions\nandBsublattice for Fe ions, and neglect Co ions for\nsimplicity. We set SA= 3.85,SB= 3.5,gA= 2.0,\ngB= 2.05,H0= 0.04 T,Han\nA= 0.0 T,Han\nB= 0.02 T,\nαA= 0.004,αB= 0.0039. The saturation magnetization\nand the total spin are calculated by using the mean field\napproximation by using JAB= 0.28 meV, JA= 0 meV,\nandJB= 0.34 meV to reproduece TN´ eel= 500 K. These\nparameters give TM≈0.079TN´ eelandTA≈0.091TN´ eel.\nFrom the figure, we see that Gilbert damping constant is\nenhanced around the angular momentum compensation\npointTA, but does not show any divergences. A small\ndiscontinuity at TMstems from the fact that the spin\nquantization axis is reversed at TMbecause of the pin-\nning by the external magnetic field. The result obtained\nhere justifies the statement in Sec. IV that the presumed\ndivergence of the magnon damping parameter at TA41\ndoes not exist when we recall the condition justifying the\nexpansion used in Ref. 42, where such an effect mani-\nfests itself as an enhancement of the damping parameter\nwithout any divergence.\n∗Electronic address: y-ohnuma@imr.tohoku.ac.jp\n1Spin Caloritronics, edited by G. E. W. Bauer, A. H. Mac-Donald, and S. Maekawa, special issue of Solid State Com-\nmun.,150, 459 (2010).7\n2G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature\nMaterials 11391-399 (2012).\n3K. Uchida, S. Takahashi, K. 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Tokura5,7\n1Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n2Department of Physics, Budapest University of Technology a nd Economics and MTA-BME\nLend¨ ulet Magneto-optical Spectroscopy Research Group, 1 111 Budapest, Hungary\n3National Institute of Chemical Physics and Biophysics, Aka deemia tee 23, 12618 Tallinn, Estonia\n4High Field Magnet Laboratory (HFML-EMFL), Radboud Univers ity Nijmegen,\nToernooiveld 7, 6525 ED Nijmegen, The Netherlands\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Sait ama 351-0198, Japan\n6Department of Advanced Materials Science, University of To kyo, Kashiwa 277-8561, Japan and\n7Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113-8656, Japan\n(Dated: September 17, 2021)\nCompetingexchangeinteractions canproducecomplexmagne ticstates together withspin-induced\nelectric polarizations. With competing interactions on al ternating triangular and kagome layers,\nthe swedenborgite CaBaCo 4O7may have one of the largest measured spin-induced polarizat ions of\n∼1700 nC/cm2below its ferrimagnetic transition temperature at 70 K. Upo n rotating our sample\naboutc= [0,0,1] while the magnetic field is fixed along [1 ,0,0], the three-fold splitting of the spin-\nwave frequencies indicates that our sample is hexagonally t winned. Magnetization measurements\nthen indicate that roughly 20% of the sample is in a domain wit h theaaxis along [1 ,0,0] and that\n80% of the sample is in one of two other domains with the aaxis along either [ −1/2,√\n3/2,0] or\n[−1/2,−√\n3/2,0]. Powder neutron-diffraction data, magnetization measur ements, and THz absorp-\ntion spectroscopy reveal that the complex spin order in each domain can be described as a triangular\narray of bitetrahedral c-axis chains ferrimagnetically coupled to each other in the abplane. The\nelectric-field dependence of bonds coupling those chains pr oduces the large spin-inducedpolarization\nof CaBaCo 4O7.\nPACS numbers: 75.25.-j, 75.30.Ds, 75.50.Ee, 78.30.-j\nI. INTRODUCTION\nCompeting exchange interactions produce complex\nmagnetic states with a wide range of interesting behav-\nior found in spin glass [1], spin ice [2], and magnetic\nskyrmions [3]. In multiferroic materials, complex spin\nstates can exibit a spin-induced electric polarization P\ndue to either the spin current, p-dorbital hybridization,\nor magnetostriction [4,5]. Because the coupling between\nthe electrical and magnetic properties in multiferroic ma-\nterials is both scientifically and technologically impor-\ntant, the effects of competing exchange interactions have\nbeen investigated in a wide range of multiferroic materi-\nals such asRMnO3[6], CoCrO 4[7], CuCrO 2[8], CuFeO 2\n[9], and MnWO 4[10]. While the first four materials [6–\n9] are geometrically frustrated due to competing interac-\ntions on a triangular lattice, MnWO 4[10] exhibits long-\n∗Copyright notice: This manuscript has been authored by UT-\nBattelle, LLC under Contract No. DE-AC05-00OR22725 with th e\nU.S. Department of Energy. The United States Government re-\ntains and the publisher, by accepting the article for public ation,\nacknowledges that the United States Government retains a no n-\nexclusive, paid-up, irrevocable, world-wide license to pu blish or re-\nproduce the published form of this manuscript, or allow othe rs to\ndo so, for United States Government purposes. The Departmen t of\nEnergy will provide public access to these results of federa lly spon-\nsored research in accordance with the DOE Public Access Plan\n(http://energy.gov/downloads/doe-public-access-plan ).range competing interactions [11] on a highly-distorted\nmonoclinic lattice.\nCompounds in the “114” swedenborgite family [12]\nRBaM4O7(M= Co or Fe) contain alternating triangu-\nlar and kagome layers, both of which are geometrically\nfrustrated when undistorted. The “114” cobaltites [13–\n15] were initially studied to find charge ordering among\nthe Co2+and Co3+ions. An important member of this\nfamily, YBaCo 4O7exhibits antiferromagnetic ordering\n[16,17] below 110 K and diffuse scattering [13,14] indica-\ntive of spin disorder below 60 K. The magnetic state be-\ntween 110 K and 60 K is stabilized by a structural tran-\nsition [18] that relieves the geometric frustration. Both\nstructural and magnetic transitions are quite sensitive\nto excess oxygen and no magnetic order [19,20] appears\nin YBaCo 4O7+δforδ≥0.12. Another family member,\nYbBaCo 4O7undergoes a structural transition at 175 K\nthatstabilizesanantiferromagneticstatebelow80K[21].\nAparticularlyinteresting“114”cobaltite, CaBaCo 4O7\nundergoes an orthorhombic distortion [22,23] that re-\nlieves the geometric magnetic frustration on both the\nkagome and triangular layers sketched in Fig. 1 above\nthe magnetic transition temperature Tc= 70 K. Below\nTc, CaBaCo 4O7develops a very large spin-induced po-\nlarization ∼1700nC/cm2[24], secondonly to the conjec-\ntured [25] spin-induced polarization ∼3000 nC/cm2of\nBiFeO 3. Also unusual, CaBaCo 4O7displays a substan-\ntial ferrimagnetic moment of about 0.9 µBper formula\nunit (f.u.) [26], which could allowmagnetic controlof the2\n!\"#$\n%\"&'($)$ !*#$\n+,) $\n+,- $+.$!\"#\n+.$\n+,- $\n$\"#+.$\n+.$$\"#+.$\n%\"#\n%\"#&\"#\n&'#+.$\n(\"#(\"#)\"#+/$\n+/$\n+/$+/$+/$\n%\"#*\"#\n%\"#\n!\"#)\"#)\"#\n*\"#+.$+0$\n+0$+,, $+,) $+,, $\n+,, $+,- $\n+,- $+,, $+,) $+,, $+,) $+,- $\n+0$+,) $*\"#\n!\"#\n$\"#$\"#\n+0$+0$\n*\"#+\"#\n+\"# +,) $+,, $+,) $+,, $\n+,, $+,- $\n+,- $*$\n\"$%\"&'($,$\n+,- $+,- $%#!#\n$#\n$#!#\n+,- $\n+.$+.$\n+.$\n&#\n&#\n+/$+/$+/$\n&#+,) $\n+.$*#+.$\n%#(#\n+#\n+#)#\n)#)#\n+0$+/$+,- $\n+,, $+,) $+,, $+,) $\n+0$+/$\n+,, $+,) $+,, $\n+,, $+,) $+,, $+,) $+,, $+,) $\n+,- $+,) $&#\n(# (#\n!#(#\n*#\n+0$\n!#!#\n!#+0$\n+0$\nFIG. 1: (Color online) (a) and (b) The predicted spin config-\nuration for layers 1 and 2 in zero field. Spins 1 and 5 lie on\na triangular layer above the first kagome layer in (a); spins 1′\nand 5′lie on a triangular layer above the second kagome layer\nin (b). Layers are arranged so that spins 1′and 5′lie directly\nabove spins 1 and 5.\nelectric polarization. Although its ferroelectrictransition\nis inaccessible and its permanent electric polarization is\nnot switchable [27], applications of CaBaCo 4O7might\nutilize the large spin-induced polarization produced by a\nmagnetic field just below Tc[24].\nThis paper examines the magnetic properties of\nCaBaCo 4O7basedonaHeisenbergmodelwith12nearest\nneighbor interactions and associated anisotropies. The\nmagnetic state of CaBaCo 4O7can be described as a tri-angular array of ferrimagnetically aligned, bitetrahedral\nc-axis chains with net moment along b. Competing in-\nteractions within each chain produce a non-collinear spin\nstate. The strongelectric polarizationofCaBaCo 4O7be-\nlowTcis induced by the displacement of oxygen atoms\nsurrounding bonds that couple those chains.\nThis paper has six sections. Section II proposes a mi-\ncroscopic model for CaBaCo 4O7. New magnetization\nand optical measurements are presented in Section III.\nFitting results are discussed in Section IV. In Section V,\nwe predict the spin-induced electric polarization. Section\nVI contains a conclusion.\nII. MICROSCOPIC MODEL\nEach magnetic unit cell of CaBaCo 4O7contains 16 Co\nions on two kagome and two triangular layers with or-\nthorhombic lattice constants a= 6.3˚A,b= 11.0˚A, and\nc= 10.2˚A. Four crystallographically distinct Co ions\nhave three different valences [22,28]. Triangular layers\ncontain mixed-valent Co3+/Co2+L(Lis a ligand hole)\nspins1, 5, 9, and13withmoments M1= 2.9µB. Kagome\nlayers contain Co2+spins 2, 3, 6, 7, 10, 11, 14, and\n15 with moments M2=M3= 2µBand mixed-valent\nCo3+/Co2+Lspins 4, 8, 12, and 16 with M4= 2.4µB.\nBecause adjacent kagome or triangular layers are related\nby symmetry, Si′=Si+8on layer two is identical to Si\non layer one. With Si=Si(cosφi,sinφi,0) constrained\nto theabplane, the ferrimagnetic moment lies along bif\nφi+4=π−φi(i= 1,...,4).\nThe 12 different nearest-neighbor exchange couplings\nJiare drawn in Figs. 1(a-b) and 2. Six of these ( J1\nthroughJ6) couple the kagome and triangular layers as\nshown in Fig. 2; the other six ( J7throughJ12) couple\nthe spins within a kagome layer as shown in Figs. 1(a)\nand (b). The dominance of nearest-neighbor exchange\nover next-nearest neighbor exchange [27] justifies setting\nthe exchange interactions between spins on the triangu-\nlar layers to zero. Our model also includes easy-plane\nanisotropies D, easy-axis anisotropies Cwithin both\nkagome and triangular layers, and hexagonal anisotropy\nAon the triangular layers.\nWith magnetic field Balongm, the Hamiltonian is\nH=−/summationdisplay\n/angbracketlefti,j/angbracketrightJijSi·Sj+Dtri/summationdisplay\ni,triSic2+Dkag/summationdisplay\ni,kagSic2\n−Ckag/summationdisplay\ni,kag(oi·Si)2−Ctri/summationdisplay\ni,tri(ni·Si)2\n−AtriRe/summationdisplay\ni,tri/parenleftbig\nSia+iSib/parenrightbig6−gµBB/summationdisplay\nim·Si,(1)\nwhereSiis a spinSoperator on site i. For simplicity,\nwe setg= 2 for all spins.\nThe easy-axis anisotropy terms proportional to Ckag\nandCtriinvolve unit vectors oialong the “bowtie” di-\nrectionsφi=π/2 (spins 2 and 6), 5 π/6 (spins 3 and 8)3\n!\"\n#\"$\"\n!\"\n%&\"%'\" %(\"\n%)\"%*\"\n#\"%+\"!$\"\n%$\"\n#$\"\n!$\"%&\"%)\"\n%+\"\n%'\"%(\"%*\"%\"&$\"\n&\"%*\"\n%)\"\n'\"(\")$\"\n%&\"%'\"\n%+\"%(\"($\"\n($\"%&\" %+\"%)\"\n%'\"%*\"%(\")\"'$\"\n*$\"\n*\"\n$,#-.\"α $,#-.\"β\nFIG. 2: (Color online) A sideways view of the zero-field spin\nconfiguration showing bitetrahedral c-axis chains αandβ.\nand 7π/6 (spins 4 and 7) for the kagome layers and ni\nalong theφi=π/6 (spin 1) and −π/6 (spin 5) directions\nfor the triangular layers. The hexagonal anisotropy on\nthe triangular layers has expectation value\n−AtriS16/summationdisplay\ni,trisin6θicos6φi.\nAll anisotropy terms may act to constrain the spins to\ntheabplane.\nSpin amplitudes Snare fixed at their observed val-\nuesMn/2µBafter performing a 1 /Sexpansion about the\nclassical limit. Alternatively, the spins Sncould all have\nbeentakenas3/2butwithdifferent g-factorsfordifferent\nsets of spins. As discussed below, that would reduce the\nestimated exchange coupling Jijby a factor of 4 SiSj/9.\nStaticpropertiesareobtainedbyminimizingtheclassi-\ncal energy /an}bracketle{tH/an}bracketri}ht(the zeroth-order term in this expansion)\nwith respect to the 16 spin angles. The eigenvalues and\neigenvectors of a 32 ×32 equations-of-motion matrix [29]\nproduced by the second-order term in the 1 /Sexpan-\nsion give the optical mode frequencies and absorptions,\nrespectively.\nIII. MAGNETIZATION AND OPTICAL\nMEASUREMENTS\nPerhaps due to excess or deficient oxygen [30] or dif-\nferent domain populations (see below), previous magne-\ntization measurements [22,26,31–33] on CaBaCo 4O7are\nrather scattered. Consequently, new magnetization mea-\nsurementswereperformedat4Konhexagonally-twinned00.25 0.5 0.75 11.25 1.5 \n0 4 8 12 16 20 24 28 32 M ( µ\nB/f.u.) \nB (T) m = [1,0,0] \n[0,1,0] \n[0,0,1] \nFIG.3: (Color online)Themeasured(circles andsquares)an d\npredicted (solid curves) magnetizations for field along [1 ,0,0],\n[0,1,0], or [0,0,1].\ncrystals with a common c=z= [0,0,1] axis. In domain\nI,alies along the laboratory direction x= [1,0,0] andb\nlies along y= [0,1,0]; in domain II, a= [−1/2,√\n3/2,0]\nandb= [−√\n3/2,−1/2,0]; and in domain III, a=\n[−1/2,−√\n3/2,0] andb= [√\n3/2,−1/2,0]. Ifplare the\ndomain populations, then the magnetizations Mxand\nMymeasured with fields along xandyonly depend on\np1andp2+p3= 1−p1. Of course, Mzmeasured with\nfield along zis independent of pl. Fig. 3 indicates that\nall three magnetizations increase monotonically up to at\nleast 32 T.\nPrevious optical measurements [34] at the ordering\nwavevector Qfound two conventional spin-wave modes\nthat couple to the ground state through the magne-\ntization operator M= 2µB/summationtext\niSi. These magnetic-\nresonance (MR) modes are degenerate in zero field with\na frequency of1.07 THz and split almost linearly with in-\ncreasingfield along y, as shown in Fig. 4. For m=y, the\nMR modes are excited in two geometries: ( i) with THz\nfieldsEω||xandBω||zand (ii) withEω||zandBω||x.\nThose measurements also found an electromagnon (EM)\nthat couples to the ground state through the polarization\noperator P. The EM with zero-field frequency 1.41 THz\nis only excited in geometry ii.\nBecause the exchange couplings already break every\ndegeneracy in the unit cell, the 16 predicted modes for a\nsingle domain are non-degenerate. Therefore, the split\nMR modes must come from different domains. This\nwas verified by measuring [35] the MR mode frequen-\ncies as a function of the rotation angle θfor field B=\nB(cosθ,sinθ,0) =B(xcosθ+ysinθ) in the laboratory\nreference frame. In practice, this is accomplished by ro-\ntating the sample about cwhile keeping the field fixed4\nTABLE I: Exchange and anisotropy parameters in meV.\np1J1=J5J2=J4J3=J6J7J8J9J10J11=J12DkagDtriCkagCtriS14Atri\n0.185 −91.5−10.8 41.4−29.9 187.8 7.9 108.0−6.7−0.67−1.24 3.70 0.77 0.0064\nerror±0.071±3.6±0.5±0.7±1.9±7.5±0.1±4.5±0.1±0.06±0.03±0.15±0.09±0.0004\n1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0510 15 \nν(THz) B (T) MR EM \nFIG. 4: (Color online) The predicted MR (solid) and EM\n(dashed) modes for domain I (thick) and domains II and III\n(thin). Measured modes are indicated by solid points.\nalongx. As shown in Fig. 5 for 12 and 15 T, each hexag-\nonal domain then contributes one MR branch with a pe-\nriod ofπ.\nWith field Bloc=B(cosψ,sinψ,0) =B(acosψ+\nbsinψ) in the domain reference frame, the upper MR\nmode in Fig. 4 corresponds to the ψ=π/2 mode for\ndomain I while the lower MR mode corresponds to the\ndegenerate ψ=±π/6 modes for domains II and III. Pre-\nviously measured MR frequencies plotted in Fig. 4 at 12\nT correspond to the diamond and triangular points in\nFig. 5(b) at θ=π/2. Cusps in the MR curves for each\ndomain at ψ= 0 andπare caused by flipping the b\ncomponent of the magnetization (see inset to Fig. 5(b)).\nIV. FITTING RESULTS\nFits for the coupling parameters utilize the field de-\npendence of M, the zero-field powder-diffraction data\n[22], the field dependence of the MR and EM modes at\nθ=π/2 [34], and the MR mode frequencies at θ= 0\nandπ/3 for 7, 12 and 15 T. The resulting exchange and\nanisotropy constants are provided in Table I and the cor-\nresponding zero-field spin state is plotted in Figs. 1(a-b).\nIncontrasttothepreviouslyproposed[22]spinstatewith\n!\"#$ %&' !(#$ %)' **# ***# *#\n!\"ψ+#0\u0001 !\"\nψ+#0+\nFIG. 5: (Color online) The measured (solid circles) and pre-\ndicted (blue, red, and green curves for domains I, II, and III ,\nrespectively) angular dependence of the MR mode frequen-\ncies for 12 and 15 T. The inset to (a) sketches the angular\ndependence of BSF(dashed curve), which separates low-field\n(LF) and high-field (HF) states. The inset to (b) shows the\nnet magnetization of any domain for angles ψon either side\nof 0. Flips of the b-axis spin at ψ= 0 andπproduce cusps in\nthe mode frequencies.\nzig-zagchainsinthe abplanecontainingspins2, 3, 6, and\n7, our spin state can be better described as an array of\nc-axis chains or connected bitetrahedra [16,36] contain-\ning spins {1,2,3,4}(chainα) or{5,6,7,8}(chainβ) as\nsketched in Fig. 2. Chains are coupled by exchanges J9,5\nJ10, andJ12in theabplane.\nWhat explains the wide range of Jivalues? An or-\nthorhombic distortion [22,23] with b/(a√\n3)−1→0.018\nasT→0 breaks the hexagonal symmetry of the abplane\nand explains the difference between the pairs {J1,J2},\n{J8,J11}, and{J10,J12}. The difference between cou-\nplings like {J7,J8}is caused by charge ordering: whereas\nJ7couples moments 2 and 3 with M2=M3,J8couples\nmoments 2 and 4 with M2/ne}ationslash=M4. Charge ordering also\nexplains the difference between the pairs {J2,J3}and\n{J9,J10}. Although not demanded by symmetry, we set\nJ1=J5,J2=J4,J3=J6, andJ11=J12because the\nspin state and excitations at Qonly depend on their av-\nerages [37].\nGiven other conditions, our fit chooses the spin state\nthat matches the powder-diffraction data [22] as closely\nas possible. At zero field, the predicted spin state has\nanglesφ1=−0.83π,φ2= 0.40π,φ3=−0.23π, and\nφ4= 0.62π. Basedexclusivelyonpowder-diffractiondata\nand symmetry constraints, the previously proposed spin\nstate [22] had φ1=−0.24π,φ2=φ3= 0.67π, andφ4=\n−0.44π. Inbothcases, φi+4=π−φi(i= 1,...,4)sothat\nthe moment Mblies along the baxis. As shown in Table\nII, our spin state does not satisfy the powder diffraction\ndata quite as well as the earlier state, primarily because\nit underestimates the powder diffraction peak I(112).\nFor the previous spin state, χ2is minimized by\nLorentzian form factors with Q1/4π= 0.088˚A−1,\nQ2/4π=Q3/4π= 0.095˚A−1, andQ4/4π= 0.088˚A−1\nfor spinsSn. Forthe new spin state, Q1/4π= 0.052˚A−1,\nQ2/4π=Q3/4π= 0.224˚A−1, andQ4/4π= 0.102˚A−1.\nAll are smallerthan the scale Q0/4π≈0.3˚A−1measured\nby Khan and Erickson [38] for Co2+in CoO.\nOur results indicate that the exchange coupling J8≈\n188 meV between moments 2 (Co2+,S2= 1) and 4\n(Co3+/Co2+L,S4= 1.2) is strongly ferromagnetic and\nlarger in magnitude even than the 155 meV antiferro-\nmagnetic coupling found in the cuprate Nd 2CuO4[39].\nThe strength of this coupling might be explained by the\ndouble-exchange mediated hopping of ligand holes L[19]\nfrom site 4 to 2. Bear in mind, however, that the es-\ntimated exchange parameters would be significantly re-\nduced if the fits were performed with S= 3/2 for all Co\nspins. In particular, J8would then fall from 188 to 100\nmeV.\nExceptforJ10, thefivelargestexchangecouplings J1=\nJ5≈ −92 meV,J3=J6≈41 meV, and J8≈188 meV\nlie within connected bitetrahedral, c-axis chains. Inside\neach chain, competing interactions between spins 1, 2,\nand 4 produce a non-collinear spin state.\nAlthough occupying a triangular lattice, chains α\nandβare magnetically ordered with moments Mch=\n(±1.18,1.33,0)µB/f.u.. These chains are primarily cou-\npled by the strongly ferromagnetic interaction J10≈108\nmeV between nearly parallel spins {4,6}(φ4= 0.62π,\nφ6= 0.60π) and{2,8}(φ2= 0.40π,φ8= 0.38π). Above\nTc= 70 K, the short-range order within each chain may\nbe responsible for the large, negative Curie-Weiss tem-perature Θ CW≈ −1720 [26] or −890 K [31], the larger\nthan expected Curie constant [26], and the susceptibility\nanomaly[31] at 360K suggestiveof short-rangemagnetic\norder far above Tc.\nComparison between the theoretical and experimen-\ntal results for the magnetization in Fig. 3 suggests that\nroughly 20% of the sample is in domain I. Different do-\nmainpopulationsorevenorthorhombictwinningin other\nsamples may explain the discrepancies between the re-\nported magnetization measurements [22,26,31–33].\nEasy-axis anisotropies AandCfavor ferrimagnetic\nalignment along brather than a. The spin-flop (SF) field\nrequired to flip the spins towards the adirection must\nincrease as the field along bincreases [40]. As shown in\nthe inset to Fig. 5(a), BSF(ψ) then increases with ψ. If\nBSF(ψ= 0)<15 T, then the MR spectrum for 15 T\nwould show a discontinuity at the transition from a low-\nfield (LF) to a high-field (HF) state below some critical\nvalue ofψ. Since the MR mode frequencies in Fig. 5(a)\ndo not exhibit any discontinuities as a function of ψ, we\nconclude that BSF(ψ= 0) exceeds 15 T and probably,\nbased on the smooth dependence of the magnetizations\non field, exceeds 32 T as well. The apparent small size\nofBSF[24,41] must reflect the net magnetization of all\nthree domains.\nPredicted modes below 5 THz are plotted in Fig. 4.\nThe Goldstone modes for all three domains are lifted by\nin-planeanisotropiestobecometheMRmodeswith zero-\nfield frequencies of 1.07 THz. As remarked earlier, the\nlower MR mode comes from domains II and III while the\nupper MR mode comes from domain I. Below 3.5 THz,\none EM mode is produced in domain I and another in\ndomains II and III. The degenerate EM modes from do-\nmains II and III dominate the optical absorption. The\npredictedfielddependenceoftheupperMRmodeisquite\nclose to the observed dependence. But the predicted cur-\nvatures of the lower MR mode and the EM mode, both\nfrom domains II and III, is not observed.\nV. SPIN-INDUCED ELECTRIC\nPOLARIZATION\nBelow the ferrimagnetic transition, CaBaCo 4O7is re-\nported [24] to develop a very large spin-induced polar-\nization∼1700 nC/cm2, which is surpassed in type I\nmultiferroics only by the conjectured [25] spin-induced\npolarization ∼3000 nC/cm2of BiFeO 3. Other mea-\nsurements indicate that the spin-induced polarization of\nCaBaCo 4O7rangesfrom320nC/cm2[32] to 900nC/cm2\n[42].\nThe electric-field dependence of any interaction term\nin the spin Hamiltonian Hcan induce an electric po-\nlarization below Tc. However, the electric-field depen-\ndence of the easy-plane anisotropy Dcannot explain the\nspin-induced polarization along cbecause the expecta-\ntion value of Pi=κSic2withκ=−∂D/∂E cwould van-\nish in zero magnetic field when all the spins lie in the6\nTABLE II: Ratios of powder-diffraction peak intensities\nI(002)/I(101)I(012)/I(101)I(111)/I(101)I(112)/I(101)I(121)/I(101)I(122)/I(101)Mb(µB/f.u.)χ2\nexperimental [22] 0.344 0.326 0.322 0.477 0.262 0.404\nprevious [22] 0.286 0.414 0.384 0.411 0.232 0.427 0.88 0.021\ncurrrent 0.360 0.380 0.378 0.286 0.313 0.449 1.33 0.047\nabplane. Easy-axis anisotropy AorCin theabplane\ncould produce a spin-induced electric polarization per-\npendicular to c. But the EM mode would then become\nobservable for a THz electric field in the abplane, con-\ntrary to measurements.\nAs conjectured previously [24], the spin-induced po-\nlarization in CaBaCo 4O7must then be generated by the\ndependence of the exchange interactions Jijon an elec-\ntric field, called magnetostriction. Coupling constant\nλij=∂Jij/∂Ecfor bond {i,j}is associated with a spin-\ninduced polarization [43] per site of Pij\nc=λijSi·Sj/4,\nwhich accounts for the four equivalent bonds per unit\ncell. Expanding in the electric field Ecyields an interac-\ntion term −EcλijSi·Sj, linear in the electric field and\nquadratic in the spin operators.\nTaking|0/an}bracketri}htas the ground state and |n/an}bracketri}htas the excited\nspin-wave state, the MR matrix element /an}bracketle{tn|Ma|0/an}bracketri}htmixes\nwith the EM matrixelement /an}bracketle{tn|Pij\nc|0/an}bracketri}htfor domainsII and\nIII but not for domain I. Therefore, our model can ex-\nplain the strong asymmetry [44] ∼Re/braceleftbig\n/an}bracketle{tn|M·Bω|0/an}bracketri}ht/an}bracketle{t0|P·\nEω|n/an}bracketri}ht/bracerightbig\nin the absorption of counter-propagating light\nwaves [34] for the lower observed MR mode in geometry\niiwithEω||c. But it cannot explain the observed asym-\nmetry of this mode in geometry iwithEω⊥cif only\n/an}bracketle{t0|Pc|n/an}bracketri}htis significant.\nHow can we estimate the coupling constants λijand\nthe spin-induced electric polarization? The optical ab-\nsorption of any mode in domain lis proportional to pl2\nbecauseeachmatrixelementisseparatelyproportionalto\npl. At nonzero field, the EM mode absorption is propor-\ntional top22+p32while the upper MR mode absorption\nis proportionalto p12. At zero field, all domains have the\nsame mode spectrum so that both the MR and EM mode\nabsorptions are proportional to p12+p22+p32. Experi-\nmentally, the ratio rof the absorptionof the EM mode to\nthe absorption of the upper MR mode rises from r= 7.5\nat 0 T tor= 35 at 10 T. This growth is explained by the\nB >0ratio(p22+p32)/p12≈10±5forp1= 0.185±0.071\nandp2=p3= (1−p1)/2.\nAt both 0 and 10 T, the only sets of bonds that gen-\nerate spin-induced polarizations of the right magnitude\nare{2,7}and{3,4}. Each of those bonds couples adja-\ncentc-axis chains through pairs of spins that are almost\nanti-parallel. From the relative absorptions rat 0 or\n10 T, we estimate that /an}bracketle{tP27\nc/an}bracketri}ht ≈2350 or 1960 nC/cm2\nand/an}bracketle{tP34\nc/an}bracketri}ht ≈2110 or 1730 nC/cm2. Results for both\nsets of bonds are consistent with the recently observed[24] polarization of 1700 nC/cm2. By contrast, density-\nfunctional theory [27] predicts that the spin-induced po-\nlarization along cis 460 nC/cm2. The spin-induced\npolarization should remain fairly constant with applied\nmagnetic field, decreasing by about 1% for a 10 T field\nalongb.\nVI. CONCLUSION\nWe have presented a nearly complete solution for the\nmagnetization, spin state, and mode frequencies of the\nswedenborgite CaBaCo 4O7. An orthorhombic distor-\ntion aboveTcpartially relieves the geometric frustration\non the kagome and triangular layers and allows ferri-\nmagnetism and ferroelectricity to coexist below Tc. Al-\nthough occupying a triangular lattice, bitetrahedral c-\naxis chains are ferrimagnetically ordered in the abplane.\nCompeting interactions within each chain produce non-\ncollinear spin states. Sets of bonds coupling those chains\nare responsible for the large spin-induced polarization of\nCaBaCo 4O7.\nDespite its fixed permanent electric polarization, this\nswedenborgite may yet have important technological ap-\nplications utilizing the large changes [24] in the spin-\ninduced polarizationwhen a modest magnetic field <1T\nis applied along bjust belowTc. Abig jump in the polar-\nization should also be produced just below Tcby rotating\na fixed magnetic field about the caxis. Above all, our\nwork illuminates a pathway to develop other functional\nmaterials with sizeable magnetic moments and electrical\npolarizations.\nAcknowledgments\nResearchsponsoredby the U.S. Department ofEnergy,\nOffice of Science, Basic Energy Sciences, Materials Sci-\nences and Engineering Division (RF), by the Hungarian\nResearch Funds OTKA K 108918, OTKA PD 111756,\nand Bolyai 00565/14/11 (SB, VK, and IK), and by the\ninstitutional research funding IUT23-3 of the Estonian\nMinistry of Education and Research and the European\nRegional Development Fund project TK134 (TR and\nUN).7\n1K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801\n(1986).\n2M.J. Harris, S.T. Bramwell, D.F. McMorrow, T. Zeiske,\nand K.W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997).\n3S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B¨ oni, Science323, 915\n(2011).\n4D.I. Khomskii, J. Magn. Magn. Mater. 306, 1 (2006).\n5S.-W. Cheong and M. Mostovoy, Nat. 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Caignaert, and\nB. Raveau, Phys. Rev. B 86, 184403 (2012).\n34S. Bord´ acs, V. Kocsis, Y. Tokunaga, U. Nagel, T. R˜ o˜ om,\nY. Takahashi, Y. Taguchi, and Y. Tokura, Phys. Rev. B\n92, 214441 (2015).\n35The THz spectra of CaBaCo 4O7were measured at 2.5 K\nwith linear light polarization using Fourier-transform sp ec-\ntroscopy [34]. The reference spectrum was taken in zero\nfield.\n36M. Valldor, J. Phys.: Cond. Mat. 16, 9209 (2004).\n37Spin excitations away from q=Qdepend separately on\nthese exchange constants. The stability of the coplanar\nground state requires that each pair of exchange constants\nbe sufficiently close to one another.\n38D.C. Khan and R.A. Erickson, Phys. Rev. B 1, 2243\n(1970).\n39P. Bourges, H. Casalta, A.S. Ivanov, and D. Petitgrand,\nPhys. Rev. Lett. 79, 4906 (1997).\n40While the coplanar LF spin state has a lower energy than\nthe HF state below 32 T, it becomes locally unstable to a\nbuckledstate with spins cantedoutof the abplane between\n16 and 32 T for field along a. Based on the smooth depen-\ndence of the magnetizations on field, the transition from\nthe coplanar to the buckled state must be second order.\n41V. Pralong, V. Caignert, T. Sarkar, O.I. Lebedev, V. Duf-\nfort, and B. Raveau, J. Sol. St. Chem. 184, 2588 (2011).\n42V. Koscis, S. Bord´ acs, and I. K´ ezsm´ arki, (unpublished).\n43Based on the symmetry of adjacent layers, it is easy to\nshow that the polarization operator P12associated with\nbond{1,2}has components\nP12\na∝C12−C56−C1′2′+C5′6′,\nP12\nb∝C12+C56−C1′2′−C5′6′,\nP12\nc∝C12+C56+C1′2′+C5′6′,\nwhereCij=Si·Sj. Similar relations hold for other bonds.\nOnly theccomponent Pij\nccan produce a static polariza-\ntion but all three components may contribute to the off-\ndiagonal polarization matrix elements ∝an}bracketle{t0|Pij\nα|n∝an}bracketri}ht(n∝ne}ationslash= 0).\n44S. Miyahara and N. Furukawa, J. Phys. Soc. Japan 80,\n073708 (2011)."    },    {        "title": "1112.1646v1.Magnetization_and_spin_gap_in_two_dimensional_organic_ferrimagnet_BIPNNBNO.pdf",        "content": "arXiv:1112.1646v1  [cond-mat.mes-hall]  7 Dec 2011Magnetization and spin gap in two-dimensional organic\nferrimagnet BIPNNBNO\nV.E. Sinitsyn, I.G. Bostrem, A.S. Ovchinnikov\nDepartment of Physics, Ural State University, 620083, Ekat erinburg, Russia\nY. Hosokoshi\nDepartment of Physical Science, Osaka Prefecture Universi ty, Osaka, Japan\nK. Inoue\nDepartment of Chemistry, Hiroshima University, Hiroshima, Japan\n(Dated: November 9, 2018)\nA magnetization process in two-dimensional ferrimagnet BI PNNBNO is analyzed.\nThe compound consists of ferrimagnetic (1,1/2) chains coup led by two sorts of anti-\nferromagnetic interactions. Whereas a behavior of the magn etization curve in higher\nmagnetic fields can be understood within a process for the sep arate ferrimagnetic\nchain, an appearance of the singlet plateau at lower fields is an example of non-Lieb-\nMattis typeferrimagnetism. By usingthe exact diagonaliza tion technique for a finite\nclusters of sizes 4 ×8 and 4×10 we show that the interchain frustration coupling\nplays an essential role in stabilization of the singlet phas e. These results are comple-\nmented by an analysis of four cylindrically coupled ferrima gnetic (1,1/2) chains via\nan abelian bosonization technique and an effective theory bas ed on the XXZ spin-\n1/2 Heisenberg model when the interchain interactions are s ufficiently weak/strong,\nrespectively.\nI. INTRODUCTION\nDuring the last fifteen years, two-dimensional (2D) quantum spin s ystems have attracted\na lot of attention both from theoretical and experimental physicis ts. A competition between\nconventional classically ordered phases and more exotic quantum o rdered phases lies in\nthe focus of the investigations. Magnetic systems with a finite corr elation length at zero\ntemperatureandafinitespingapabovethesingletgroundstate, s pinliquids, realizeHaldane2\nprediction at the level of two space dimensions1.\nTo date, one can distinguish two main routes in studies of 2D spin gap c ompounds. A\nformation of spin gap in spin dimer systems, for example SrCu 2(BO3)22and CaV 4O93, is\nexplained by a modified exchange topology similar to Shastry-Suther land lattice4. Another\nway to increase quantum fluctuations and stabilize a spin liquid ground state is realized\nin kagome antiferromagnet5. Experimental candidates for 2D kagome antiferromagnets are\ncurrently available: herbertsmithite6–8and volborthite9,10. Both these strategies deal with\nantiferromagnetic compounds. In view of this, an observation of a singlet ground state with\na pronounced spin gap in 2D ferrimagnetic material BIPNNBNO seems exotic12.\nThe crystal structure of BIPNNBNO is shown in Fig. 1. A magnetic un it of the spin\nsystem presents organic triradical BIPNNBNO. Each of the molecu les includes three s=1/2\na\nb\nFIG. 1: Magnetic model of BIPNNBNO crystal. The black (white ) circles denote spins S=1\n(s=1/2).\nspins (see Fig. 2) with intramolecular ferromagnetic JFand antiferromagnetic JAFinterac-\ntions. The magnitude of |JF| ∼300 K is very large, and two spins coupled ferromagnetically\nbehave as a S - 1 moiety. Ferrimagnetic chains are stretched along t he b-axis. There are two\nkinds of antiferromagnetic interchain interactions along the a-axis . One is between the s-1/2\nspins, which connects the nearest neighboring chains. The other is between S-1 species,\nwhich connects the next nearest neighboring chains and introduce s spin frustration.3\nS=1S=1/2 S=1/2\nJAFJAFJAF\nJFN+\nO- NO\nN N\nO OJAFJAF\nJF\nFIG. 2: Molecular structure of BIPNNBNO and the elementary m agnetic cell at JF≫ |JAF|.\nThe puzzle is following. It iswell known that a low-energy physics of an isolated ferrimag-\nnetic (S,s) chain corresponds to a gapless (S-s) ferromagnet13. Quite predictably one might\nexpect an appearance of an ordered state with fluctuations in the form of spin waves near\nthe classical state. However, measurements of magnetization sh ows that an opening of a gap\nby analogy with the Haldane chain is likely scenario. Such a behavior is a m anifestation of\nnon-Lieb-Mattis type ferrimagnetism11. Namely, the magnetization measured at 400 mK is\nnearly zero below 4.5 T, increases rapidly above 4.5 T and exhibits a bro ad 1/3 plateau and\na narrow 2/3 one at 7-23 T and around 26 T, respectively. Above 29 T, the magnetization\nis completely saturated12.\nThepurposeofthepaperistoinvestigatethemagnetizationproce ss. Theproblemiscom-\nplicated by a lack of reliable information about intra- and interchain ex change interactions.\nSo, before studying of the 2D ferrimagnetic system BIPNNBNO we d evelop a simple quan-\ntum mechanical approach that models a magnetization process of t he ferrimagnetic chain\n(1,1/2). The treatment agrees qualitatively with a predictions of th e theory for quantum\nspin chains14and provides reasonable estimations of the exchange intrachain couplings. In\naddition, it captures a peculiarity of a magnetization process in the p rototype 2D material,\ni.e. an appearance of the intermediate 2/3 plateau. Given these est imations we examine the\nmagnetization process in BIPNNBNO by analyzing exact diagonalizatio n (ED) calculations\nfor a finite clusters of size N= 32 andN= 40. A main conclusion to be drawn from these\ncalculations that an emergence of the anomalous singlet plateau is a c onsequence of the\nfrustrating interchain interaction.\nTwo different mechanisms of formation of the plateau may be likely can didates: a gener-4\nalization of Haldanes conjecture to the weakly coupled ferrimagnet ic chains, and a valence-\nbond (dimerized) type ground state in the strong-coupling limit. To d etermine what of these\nscenarios is relevant we develop low-energy effective theories for t he 4-legs spin tube, which\nforms a minimal setup including the interchain couplings. In the regime of weakly coupled\nspin tube legs we apply abelian bosonization technique. The opposite lim it of a strong-ring\ninteraction is analyzed in terms of an effective Heisenberg XXZ model where the intrachain\ncoupling is perturbatively taken into account. Our analytical treat ment shows that only the\nfirst approach confirms an important role of frustration in stabiliza tion of the singlet phase.\nNotethatastudyofspintubesisofinterest byitselfbecauseboth offrustrationandquan-\ntum fluctuation arestrong15. Our model isdirectly related with thecompound BIPNNBNO,\nbut the main results are expected to apply to other frustrated sp in tubes as well. Re-\ncently, it has been reported that the experimental candidate for the four-leg spin tube,\nCu2Cl4·D8C4SO2, is available16.\nThe paper is organized as follows: In Sec. II we consider a magnetiza tion process in the\nferrimagnetic chain (1 ,1/2). In Sec. III we discuss results of magnetization process in two-\ndimensional ferrimagnetic system obtained via the exact diagonaliza tion method on a finite\ncluster. In Sec. IV we derive effective low-energy spin-1/2 Hamilton ian. A bosonisation\nstudy of the spin tube is carried out in Sec. V. A discussion of these r esults is relegated to\nthe Conclusion part.\nII. MAGNETIZATION OF AN ISOLATED FERRIMAGNETIC CHAIN\nThe issue that we address below is whether a calculation for an isolate d ferrimagnetic\nchain partially reproduces features of the magnetization curve ob served in the BIPNNBNO\ncrystal. We demonstrate that both the 1/3 plateau and the 2/3 pla teau can be recovered\nwithin a simple quantum mechanical analysis of a magnetization proces s of an isolated\nquantum (1 ,1/2) ferrimagnetic chain under an applied magnetic field. This enables to\nestimate the intrachain exchange parameters JAFandJ1.5\nS=N/2\nS=N/2+1\nS=N\nS=N-1\nS=N+1\nS=3N/2-1\nS=3N/2\nFIG. 3: Schematic picture of states of the ferrimagnetic cha in used in construction of a magneti-\nzation curve. Excited blocks are marked by the gray shadow.\nWe start from the values of the critical fields derived from the ED da ta17\n\n\nB1=∂ε\n∂m/vextendsingle/vextendsingle\nm→1/3+0≈E(N/2+1)−E(N/2),\nB2=∂ε\n∂m/vextendsingle/vextendsingle\nm→2/3−0≈E(N)−E(N−1),\nB3=∂ε\n∂m/vextendsingle/vextendsingle\nm→2/3+0≈E(N+1)−E(N),\nBsat=∂ε\n∂m/vextendsingle/vextendsingle\nm→1−0≈E(3N/2)−E(3N/2−1),(1)\nwhereε,mare the energy and the magnetization per an elementary cell of the (1,1/2)\nferrimagnetic chain, Nis a number of the elementary cells.\nTo estimate the energies E(S) in the right-hand side of Eqs.(1), where Sis a total spin\nof the chain, we use the Hamiltonian of the one-dimensional quantum (1,1/2) ferrimagnet\nˆHc=JAFN/summationdisplay\ni=1/vectorS1i/vector s2i+J1N/summationdisplay\ni=1/vector s2i/vectorS1i+1(S1= 1, s2= 1/2), (2)6\nand construct the required quantum states |SM/an}bracketri}htwith the given quantum numbers of the\ntotal spinSand itsz-projection M.\nWe suppose that the 1 /3 magnetization plateau corresponds to the ground state of the\n(1,1/2) ferrimagnetic chain with S=N/2. The wave function of the polarized state is given\nby\n|N/2,N/2/an}bracketri}ht=N/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\ni, (3)\nand presents a direct product of the spin states of the magnetic e lementary cells (1,1/2) [see\nFig. 3]. The corresponding energy eigenvalue equals to\nE(N/2) =−JAFN−1\n9J1N. (4)\nWith an increasing of a magnetic field the ground state (3) is destroy ing and the state\nwithS=N/2 + 1 stabilizes. A low-lying excitation may be qualitatively considered as a\nforming of one triplet bond. The trial new wave function is\n|N/2+1,N/2+1/an}bracketri}ht=1√\nNN/summationdisplay\nk=1\nN/productdisplay\ni(/negationslash=k)=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\ni\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\nk=N/summationdisplay\nk=1αkΨk.(5)\nIt is composed from all arrangements of the excited block within the chain taken with equal\nweightsαk.\nBy introducing the state and calculating the matrix element\n/an}bracketle{tΨk|ˆHc|Ψk′/an}bracketri}ht=/bracketleftbigg\nE(N/2)+3\n2JAF+7\n18J1/bracketrightbigg\nδkk′−1\n3J1δk,k′±1 (6)\none obtains the relationship for the coefficients αk\n−1\n3J1αk−1+/bracketleftbigg\nE(N/2)+3\n2JAF+7\n18J1−E(N/2+1)/bracketrightbigg\nαk−1\n3J1αk+1= 0,(7)\nwhich is tantamount to\nE(N/2+1) =E(N/2)+3\n2JAF+7\n18J1−1\n3J1/parenleftbiggαk−1\nαk+αk+1\nαk/parenrightbigg\n. (8)\nThis expression includes two independent variational parameters αk−1/αkandαk+1/αk. The\nminimal value\nEmin(N/2+1) =E(N/2)+3\n2JAF−5\n18J1 (9)7\nis reached provided αk−1/αk=αk+1/αk= 1. This yields the critical magnetic field B1\ndestroying the 1 /3 plateau\nB1=3\n2JAF−5\n18J1. (10)\nTo find the critical fields B2andB3of the beginning and the end of the 2/3 plateau,\nrespectively, we construct the trial states\n|N,N/an}bracketri}ht=N/2/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i, (11)\n|N−1,N−1/an}bracketri}ht=1√\nNN/2/summationdisplay\nk=1/bracketleftBiggk−1/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i/bracketrightBigg\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2k−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2k\nN/2/productdisplay\ni=k+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i\n,(12)\n|N+1,N+1/an}bracketri}ht=1√\nNN/2/summationdisplay\nk=1/bracketleftBiggk−1/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i/bracketrightBigg\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2k−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2k\nN/2/productdisplay\ni=k+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i\n,(13)\nwhich are schematically shown in Fig. 3.\nBy the same manner we obtain\nB2=3\n2JAF+7\n18J1, B3=3\n2JAF+5\n6J1. (14)\nThe saturation field Bsatis determined with an aid of the trial wave functions\n|3N/2,3N/2/an}bracketri}ht=N/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\ni, (15)\n|3N/2−1,3N/2−1/an}bracketri}ht=1√\nNN/summationdisplay\nk=1\nN/productdisplay\ni(/negationslash=k)=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\ni\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\nk(16)\nthat results in\nBsat=3\n2JAF+3\n2J1. (17)\nGiven the experimental estimations for the 2D BIPNNBNO system, B1∼31K,B2≈\nB3∼35K, andBsat= 39K, we obtain from Eqs.(10,17) the values of the intrachain\nexchange couplings, JAF≈21KandJ1≈3.5K. By substituting them into Eq.(14) we get8\nthe critical fields of the 2/3 plateau, B2≈32.8K(24.4T) andB3≈34.4K(25.6T). A\nqualitativebehavioroftheferrimagneticchainmagnetizationcurve builtfromthesereference\npoints is depicted in Fig. 4. We emphasize especially that an emergence of the intermediate\n2/3 plateau is not related with interchain frustration effects.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s49/s47/s51/s50/s47/s51/s77/s47/s77\n/s115/s97/s116\n/s66/s40/s84/s41\nFIG. 4: Qualitative magnetization curve of the ferrimagnet ic (1,1/2) chain.\nIII. MAGNETIZATION: EXACT DIAGONALIZATION\nnnJ\nnnnJAFJ 1J\nFIG. 5: Cluster of N= 32 sites used in the exact diagonalization. The fixed exchan ge couplings\nareJAF= 21K,J1= 3.5K,Jnn= 0.5J1= 1.75K.\nIn order to understand a role of the interchain couplings the magne tization process of\nthe BIPNNBNO ferrimagnet was examined by the variant of a numeric al diagonalization\nmethod with conservation of a total cluster spin18,19.9\nThe model Hamiltonian is given by\nˆHclust=JAF/summationdisplay\nij/vectorSi/vector sj+J1/summationdisplay\nij/vectorSi/vector sj+Jnn/summationdisplay\nij/vector si/vector sj+Jnnn/summationdisplay\nij/vectorSi/vectorSj, (18)\nwhere/vectorSi(/vector si) denotes spin-1 (spin-1/2) operator at site i. The sublattices and the network\nof the antiferromagnetic interactions, JAF,J1,JnnandJnnn, are shown in Fig. 5. We\nperform calculation of the N-step magnetization curve for the N=3 2 cluster depicted in the\nsame Figure. The intrachain parameters, JAFandJ1, have been estimated in the previous\nSection whereas the interchain ones, JnnandJnnn, are assumed to be less than J1. The open\nboundary conditions are used for the numerical calculations.\nThe magnetization process is compared with the results of the mode l of non-interacting\n(1,1/2) ferrimagnetic chains. A standard way to build magnetization curve atT= 0\nis to define the lowest energy E(N,M) of the Hamiltonian (2) in the subspace where\n/summationtextN\nj=1/parenleftbig\nSz\nj+sz\nj/parenrightbig\n=Mfor a finite system of Nelementary ( S,s) blocks. Applying a magnetic\nfieldBleads to a Zeeman splitting of the energy levels, and therefore level crossing occurs on\nincreasing the field. These level crossing correspond to jumps in th e magnetization until the\nfully polarized state is reached at a certain value of the magnetic field . The magnetization\nof four independent chains is then derived from\nm= 4M/N, M = max[M|E(N,M+1)−E(N,M)>B], (19)\nwhich gives a step curve.\nAn importance of the frustrating coupling is seen from comparison o f two magnetization\ncurves displayed in Figs. 6, 7. They correspond to no frustration c ase and a pronounced\nfrustrating coupling, respectively. The magnetization curves exh ibit several interesting fea-\ntures. For instance, the magnetization behavior in higher magnetic fields (B > B 1) is well\nreproduced within the model of non-interacting ferrimagnetic cha ins. Another remarkable\nfeature revealed by Figs. 6, 7 that the singlet ground state platea u emerges at non-zero\nfrustration interaction whereas the narrow 2 /3 plateau appears regardless of the frustration.\nWe numerically found that the width ∆ Sof the singlet plateau scales almost linearly with\naJnnnvalue (Fig. 8). To check into the case of the dependence we repeat cacluations on\na cluster of larger size, N= 40, with the same set of parameters that support the finding.\nThe observation points out that the zero magnetization plateau ha s a quantum origin with\na crucial role of frustration which destroys a long-range order an d drives the system into the10\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48\n/s50/s47/s51\n/s49/s47/s51\n/s66/s40/s84/s41/s77/s47/s77\n/s115/s97/s116\nFIG. 6: Magnetization curve for the 32-site cluster. The exc hange couplings are taken as JAF=\n21K,J1= 3.5K,Jnn= 0.5J1, andJnnn= 0 (no frustration). The dotted line marks a calculation\nvia the model of non-interacting (1 ,1/2) chains.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s77/s47/s77\n/s115/s97/s116\n/s66/s40/s84/s41/s49/s47/s51/s50/s47/s51\nFIG. 7: Magnetization curve for the 32-site cluster. The exc hange couplings are taken as JAF=\n21K,J1= 3.5K,Jnn= 0.5J1,Jnnn= 0.075J1. The dotted line marks a calculation via the model\nof non-interacting (1 ,1/2) chains.11\n0□00. 0□02 . 0□04 . 0□06 . 0□08 . 0□10 . 0□12 . 0□14 .0□00.0□02.0□04.0□06.0□08. /c68S/JAF\nJnnn/JAF\nFIG. 8: Value of the singlet plateau ∆ Sas a function of the frustrating coupling Jnnnobtained on\na cluster of size N= 32 (black circles) and N= 40 (white circles).\nsinglet phase. Below, we address analytically the issue in the regimes o f strong and weak\ninterchain couplings.\nIV. A FORMATION OF THE SINGLET PLATEAU\nIn low-dimensional Heisenberg systems frustrating couplings can d rive transitions to gap-\nfull quantum states, where local singlets forma groundstate. Th ese quantum gappedphases\nmay have long-rangedsinglet order (valence bondstate), or realiz ea resonating valence band\nspin liquid. In last case, a ground state is a coherent superposition o f all lattice-coverings\nby local singlets21.\nTorecognizefeaturesofthesephasesintheEDresultsweundert akeanalytical treatments\nof the four-legs spin tube shown in Fig. 9. The new system is infinite alo ng theb-axis,\nand periodic with the 4-site period along the a-axis. The tube forms a minimal setup\nincluding the interchain nearest- and next-to-nearest neighbor c ouplings and contains the\nsame number of ferrimagnetic chains parallel to the b-axis as the clu sters in the ED study.\nAs we demonstrate below, the simplified model elucidates an importan t role of frustration\nin stabilization of the singlet phase.12\n1J\nnnJ\nnnnJ\nJ\nJJ’1\n2\n3412\n34\nFIG. 9: 4-leg spin tube structure used in the limit of strong r ing coupling (up) and in the case\nof weakly interacting chains (below). The black (white) cir cles mean spin-1 (spin-1/2) sites. The\ngray circles denote renormalized spin-1/2 blocks.\nThe singlet phase may arise in the limit of strong ring coupling Jnn,Jnnn≫J1. In this\ncase, the problem can be analyzed in terms of Heisenberg XXZ model similar to ladders in a\nmagnetic field20. The opposite limit ( J1≫Jnn,Jnnn) results in a scenario of weakly interact-\ning chains. Based on a block renormalization procedure the original s ystem is then mapped\nonto the model of a spin tube with four ferromagnetic spin-1/2 legs . We mention that the\nground state properties of two-leg spin ladders with ferromagnet ic intrachain coupling and\nantiferromagnetic interchain couplings have been discussed in Refs .30,31in an absence of an\nexternal field. A magnetization process of those spin ladders with a n even number of legs\n(2 and 4) has been studied in Ref.32in the regime of weak ferromagnetic coupling along the\nlegs and strong antiferromagnetic coupling along the rungs.\nAn appearance of the singlet phase in the frustrated spin tube with four weakly coupled\nferromagnetic spin-1/2 legs can be studied through the bosonizat ion technique which proves\neffectiveness for quasi-one-dimensional spin-one-half systems. To the best of our knowledge,\nthe system has never been previously reported, however our fur ther analysis follows closely\nto that of given in Ref.24, where 4-legs spin tube with antiferromagnetic chains and a specific13\nform of diagonal rung interactions (but with no frustration) has b een treated. Note as well\nthat spin ladders with ferromagnetic and ferrimagnetic legs are muc h less studied26,27by the\nbosonization approach in comparison with ladder systems with antife rromagnetic legs. The\nmain problem arising here is that the formalism is well defined only if ther e is an easy-plane\nexchange anisotropy. In this regard, we note that measurement s of the angular dependence\nof the ESR linewidth for the BIPNNBNO system showed that the large st linewidth was\nobserved for the field direction perpendicular to the ab plane28. Due to the theoretical\nconsideration by Oshikawa and Affleck29a critical regime of XY-anisotropy is expected in\nthe compound. In addition we point out that the ED algorithm invoked in the previous\nSection enables to treat clusters of sufficiently large sizes due to us e of the rotational SU(2)\nsymmetry. The latter is broken by the anisotropy whose role in a sing let gap formation is a\nsubject of future ED studies.\nA. Spin tube: weakly interacting rings and a model of a single XXZ chain\nWe study the Hamiltonian of the spin tube (see Fig. 9)\nH=N/summationdisplay\nn=1Hring\nn+J1N/summationdisplay\nn=1(Sn,1sn+1,1+sn,2Sn+1,2+Sn,3sn+1,3+sn,4Sn+1,4)−BN/summationdisplay\nn=14/summationdisplay\ni=1/parenleftbig\nSz\nn,i+sz\nn,i/parenrightbig\n,\n(20)\nwhere the Hamiltonian of the separate ring is\nHring\nn=Jnn(sn,1sn,2+sn,2sn,3+sn,3sn,4+sn,4sn,1)+JnnnN/summationdisplay\nn=1(Sn,1Sn,3+Sn,2Sn,4).\nHere,S= 1 ands= 1/2,nis the index of the ring, Nis the total number of rings, and the\nindeximarks the (1,1/2) blocks inside the rings. Periodic boundary conditio ns along the\ntube direction are imposed. In our model it is suggested that Jnn,Jnnn≫J1.\nIn the limit J1= 0 the system decouples into a collection of nonineracting rings. At z ero\nmagnetic field the singlet and triplet states\n|ψ0/an}bracketri}ht=−√\n3\n2|00;00/an}bracketri}ht+1\n2|11;00/an}bracketri}ht,\n|ψ1/an}bracketri}ht=1√\n2|01;11/an}bracketri}ht+1√\n2|10;11/an}bracketri}ht (21)\nhave the lowest energies E0=−2Jnn/9+8Jnnn/9 andE1=−Jnn/9+8Jnnn/9, respectively.\nThestatesofthering |S12S34;SM/an}bracketri}htareobtainedviathecommonruleofadditionofmoments,14\nwhereS12(S34) is spin of dimer composed of the spins of the 1 and 2 (3 and 4) blocks. The\nsinglet and triplet states of the ring that enter into (21) are given in Appendix A.\nUpon increasing the magnetic field a transition between the singlet an d triplet states\noccurs atB=Jnn/9 and the the total magnetization jumps abruptly from zero to M=N.\nAtnon-zeroringcouplingthesharptransitionisbroadenedandsta rtsfromacriticalvalue\nB0. To find the field we derive the XXZ spin chain Hamiltonian by we using the standard\napproach that is analogous to study of spin-1/2 ladder with strong rung exchange20.\nThe Hamiltonian (20) is splitted into two parts\nH=H0+H1.\nH0=N/summationdisplay\nn=1Hring\nn−Bc4/summationdisplay\ni=1N/summationdisplay\nn=1(Sz\nin+sz\nin),\nH1=J1N/summationdisplay\nn=1[Sn,1sn+1,1+sn,2Sn+1,2+Sn,3sn+1,3+sn,4Sn+1,4]−(B−Bc)4/summationdisplay\ni=1N/summationdisplay\nn=1(Sz\nin+sz\nin),\nwhereBc=E1−E0. TheH1lifts the 2N-fold degeneracy of the ground state of the\nHamiltonian H0. The later can be either in the state |ψ0/an}bracketri}htor|ψ1/an}bracketri}ht. By using the standard\nmany body perturbation theory22the effective Hamiltonian can be derived\nHeff=Jeff\nxyN/summationdisplay\nn=1/parenleftBig\n˜Sx\nn˜Sx\nn+1+˜Sy\nn˜Sy\nn+1/parenrightBig\n+Jeff\nzN/summationdisplay\nn=1˜Sz\nn˜Sz\nn+1−Heff\nzN/summationdisplay\nn=1˜Sz\nn, (22)\nwhereJeff\nxy=−16J1/27,Jeff\nz=−J1/9 andBeff\nz=J1/9+B−Bc.\nTo get the expression the pseudo-spin ˜Si= 1/2 operators that act on the states |ψ0/an}bracketri}htand\n|ψ1/an}bracketri}htare introduced\n˜Sz\nn|ψ0/an}bracketri}htn=−1\n2|ψ0/an}bracketri}htn,˜Sz\nn|ψ1/an}bracketri}htn=1\n2|ψ1/an}bracketri}htn,\n˜S+\nn|ψ0/an}bracketri}htn=|ψ1/an}bracketri}htn,˜S+\nn|ψ1/an}bracketri}htn= 0,\n˜S−\nn|ψ0/an}bracketri}htn= 0,˜S−\nn|ψ1/an}bracketri}htn=|ψ0/an}bracketri}htn. (23)\nThe starting spin-1 operators and the pseudo-spin operators in t he restricted space are\nrelated by\nSz\nin=1\n6+1\n3˜Sz\nn,\nS+\nin= (−1)i−14\n3√\n3˜S+\nn, S−\nin= (−1)i−14\n3√\n3˜S−\nn. (24)15\nThe corresponding map for the spin-1/2 operators is\nsz\nin=−1\n24−1\n12˜Sz\nn,\ns+\nin=(−1)i\n3√\n3˜S+\nn, s−\nin=(−1)i\n3√\n3˜S−\nn. (25)\nThe Jordan-Wigner transformation maps the Hamiltonian (22) onto a system of inter-\nacting spinless fermions\nHsf=tN/summationdisplay\nn=1/parenleftbig\nc+\nici+1+c+\ni+1ci/parenrightbig\n+VN/summationdisplay\nn=1nini+1−µN/summationdisplay\nn=1ni, (26)\nwheret=Jeff\nxy/2,V=Jeff\nzandµ=Jeff\nz+Beff\nz.\nThe lowest critical field B0corresponds to that value of thechemical potential µfor which\nthe band of spinless fermions starts to fill up. This yields the conditio nµ=−2tand leads\nto the result\nB0=1\n9Jnn+16\n27J1. (27)\nThe critical value involves no frustration parameter Jnnnthat is clearly contrary to the ED\nresults.\nA similar analysis can be carried out for the saturation field. The deta ils of the calcula-\ntions are relegated to Appendix B.\nB. Spin tube: a model of weakly interacting ferromagnetic legs and abelian\nbosonization\nTo apply the bosonization we should map the initial system, consisting of two sorts of\nspins, spin-1/2 and spin-1, to the spin-1/2 system by using the qua ntum renormalization\ngroup (QRG) in real space based on the block renormalization proce dure23. To exploit\nthe real-space QRG technique one divide the spin lattice into small bloc ks, namely, the\nintrachain dimers (1,1/2), and obtains the lowest energy states {|α/an}bracketri}ht}of each isolated block.\nThe effect of inter-block interactions is then taken into account by constructing an effective\nHamiltonian Heffwhich now acts on a smaller Hilbert space embedded in the original one.\nIn this new Hilbert space each of the former blocks is treated as a sin gle site. The effective\nHamiltonians Heff=Q†HQis constructed via the projection operator Q=N/producttext\ni=1Qiwith16\nQi=m/summationtext\nα=1|α/an}bracketri}ht/an}bracketle{tα|of eachi-th block where mis the number of low energy states that are kept\nandNis a number of lattice cells.\nWe hold the lowest doublet S−1/2 to find an effective low-energy Hamiltonian. The\nhigher energy S−3/2 states are neglected. One can check that the reduced matrix ele ments\nare (S1= 1,s2= 1/2)\n/an}bracketle{t11/2;1/2||S1||11/2;1/2/an}bracketri}ht= 2/radicalbigg\n2\n3,\n/an}bracketle{t11/2;1/2||s2||11/2;1/2>=−1√\n6.\nTherefore the effective spin-1/2 operators of the renormalized c hain are\nQ†\ni/vectorS1iQi=4\n3/vectorSi, Q†\ni/vector s2iQi=−1\n3/vectorSi(S= 1/2). (28)\nThe renormalized Hamiltonian of the intrachain interactions corresp onds to the ferromag-\nnetic Heisenberg spin-1/2 model with the exchange coupling J=−4J1/9 (Fig. 9). The\ninterchain interactions between the nearest neighbors, spins -1/ 2, and next to the nearest\nneighbors, spins -1, are renormalized as J⊥=Jnn/9 andJ′\n⊥= 16Jnnn/9, respectively.\nConsider a four-legs spin tube consisting of spin-1 /2 chains. The Hamiltonian of the\nsystem is\nˆHtube=4/summationdisplay\nλ=1ˆHλ+ˆH⊥\n12+ˆH⊥\n23+ˆH⊥\n34+ˆH⊥\n14+ˆH′⊥\n13+ˆH′⊥\n24. (29)\nThe spins along the chains are coupled ferromagnetically, the Hamilto nian for the separate\nλ-th chain is\nˆHλ=−JxyN/summationdisplay\ni=1/parenleftbig\nSx\nλ,jSx\nλ,j+1+Sy\nλ,jSy\nλ,j+1/parenrightbig\n−JzN/summationdisplay\ni=1Sz\nλ,jSz\nλ,j+1,\nwhereSx,y,z\nλ,jare the spin S=1/2 operators at the jth site, the intraleg coupling is ferromag-\nnetic,J >0.\nThe interaction parts are given by\nˆH⊥\nλλ′=Jxy\n⊥,λλ′N/summationdisplay\nj=1/parenleftbig\nSx\nλ,jSx\nλ′,j+Sy\nλ,jSy\nλ′,j/parenrightbig\n+Jz\n⊥,λλ′N/summationdisplay\ni=1Sz\nλ,jSz\nλ,j, (30)\nˆH′⊥\nλλ′=J′xy\n⊥,λλ′N/summationdisplay\nj=1/parenleftbig\nSx\nλ,jSx\nλ′,j+Sy\nλ,jSy\nλ′,j/parenrightbig\n+J′z\n⊥,λλ′N/summationdisplay\ni=1Sz\nλ,jSz\nλ,j (31)17\nandincludes thenearest, J⊥>0, andthenext-to-nearest, J′\n⊥>0, antiferromagneticinterleg\ncouplings.\nThe unitary transformation keeping spin commutation relations\nSx,y\nλ,j→(−1)jSx,y\nλ,j, Sz\nλ,j→Sz\nλ,j\nmaps the Hamiltonian (29) to the Hamiltonian with antiferromagnetic le gs. It changes\nJxy→ −JxyandJz→Jz, and the ferromagnetic isotropic point is ∆ = Jz/Jxy=−1 in\nthe Hamiltonian with the antiferromagnetic legs.\nFollowing the general procedure of transforming a spin model to an effective model of\ncontinuum field, we convert the spin Hamiltonian of the spin tube with antiferromagnetic\nlegs to a Hamiltonian of spinless fermions using Jordan-Wigner transf ormation, then map\nit to a modified Luttinger model. The bosonic expressions for spin ope rators are\nS+\nλ(x) =S+\njλ\na=e−i√πΘλ\n√\n2πa/bracketleftBig\ne−i(πx/a)+cos/parenleftBig√\n4πΦλ/parenrightBig/bracketrightBig\n,\nSz\nλ(x) =Sz\njλ\na=1√π∂xΦλ+1\nπaei(πx/a)sin/parenleftBig√\n4πΦλ/parenrightBig\n, (32)\nwhere Φ and Θ are the bosonic dual fields, and xis defined on the lattice, xj=ja,ais a\nshort-distance cutoff.\nThe bosonized form of the Hamiltonian of the non-interacting chains is\nHλ=u\n2/integraldisplay\ndx/bracketleftbigg\nKΠ2\nλ+1\nK(∂xΦλ)2/bracketrightbigg\n, (33)\nwhere Π λ(x) =∂xΘλis canonically conjugate momentum to Φ λ. The Luttinger liquid\nparameters are fixed from the Bethe ansatz solution33\nK=π\n2(π−arccos∆), u=Jxyπ√\n1−∆2\n2arccos∆. (34)\nThe velocity uvanishes and Kdiverges for ∆ = −1. This corresponds to the ferromagnetic\ninstability point of a single chain.\nThe interchain interactions (30) between the nearest neighbor ch ains reads as\nH⊥\nλλ′=Jz\n⊥,λλ′/integraldisplaydx\nπ(∂xΦλ)(∂xΦλ′)+g1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4π(Φλ+Φλ′)/parenrightBig\n+g2/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4π(Φλ−Φλ′)/parenrightBig\n+g3/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig18\n+g4/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig\ncos/parenleftBig√\n4π(Φλ+Φλ′)/parenrightBig\n+g5/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig\ncos/parenleftBig√\n4π(Φλ−Φλ′)/parenrightBig\n, (35)\nwhereg1=−2Jz\n⊥,λλ′,g2= 2Jz\n⊥,λλ′,g3= 2πJxy\n⊥,λλ′,g4=g5=πJxy\n⊥,λλ′. The Hamiltonian (31)\nof the next-to-nearest couplings has a similar form with a formal ch angeJz\n⊥,λλ′→J′z\n⊥,λλ′,\ng1→g′\n1=−2J′z\n⊥,λλ′,g2→g′\n2= 2J′z\n⊥,λλ′etc.\nFollowing the route of Refs.24,25it is convenient to introduce a symmetric mode Φ sand\nthree antisymmetric ones Φ a1, Φa2, Φa3\nΦs=1\n2(Φ1+Φ2+Φ3+Φ4),\nΦa1=1\n2(Φ1+Φ2−Φ3−Φ4),\nΦa2=1\n2(Φ1−Φ2−Φ3+Φ4),\nΦa3=1\n2(Φ1−Φ2+Φ3−Φ4).(36)\nIn terms of the new fields the quadratic part of the Hamiltonian (29) is diagonalized to\nH0=us\n2/integraldisplay\ndx/bracketleftbigg\nKsΠ2\ns+1\nKs(∂xΦs)2/bracketrightbigg\n+3/summationdisplay\ni=1uai\n2/integraldisplay\ndx/bracketleftbigg\nKaiΠ2\nai+1\nKai(∂xΦai)2/bracketrightbigg\n(37)\nwith\nus=u/parenleftbigg\n1+2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K s=K/parenleftbigg\n1+2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n,\nua1=ua2=u/parenleftbigg\n1−KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K a1=Ka2=K/parenleftbigg\n1−KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n,\nua3=u/parenleftbigg\n1−2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K a3=K/parenleftbigg\n1−2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n.(38)\nThe relevant and marginally relevant terms of the interchain coupling s are given by\nHint= 2g12/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦs/parenrightBig\ncos/parenleftBig√\n4πΦai/parenrightBig\n+2g′\n1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦs/parenrightBig\ncos/parenleftBig√\n4πΦa3/parenrightBig\n+2g22/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦai/parenrightBig\ncos/parenleftBig√\n4πΦa3/parenrightBig\n+2g′\n2/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦa1/parenrightBig\ncos/parenleftBig√\n4πΦa2/parenrightBig\n+2g32/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftbig√πΘai/parenrightbig\ncos/parenleftbig√πΘa3/parenrightbig\n+2g′\n3/integraldisplaydx\n(2πa)2cos/parenleftbig√πΘa1/parenrightbig\ncos/parenleftbig√πΘa2/parenrightbig\n+g4/integraldisplaydx\n(2πa)2/bracketleftBig\ncos/parenleftbig√π(Θa2+Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa1)/parenrightBig19\n+cos/parenleftbig√π(Θa1−Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa2)/parenrightBig\n+cos/parenleftbig√π(Θa2−Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa1)/parenrightBig\n+cos/parenleftbig√π(Θa1+Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa2)/parenrightBig/bracketrightBig\n+g′\n4/integraldisplaydx\n(2πa)2/bracketleftBig\ncos/parenleftbig√π(Θa1+Θa2)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa3)/parenrightBig\n+cos/parenleftbig√π(Θa1−Θa2)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa3)/parenrightBig/bracketrightBig\n. (39)\nTheg5terms are irrelevant and are omitted.\nThe Hamiltonian (37) describes four independent gapless spin-1 /2 chains coupled by the\ninterchain interaction in the form of Eq.(39). It is expected that th e interleg coupling results\nin the Haldane gap in the excitation spectrum. Note that in the vicinity of the single chain\nferromagnetic instability, ∆ = −1, the effective bandwidth collapses, u→0, and the effect\nof the interleg couplings becomes crucial. To find a detail behavior of the gap in the phase\ndiagram with antiferromagnetic ( J⊥,J′\n⊥>0) interleg coupling and ferromagnetic leg regime\n(∆<0) we use the renormalization group analysis.\nThe RG equations are derived through the standard technique (se e Ref.34, for example).\nThe result is\ndg1\ndl= [2−(Ks+Ka1)]g1,\ndg′\n1\ndl= [2−(Ks+Ka3)]g′\n1,\ndg2\ndl= [2−(Ka3+Ka1)]g2,\ndg′\n2\ndl= [2−2Ka1]g′\n2,\ndg3\ndl=/bracketleftbigg\n2−1\n4/parenleftbigg1\nKa1+1\nKa3/parenrightbigg/bracketrightbigg\ng3,\ndg′\n3\ndl=/bracketleftbigg\n2−1\n2Ka1/bracketrightbigg\ng′\n3,\ndg4\ndl=/bracketleftbigg\n2−/parenleftbigg\nKs+Ka1+1\n4Ka1+1\n4Ka3/parenrightbigg/bracketrightbigg\ng4,\ndg′\n4\ndl=/bracketleftbigg\n2−/parenleftbigg\nKs+Ka3+1\n2Ka1/parenrightbigg/bracketrightbigg\ng′\n4,\ndKs\ndl=−4g2\n1/parenleftbiggKs\n4πus/parenrightbigg2\n−2g′\n12/parenleftbiggKs\n4πus/parenrightbigg2\n−2g2\n4/parenleftbiggKs\n4πus/parenrightbigg2\n−g′\n42/parenleftbiggKs\n4πus/parenrightbigg2\n,\ndKa1\ndl=−1\n2g2\n1/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n2g2\n2/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n2g′\n22/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n4g2\n4/parenleftbiggKa1\n2πua1/parenrightbigg220\n+1\n2/parenleftbiggg3\n4πua1/parenrightbigg2\n+1\n2/parenleftbiggg′\n3\n4πua1/parenrightbigg2\n+1\n4/parenleftbiggg4\n4πua1/parenrightbigg2\n+1\n4/parenleftbiggg′\n4\n4πua1/parenrightbigg2\n,\ndKa3\ndl=−1\n2g′\n12/parenleftbiggKa3\n2πua3/parenrightbigg2\n−g2\n2/parenleftbiggKa3\n2πua3/parenrightbigg2\n−1\n4g′\n42/parenleftbiggKa3\n2πua3/parenrightbigg2\n+/parenleftbiggg3\n4πua3/parenrightbigg2\n+1\n2/parenleftbiggg4\n4πua3/parenrightbigg2\n. (40)\nOne sees that the g1terms are relevant for Ks+Ka1<2; theg′\n1term is relevant for\nKs+Ka3<2; theg2terms are relevant for Ka3+Ka1<2; theg′\n2term is relevant for\nKa1<1; theg3term is relevant for K−1\na1+K−1\na3<8; theg′\n3term is relevant for Ka1>1/4.\nDespite the g4andg′\n4terms are irrelevant they are the most relevant terms which couple\nthe symmetric and antisymmetric modes24.\nUsing the RG equations the behavior of the gap in the whole phase diag ram can be\nestablished. Following to standard routine, we analyze the effect of the transversal ( Jxy\n⊥)\nand the longitudinal ( Jz\n⊥) parts of the interleg coupling separately.\n1. Transversal part of the interleg interactions\nIn this case Jxy\n⊥,J′xy\n⊥/ne}ationslash= 0 andJz\n⊥,J′z\n⊥= 0, the initial values of the coupling constants are\ngiven byg1(l= 0) =g′\n1(l= 0) = 0,g2(l= 0) =g′\n2(l= 0) = 0,g3(l= 0) = 2πJxy\n⊥,g′\n3(l= 0) =\n2πJ′xy\n⊥,g4(l= 0) =πJxy\n⊥andg′\n4(l= 0) =πJ′xy\n⊥. The bare Luttinger parameters are us=u,\nua1=ua2=ua3=u, andKs(l= 0) =K,Ka1(l= 0) =Ka2(l= 0) =Ka3(l= 0) =K.\nThe termg3is relevant for −1≤∆≤0 while the g4term is irrelevant. It is easily checked\nnumerically that g3,g′\n3grow whereas g4,g′\n4decrease under the RG. It means that Θ 1, Θ2, Θ3\nare locked in one of the vacuum states (Θ 1= Θ2= 0, Θ 3=√πor Θ1= Θ2=√π, Θ3= 0\nprovidedJxy\n⊥>J′xy\n⊥), fluctuations of the fields Θ ai(i= 1,2,3) are completely suppressed.\nTherefore arbitrary Jxy\n⊥> J′xy\n⊥/ne}ationslash= 0 generate a gap in the antisymmetric modes (Θ iare\npinned, disordered).\nAfter the fluctuations of the fields Θ iare stopped, the infrared behavior of the symmetric\nmode is governed by the term of the coupling with the antisymmetric m odes\n˜Hint= 2˜g42/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos(√\n4πΦs)cos(√\n4πΦai)\n+2˜g′\n4/integraldisplaydx\n(2πa)2cos(√\n4πΦs)cos(√\n4πΦa3). (41)21\nwhere\n˜g4= ¯g4/an}bracketle{tcos/parenleftbig√π[Θa1+Θa3]/parenrightbig\n/an}bracketri}ht= ¯g4/an}bracketle{tcos/parenleftbig√π[Θa2+Θa3]/parenrightbig\n/an}bracketri}ht,\n˜g′\n4= ¯g′\n4/an}bracketle{tcos/parenleftbig√π(Θa1+Θa2)/parenrightbig\n/an}bracketri}ht,\nand ¯g4, ¯g′\n4are renormalized couplings provided g3,g′\n3=O(1). Here, the invariance of ˜Hint\ngiven by Eq.(39) under Θ ai→ −Θaiyields/an}bracketle{tcos/parenleftbig√π(Θai−Θaj)/parenrightbig\n/an}bracketri}ht=/an}bracketle{tcos/parenleftbig√π(Θai+Θaj)/parenrightbig\n/an}bracketri}ht.\nDespiteeiΦaihas exponentially decaying correlations due to the Θ aiare pinned, a scrupulous\nanalysis24shows that the effective Hamiltonian for Φ spresents a standard sine-Gordon\nHamiltonian\n˜Heff=us\n2/integraldisplay\ndx/bracketleftbigg\n¯KsΠ2\ns+1\n¯Ks(∂xΦs)2/bracketrightbigg\n+g/integraldisplaydx\n(2πa)2cos/parenleftBig√\n16πΦs/parenrightBig\n,(42)\nwhere¯Ksis a renormalized value of K, andgis a new effective coupling constant. From\nthe correlation function /an}bracketle{texp/bracketleftbig\ni√\n16πΦs(x)/bracketrightbig\nexp/bracketleftbig\ni√\n16πΦs(y)/bracketrightbig\n/an}bracketri}ht= (a2/|x−y|2)4¯Ksit follows\nthat thegterm has a scale dimension 4 ¯Ks. Therefore, it is relevant for ¯Ks<1/2, when Φ s\nis pinned, i.e. becomes massive35.\nTo summarize, the transversal part of the interleg coupling suppo rts gapped antisymmet-\nric modes, the symmetric sector is gapped at ¯Ks<1/2, and remains gapless at ¯Ks>1/2.\nThe condition ¯Ks= 1/2 determines a boundary between the gapless Spin Liquid XY1\nphase36, and a generalization of the gapped Rung Singlets phase37for the for-leg spin tube\n(Fig.10). (Hereinafter, we retain names of phases used in the theo ry of spin ladders with\nferromagnetic legs.) In thelast case, spins onthe same rungs, or a longthe shortest diagonals\nform singlet pairs by a dynamical way.\n2. Longitudinal part of the interleg interactions\nFor the case of the longitudinal part of the interleg exchange, Jxy\n⊥,J′xy\n⊥= 0 andJz\n⊥,J′z\n⊥/ne}ationslash=\n0, the bare values of the coupling constants are given by g1(l= 0) =−2Jz\n⊥,g′\n1(l= 0) =\n−2J′z\n⊥,g2(l= 0) = 2Jz\n⊥,g′\n2(l= 0) = 2J′z\n⊥,g3(l= 0) =g′\n3(l= 0) = 0, and g4(l= 0) =g′\n4(l=\n0) = 0.\nThe strong-coupling phase diagramin the vicinity of ferromagnetic in stability point (∆ =\n−1andJz\n⊥,J′z\n⊥= 0)obtainedintheRGanalysisisshowninFig. 11. Inthesectordenot edas\nspin liquid II phase37theg1,2,g′\n1,2terms are irrelevant. The symmetric and antisymmetric\nmodes remain gapless. In the sector marked as a Haldane phase the termsg1,2,g′\n1,2are22\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48/s45/s49/s44/s48/s48/s45/s48/s44/s57/s56\n/s82/s117/s110/s103/s32/s83/s105/s110/s103/s108/s101/s116/s115\n/s74 /s47/s74/s32/s74 /s39/s47/s74 /s61/s48/s46/s50\n/s83/s112/s105/s110/s32/s76/s105/s113/s117/s105/s100/s32/s88/s89/s49/s32/s80/s104/s97/s115/s101\nFIG. 10: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\ntransverse coupling between legs.\nrelevant. Since all of the modes are coupled and locked together, b oth the symmetric and\nantisymmetric modes are gapped.\nThe phase of a ferromagnet with antiphase interchain order arises as a result of the\nferromagnetic instability with increasing interleg antiferromagnetic coupling. The boundary\nof the transition into the phase is obtained by studying the velocity r enormalization of the\ncorresponding gapless excitations. We mark the transition at uai= 0 (i= 1,2,3).\n3. Isotropic interleg exchange\nThe initial values of the coupling constants are g1(l= 0) =−2J⊥,g′\n1(l= 0) =−2J′\n⊥,\ng2(l= 0) = 2J⊥,g′\n2(l= 0) = 2J′\n⊥,g3(l= 0) = 2πJ⊥,g′\n3(l= 0) = 2πJ′\n⊥,g4(l= 0) =πJ⊥and\ng′\n4(l= 0) =πJ′\n⊥.\nFromtheRGequations(40)itisseenthatthemostrelevantoperat orsaretheg3,g′\n3terms.\nTherefore, the antisymmetric sector is gapped, and Θ aiare locked in the disordered phase.\nAs in the case of the transversal interleg interactions an effective sine-Gordon Hamiltonian\nfor the symmetric mode determines phase boundary ¯Ks= 1/2 between gapped and gapless\nphases. Numerical analysis shows that the ground state phase dia gram consists of the\ndisordered Rung Singlet gapfull phase and the stripe ferromagnet ic phase with dominating23\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50\n/s72/s97/s108/s100/s97/s110/s101/s32/s80/s104/s97/s115/s101\n/s83/s112/s105/s110/s32/s76/s105/s113/s117/s105/s100/s32/s73/s73/s32/s80/s104/s97/s115/s101/s32/s74\n/s122/s32/s39/s47/s74\n/s122/s32/s61/s32/s48/s46/s50\n/s74\n/s32/s47/s32/s74/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s32\n/s40/s97/s110/s116/s105/s112/s104/s97/s115/s101/s32/s105/s110/s116/s101/s114/s99/s104/s97/s105/s110/s32/s111/s114/s100/s101/s114/s41\n/s122\nFIG. 11: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\nlongitudinal coupling between legs.\nintraleg ferromagnetic ordering. The sector of the rung singlet ph ase increases at J′\n⊥→J⊥\n(see Fig. 12).\nA possible physical picture that reconciles the phase diagram with th e results of the\nprevious sections might look as follows. The system BIPNNBNO is an ar ray ofloosely\ncoupled ferrimagnetic chains in a presence of an extremely weak XY anisotropy and a strong\nfrustration, J′\n⊥∼J⊥, what corresponds to the region of the disordered Rung Singlet ph ase\nclose the point of the ferromagnetic instability, ∆ = −1.\nV. CONCLUSION\nWehave studied magnetizationprocess intwo-dimensional compoun dBIPNNBNO which\nexhibits ferrimagnetism of non-Lieb-Mattis type. The investigation is complicated by a lack\nof reliable information about exchange interactions in the system. F or a start, we proposed\nthenaive model ofnon-interacting ferrimagnetic chains andshowe d that anappearance both\n1/3 and 2/3 plateaus can be explained within the model. This provides u s the intrachain\nexchange couplings JAFandJ1. By setting these parameters in the exact diagonalization24\n/s48/s44/s48 /s48/s44/s49 /s48/s44/s50 /s48/s44/s51 /s48/s44/s52 /s48/s44/s53 /s48/s44/s54 /s48/s44/s55 /s48/s44/s56 /s48/s44/s57 /s49/s44/s48/s45/s49/s44/s48/s45/s48/s44/s57/s45/s48/s44/s56/s45/s48/s44/s55/s45/s48/s44/s54/s45/s48/s44/s53/s45/s48/s44/s52/s45/s48/s44/s51/s45/s48/s44/s50\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116\n/s40/s97/s110/s116/s105/s112/s104/s97/s115/s101/s32/s105/s110/s116/s101/s114/s99/s104/s97/s105/s110/s32/s111/s114/s100/s101/s114/s105/s110/s103/s41/s68/s105/s115/s111/s114/s100/s101/s114/s101/s100/s32\n/s82/s117/s110/s103/s32/s83/s105/s110/s103/s108/s101/s116/s115/s32\n/s74 /s32/s47/s74\nFIG. 12: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\nisotropic coupling between legs. The phase boundary betwee n the disordered rung singlet gapfull\nphase and the stripe ferromagnetic phase is shown by the dott ed and solid lines for J′\n⊥/J⊥= 0.2\nandJ′\n⊥/J⊥= 1.0, respectively.\nroutine the magnetization curve for the 32 and 40-sites clusters is numerically calculated.\nWe demonstrate that the magnetization curve similar to that obser ved in the experiment\nis obtained in the regime of weak interchain coupling, J1≫Jnn> Jnnn. Another revealed\nphenomenon is that a width of the singlet plateau increases with a gro wth of the antiferro-\nmagnetic frustrating coupling between the next-to-nearest cha ins. Following these results,\nwe apply on the tube lattice two low-energy theories which could expla in an appearance of\nthe singlet phase. The first one is based on the effective XXZ Heisenb erg model in a longitu-\ndinal magnetic field in the limit where the interchain coupling dominates, Jnn≥Jnnn≫J1.\nWe derive the critical field destroying the singlet plateau, it turns ou t that it does not\ndepend on the frustration parameter Jnnn. Another analytical strategy is realized via the\nabelian bosonization formalism which is relevant for the opposite limit J1≫Jnn≥Jnnn.\nWe demonstrate that the gapfull disordered Rung Singlets phase comes up when the XY\nexchange anisotropy may tilt the balance from the long-range orde r with an antiphase in-\nterchain arrangement of ferrimagnetic chains towards the spin liqu id phase. A role of the25\nanisotropy in a formation of the spin gap in the original two-dimension al system deserves a\nfurther study.\nAcknowledgments\nWe acknowledge Grant RFBR No. 10-02-00098-a for a support. We wish to thank Prof.\nD.N. Aristov for discussions.\nAppendix A: Wave functions of the ring\nThe states of the ring |S12S34SM/an}bracketri}htare composed from the functions |m1m2m3m4/an}bracketri}ht, where\nmi=±1/2 marks the spin-1/2 state of the separate i-th (1,1/2) block in the ring\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle11\n2;1\n2±1\n2/angbracketrightbigg\n=±/radicalbigg\n2\n3|1±1/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2∓1\n2/angbracketrightbigg\n∓1√\n3|10/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2±1\n2/angbracketrightbigg\n.\nThe basic functions of the singlet states are given by\n|00;00/an}bracketri}ht=1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n2−1\n2/angbracketrightbigg\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n2−1\n21\n2/angbracketrightbigg\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n2−1\n2/angbracketrightbigg\n+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n2−1\n21\n2/angbracketrightbigg\n,\n|11;00/an}bracketri}ht=1√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2−1\n2−1\n2/angbracketrightbigg\n+1√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2−1\n21\n21\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n2−1\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n2−1\n21\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n2−1\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n2−1\n21\n2/angbracketrightbigg\n.\nThe triplet states read as\n|01;11/an}bracketri}ht=1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n21\n2/angbracketrightbigg\n−1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n21\n2/angbracketrightbigg\n,\n|10;11/an}bracketri}ht=1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n21\n2−1\n2/angbracketrightbigg\n−1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2−1\n21\n2/angbracketrightbigg\n.\nAppendix B: Saturation field in the limit of the strong ring coupling\nTo get the saturation field the functions of the ring with the total s pinsS= 5 andS= 6\n|ψ6/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4,\n|ψ5/an}bracketri}ht=−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n426\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n4.\nwith the energies E5=JAF/2−Jnn/3+Jnnn/2,E6= 2JAF+Jnn+2Jnnnare needed.\nBy introducing the pseudo-spin operators in the restricted space similar to Eq.(23) the\noriginal spin operators are presented as follows ( S= 1,s= 1/2)\nSz\nin=23\n24+1\n12˜Sz\nn, sz\nin=5\n12+1\n6˜Sz\nn,\nS+\nin= (−1)i+11\n2/radicalbigg\n2\n3˜S+\nn, s+\nin= (−1)i1\n2/radicalbigg\n2\n3˜S+\nn,\nS−\nin= (−1)i+11\n2/radicalbigg\n2\n3˜S−\nn, s−\nin= (−1)i1\n2/radicalbigg\n2\n3˜S−\nn. 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B 67, 064419 (2003)."    },    {        "title": "1909.05936v3.Magnetostrictively_induced_stationary_entanglement_between_two_microwave_fields.pdf",        "content": "Magnetostrictively induced stationary entanglement between two microwave fields\nMei Yu,1Heng Shen,2and Jie Li1, 3,\u0003\n1Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou, Zhejiang, China\n2Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK\n3Kavli Institute of Nanoscience, Department of Quantum Nanoscience,\nDelft University of Technology, 2628CJ Delft, The Netherlands\n(Dated: May 18, 2020)\nWe present a scheme to entangle two microwave fields by using the nonlinear magnetostrictive interaction\nin a ferrimagnet. The magnetostrictive interaction enables the coupling between a magnon mode (spin wave)\nand a mechanical mode in the ferrimagnet, and the magnon mode simultaneously couples to two microwave\ncavity fields via the magnetic dipole interaction. The magnon-phonon coupling is enhanced by directly driving\nthe ferrimagnet with a strong red-detuned microwave field, and the driving photons are scattered onto two\nsidebands induced by the mechanical motion. We show that two cavity fields can be prepared in a stationary\nentangled state if they are respectively resonant with two mechanical sidebands. The present scheme illustrates\na new mechanism for creating entangled states of optical fields, and enables potential applications in quantum\ninformation science and quantum tasks that require entangled microwave fields.\nMany quantum information tasks, e.g., quantum telepor-\ntation [1], quantum metrology [2], and fundamental tests of\nquantum mechanics [3], require optical entangled states. Con-\nventionally, they have been generated via parameteric down-\nconversion with nonlinear crystals [4]. Alternative e \u000ecient\napproaches have been adopted by utilizing, e.g., four-wave\nmixing in optical fibers [5] and atomic vapors [6], quantum\ndots [7], and periodically poled lithium niobate waveguide [8],\nto name but a few. In the microwave (MW) domain, the en-\ntangled fields are typically produced by using the nonlinear-\nity in Josephson parametric amplifiers [9], or by injecting a\nsqueezed vacuum through a linear MW beamsplitter [10]. In\nthe field of optomechanics, two MW fields can get entangled\nby coupling to a common mechanical resonator via radiation\npressure [11, 12]. Despite their di \u000berent forms, most of the\nmechanisms utilize the nonlinearity of the physical processes.\nIn this Letter, we present a mechanism, distinguished\nfrom all previous approaches, for creating continuous-variable\n(CV) entanglement of MW fields by using the nonlinear mag-\nnetostrictive interaction in a ferrimagnet. Specifically, two\nMW cavity fields couple to a magnon mode in a ferrimag-\nnetic yttrium-iron-garnet (YIG) sphere [13–22], and simul-\ntaneously the magnon mode couples to a phonon mode em-\nbodied by the vibrations of the sphere induced by the mag-\nnetostrictive force [23]. Due to the intrinsic low frequency\nof the phonon mode, it owns a large thermal occupation at\ntypical cryogenic temperatures. We thus drive the magnon\nmode with a red-detuned MW field (with the detuning equal\nto the mechanical frequency), which leads to the stimula-\ntion of the anti-Stokes process, i.e., a MW photon interacts\nwith a phonon and converts into a magnon of a higher fre-\nquency [24, 25]. This process corresponds to the cooling of\nthe phonon mode, a prerequisite for observing quantum ef-\nfects in the system [26]. The strong magnon drive also en-\nhances the e \u000bective magnon-phonon coupling, and when this\ncoupling is su \u000eciently strong, magnomechanical entangle-\nment is created, similar to the mechanism of creating optome-chanical entanglement with a strong red-detuned drive [27].\nThe entanglement originates from the nonlinear magnetostric-\ntive coupling, and could be distributed to two MW fields due\nto the linear magnon-photon coupling. Or more intuitively,\nthe mechanical motion scatters the MW driving photons onto\ntwo sidebands, which are entangled due to the mediation of\nmechanics [28, 29]. And if two MW cavities are respectively\nresonant with the two sidebands, the two cavity fields get en-\ntangled. Similar mechanism has been used to generate atom-\nlight entanglement [30]. We prove its validity, and two MW\nfields indeed get maximumly entangled when they are respec-\ntively resonant with the two mechanical sidebands (we assume\nthe resolved sidebands [15–23]). We verify the entanglement\nin both the quantitative and qualitative ways, i.e., by calculat-\ning the logarithmic negativity and by using the Duan criterion\nfor CV systems.\nWe first present a general model of the scheme, then solve\nthe system dynamics by means of the standard Langevin for-\nmalism and the linearization treatment, and study the entan-\nglement in the stationary state. Finally, we show strategies to\nmeasure /verify the optical entanglement, and provide possible\nconfigurations for experimental realizations.\nThe model. The system consists of two MW cavity modes, a\nmagnon mode, and a mechanical mode, as shown in Fig. 1(a).\nThe magnons, as quantized spin wave, are the collective ex-\ncitations of a large number of spins inside a massive YIG\nsphere. The magnon mode couples to two MW cavity modes\nvia magnetic dipole interaction, and, simultaneously, to a me-\nchanical vibrational mode via the magnetostrictive force [23–\n25]. The mechanical frequency we study is much smaller than\nthe magnon frequency, which yields an e \u000bective dispersive\nmagnon-phonon interaction [23, 31]. We consider the size of\nthe YIG sphere to be much smaller than the MW wavelengths,\nhence neglecting any radiation pressure on the sphere inducedarXiv:1909.05936v3  [quant-ph]  15 May 20202\nFIG. 1: (a) General model of the scheme. A magnon mode min a\nYIG sphere couples to two MW fields a1anda2via magnetic dipole\ninteraction, and to a phonon mode bvia magnetostrictive interaction.\n(b) Mode frequencies and linewidths. The magnon mode with fre-\nquency!mis driven by a strong MW field at frequency !0, and the\nmechanical motion of frequency !bscatters the driving photons onto\ntwo sidebands at !0\u0006!b. If the magnon mode is resonant with the\nblue (anti-Stokes) sideband, and the two cavity modes with frequen-\ncies!1;2are respectively resonant with the two sidebands, the two\ncavity fields get entangled.\nby the MW fields. The Hamiltonian of the system reads\nH=~=X\nj=1;2!jay\njaj+!mmym+!b\n2(q2+p2)+G0mymq\n+X\nj=1;2gj(ay\njm+ajmy)+i\n(mye\u0000i!0t\u0000mei!0t);(1)\nwhere aj(m) and ay\nj(my) are, respectively, the annihilation\nand creation operators of the cavity mode j(magnon mode),\nsatisfying [ O;Oy]=1 (O=aj;m), and qandpare the di-\nmensionless position and momentum quadratures of the me-\nchanical mode, thus [ q;p]=i.!j,!m, and!bare the res-\nonance frequencies of the cavity mode j, the magnon mode,\nand the mechanical mode, respectively, and the magnon fre-\nquency can be adjusted in a large range by altering the external\nbias magnetic field Hvia!m=\r0H, where the gyromagnetic\nratio\r0=2\u0019=28 GHz /T.G0is the single-magnon magnome-\nchanical coupling rate, and gjdenotes the coupling rate be-\ntween the magnon mode with the cavity mode j, which can be\n(much) larger than the dissipation rates \u0014jand\u0014mof the cav-\nity and magnon modes, gj>\u0014 j;\u0014m, leading to cavity-magnon\npolaritons [15–22]. The Rabi frequency \n =p\n5\n4\r0p\nNB 0[24]\ndenotes the coupling strength between the magnon mode and\nits driving magnetic field with frequency !0and amplitude B0,\nwhere the total number of spins N=\u001aVwith the spin density\nof YIG\u001a=4:22\u00021027m\u00003and the volume of the sphere V.\nFor convenience, we switch to the rotating frame with re-\nspect to the drive frequency !0, and by including input noises\nand dissipations of the system, we obtain the following quan-\ntum Langevin equations (QLEs)\n˙aj=\u0000i\u0001jaj\u0000igjm\u0000\u0014jaj+q\n2\u0014jain\nj;(j=1;2)\n˙m=\u0000i\u0001mm\u0000iX\nj=1;2gjaj\u0000iG0mq+ \n\u0000\u0014mm+p\n2\u0014mmin;\n˙q=!bp;\n˙p=\u0000!bq\u0000G0mym\u0000\rp+\u0018;\n(2)where \u0001j=!j\u0000!0,\u0001m=!m\u0000!0,\ris the mechanical\ndamping rate, and ain\nj,minare input noise operators with\nzero mean value acting on the cavity and magnon modes,\nrespectively, which are characterized by the following corre-\nlation functions [32]: hain\nj(t)ainy\nj(t0)i=\u0002Nj(!j)+1\u0003\u000e(t\u0000t0),\nhainy\nj(t)ain\nj(t0)i=Nj(!j)\u000e(t\u0000t0), andhmin(t)miny(t0)i=\u0002Nm(!m)+1\u0003\u000e(t\u0000t0),hminy(t)min(t0)i=Nm(!m)\u000e(t\u0000t0).\nThe Langevin force operator \u0018, accounting for the Brownian\nmotion of the mechanical oscillator, is autocorrelated as\nh\u0018(t)\u0018(t0)+\u0018(t0)\u0018(t)i=2'\r\u00022Nb(!b)+1\u0003\u000e(t\u0000t0), where\na Markovian approximation has been taken valid for a large\nmechanical quality factor Qm=!b=\r\u001d1 [33]. The equi-\nlibrium mean thermal photon, magnon, and phonon numbers\nareNk(!k)=h\nexp\u0010~!k\nkBT\u0011\n\u00001i\u00001(k=1;2;m;b), with kBthe\nBoltzmann constant and Tthe environmental temperature.\nBecause the magnon mode is strongly driven, it has a large\namplitudejhmij\u001d 1, and further owing to the cavity-magnon\nbeamsplitter interactions the two cavity fields are also of large\namplitudes. This allows us to linearize the system dynamics\naround semiclassical averages by writing any mode operator\nas a c-number plus its fluctuation operator O=hOi+\u000eO,\n(O=aj;m;q;p), and neglecting small second-order fluctua-\ntion terms. Substituting those linearized mode operators into\nEq. (2), the equations are then separated into two sets of equa-\ntions, respectively, for semiclassical averages and for quan-\ntum fluctuations. The solutions of the averages are obtained,\nwhich arehpi=0,hqi=\u0000G0\n!bjhmij2,haji=\u0000igj\ni\u0001j+\u0014jhmi, andhmi\nis given by\nhmi=\n\n(i\u00011+\u00141)(i\u00012+\u00142)\n(i˜\u0001m+\u0014m)(i\u00011+\u00141)(i\u00012+\u00142)+g2\n1(i\u00012+\u00142)+g2\n2(i\u00011+\u00141);(3)\nwith ˜\u0001m= \u0001 m+G0hqithe e\u000bective detuning of the magnon\nmode including the frequency shift caused by the magne-\ntostrictive interaction. It takes a simpler form\nhmi'i\n\u0001 1\u00012\n\u0000˜\u0001m\u00011\u00012+g2\n1\u00012+g2\n2\u00011; (4)\nwhenj\u0001jj;j˜\u0001mj \u001d\u0014j;\u0014m. Let us introduce the quadratures\nof the quantum fluctuations ( \u000eX1;\u000eY1;\u000eX2;\u000eY2;\u000ex;\u000ey;\u000eq;\u000ep),\nwhere\u000eXj=(\u000eaj+\u000eay\nj)=p\n2,\u000eYj=i(\u000eay\nj\u0000\u000eaj)=p\n2,\u000ex=\n(\u000em+\u000emy)=p\n2, and\u000ey=i(\u000emy\u0000\u000em)=p\n2, and the input\nnoise quadratures are defined in the same way. The QLEs of\nthe quadrature fluctuations can be cast in the matrix form\n˙u(t)=Au(t)+n(t); (5)\nwhere u(t)=\u0002\u000eX1(t);\u000eY1(t);\u000eX2(t);\u000eY2(t);\u000ex(t);\u000ey(t);\u000eq(t);\u000ep(t)\u0003T,\nn(t)=\u0002p2\u00141Xin\n1(t);p2\u00141Yin\n1(t);p2\u00142Xin\n2(t);p2\u00142Yin\n2(t);p2\u0014mxin(t);p2\u0014myin(t);0;\u0018(t)\u0003Tis the vector of noises entering the sys-3\ntem, and the drift matrix Ais given by\nA=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@\u0000\u00141\u00011 0 0 0 g1 0 0\n\u0000\u00011\u0000\u001410 0\u0000g10 0 0\n0 0\u0000\u00142\u00012 0 g2 0 0\n0 0\u0000\u00012\u0000\u00142\u0000g20 0 0\n0 g1 0 g2\u0000\u0014m˜\u0001m\u0000G0\n\u0000g10\u0000g20\u0000˜\u0001m\u0000\u0014m0 0\n0 0 0 0 0 0 0 !b\n0 0 0 0 0 G\u0000!b\u0000\r1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(6)\nwhere G=ip\n2G0hmiis the e \u000bective magnomechanical cou-\npling rate. By using the result of Eq. (4), we obtain\nG'p\n2G0\n\u0001 1\u00012\n˜\u0001m\u00011\u00012\u0000g2\n1\u00012\u0000g2\n2\u00011; (7)\nwhich shows that the coupling can be significantly enhanced\nwith a strong magnon drive.\nWe are interested in the quantum correlation of two MW\nfields in the steady state. The steady state of the system is a\nfour-mode Gaussian state due to the linearized dynamics and\nthe Gaussian nature of input noises. Such a state is fully char-\nacterized by an 8\u00028 covariance matrix (CM) Cwith its entries\ndefined asCi j(t)=1\n2hui(t)uj(t0)+uj(t0)ui(t)i(i;j=1;2;:::;8).\nIt can be obtained straightforwardly by solving the Lyapunov\nequation [34]\nAC+CAT=\u0000D; (8)\nwhereD=diag\u0002\u00141(2N1+1);\u00141(2N1+1);\u00142(2N2+1);\u00142(2N2+\n1);\u0014m(2Nm+1);\u0014m(2Nm+1);0;\r(2Nb+1)\u0003is the di \u000busion ma-\ntrix, whose entries are defined through hni(t)nj(t0)+\nnj(t0)ni(t)i=2=Di j\u000e(t\u0000t0).\nWe adopt the logarithmic negativity [35] to quantify the en-\ntanglement between the two MW cavity fields. It is a full\nentanglement monotone under local operations and classical\ncommunication [36] and an upper bound for the distillable\nentanglement [35]. The logarithmic negativity for Gaussian\nstates is defined as [37]\nEN:=max\u00020;\u0000ln 2˜\u0017\u0000\u0003; (9)\nwhere ˜\u0017\u0000=min eigji\n2˜Cmwj(with the symplectic matrix\n\n2=\b2\nj=1i\u001byand the y-Pauli matrix \u001by) is the minimum sym-\nplectic eigenvalue of the CM ˜Cmw=PC mwP, withCmwthe\nCM of the two MW fields, which is obtained by removing in\nCthe rows and columns associated with the magnon and me-\nchanical modes, and P=diag(1;\u00001;1;1) is the matrix that\nperforms partial transposition on CMs [38].\nMW entanglement and its detection . In Fig. 2 we present\nthe main results of the entanglement between two MW cav-\nity fields. The stationary entanglement is guaranteed by the\nnegative eigenvalues (real parts) of the drift matrix A. Fig-\nure 2(a) shows clearly that the maximum entanglement is\nachieved when the two cavity fields are respectively reso-\nnant with the two mechanical sidebands [see Fig. 1(b)], i.e.,\n-2-1012-2-1012Δ\n1/ωbΔ2/ωb0\n0.0350.0700.110.140\n.00.30.60.91.21.50.61.21.82.43.0(b)( a)κ\n2/κ1g2/g10\n0.0450.0900.130.18FIG. 2: (a) Density plot of the entanglement ENbetween two MW\ncavity fields vs (a) \u00011and\u00012, (b)\u00142=\u00141andg2=g1(\u00141,g1are fixed).\nWe take ˜\u0001m=0:9!b,\u00142=\u00141,g2=g1in (a), and \u00011=\u0000\u00012=!bin\n(b). See text for the other parameters.\n\u00011=\u0000\u00012'\u0006!b, where “\u0006” sign is taken due to the sym-\nmetry of the two cavity fields. And the magnon mode reso-\nnant with the blue sideband ˜\u0001m'!bcorresponds to the anti-\nStokes process, which significantly cools the phonon mode,\nthus eliminating the main obstacle for observing entangle-\nment [24]. We have employed experimentally feasible param-\neters [23]:!m=2\u0019=10 GHz,!b=2\u0019=10 MHz,\r=2\u0019=102\nHz,\u0014m=2\u0019=\u00141=2\u0019=1 MHz, g1=2\u0019=3:8 MHz, G=2\u0019=4:5\nMHz, and T=20 mK. We use a strong magnon-phonon cou-\npling G> \u0014 mto create magnomechanical entanglement. This\nmeans that a strong magnon driving field should be used, and\nin order to avoid unwanted magnon Kerr e \u000bect [39, 40] the\nbare coupling rate G0should not be too small [24, 25]. For the\noptimal case \u00011=\u0000\u00012'\u0006!bin Fig. 2(a), a driving power\nof 6.3 mW (0.57 mW) should be used to yield G=2\u0019=4:5\nMHz for G0=2\u0019=0:3 Hz (1 Hz), while keeping the Kerr ef-\nfect negligible. In Fig. 2(b), we analyse the optimal coupling\nrates g1;2and decay rates \u00141;2, and find that in both situations\n\u00011=\u0000\u00012'!band\u0000!b, close coupling rates should be\nused, and the cavity that is resonant with the red (blue) me-\nchanical sideband should have a smaller (larger) decay rate\nthan the other. Such an asymmetric feature is due to the dif-\nferent roles of the two sidebands. The entanglement is robust\n0.000 .050 .100 .150 .200.00.10.2 \n ENT\n (K)\nFIG. 3: MW entanglement ENvs temperature T. The parame-\nters are those with which the maximum entanglement is achieved\nin Fig. 2(b).4\n-2-1012-2-1012Δ\n1/ωbΔ2/ωb1\n.861.891.921.941.972.002\n4680.51.01.52.02.5(b)( a)κ\n1/2π (ΜΗz)κ2/2π (ΜΗz)1\n.801.841.881.921.962.00\nFIG. 4: Density plot of h\u000eX2\n+i+h\u000eY2\n\u0000ivs (a) \u00011and\u00012, (b)\u00141and\u00142.\nThe blank areas denote h\u000eX2\n+i+h\u000eY2\n\u0000i>2. The parameters are the\nsame as in Fig. 2(a), and we take optimal detunings \u00011=0:9!band\n\u00012=\u00001:1!bin (b).\nagainst environmental temperature and survives up to \u0018140\nmK, as shown in Fig. 3, below which the average phonon\nnumber is always smaller than 1, showing that mechanical\ncooling is thus a precondition to observe quantum entangle-\nment in the system [24]. The generated MW entanglement\ncan be detected by measuring the CM of two cavity output\nfields. Such measurement in the MW domain has been real-\nized in the experiments [12, 41].\nAlternatively, one can also verify the entanglement by using\nthe Duan criterion [42], which requires simpler experimental\noperations, i.e., one does not have to measure all the entries\nof the 4\u00024 CM, but measure only two collective quadra-\ntures [43]. Specifically, a su \u000ecient condition for entangle-\nment is that the two collective quadratures satisfy the inequal-\nity\nh\u000eX2\n+i+h\u000eY2\n\u0000i<2; (10)\nwhere X+=X1+X2, and Y\u0000=Y1\u0000Y2. Figure 4(a) shows\nthat in two areas around \u00011=\u0000\u00012'\u0006!bthe inequality is\nfulfilled, indicating that the two cavity fields are entangled.\nExperimental implementations . We now discuss possible\nconfigurations that could realize the proposal. Two MW cav-\nXY\n45°Loop antenna\nFIG. 5: A YIG sphere is placed near the maximum magnetic fields\nof two MW cavity fields in a cross-shape cavity. A loop antenna at\nthe end of a superconducting MW line is used to drive the magnon\nmode [39].ities and each cavity containing a cavity mode are preferred.\nIn this situation, the frequencies of the cavity fields can be ad-\njusted flexibly to match the two mechanical sidebands. The\ntwo cavities could be placed perpendicularly in the horizon-\ntal plane with the YIG sphere located in the intersection (near\nthe maximum magnetic fields) of the cavity fields. This can be\nrealized in a planar cross-shape cavity [44] or coplanar waveg-\nuide [45], see Fig. 5. Taking the “X”-cavity [44] as an exam-\nple, one can set the bias magnetic field along the z(vertical)\ndirection, the magnetic fields of two cavity modes along the\nxandydirection, respectively, and the driving magnetic field\nin the x-yplane and of e.g., 45 degrees with both the xand\nydirection. For directly driving the magnon mode, one may\nadopt a superconducting MW line with a small loop antenna\nat its end [39]. In this case, the loop antenna will also cou-\nple to the two cavity modes leading to increased cavity de-\ncay rates. However, owing to its relatively small dimension\ncompared with the cavity setup the influence is only moder-\nate [46]. Besides, the cross configuration of the cavity may\nalso reduce the Qfactor of the cavities induced by the damage\nto boundary conditions. Taking into account the aforemen-\ntioned e \u000bects, we study the Duan criterion for taking larger\ncavity decay rates in Fig. 4(b). It shows that with much larger\ndecay rates, the two cavity fields are still entangled. Given\nthe flexibility of the cavity resonant frequencies, the mechan-\nical frequency can be freely chosen in a large range (always\nkeeping it much smaller than the magnon frequency). The\nresults presented in this work employed a \u001810 MHz mechan-\nical mode of a 250- \u0016m-diameter YIG sphere [23]. For such\na large sphere, the bare magnomechanical coupling is small,\nbut it can be increased by using a smaller sphere such that the\npump power required is reduced, which can weaken both the\nunwanted nonlinear e \u000bect and the by-e \u000bect of the coupling of\nthe loop antenna to the cavity modes.\nConclusions. We present a new mechanism for creating\nMW entangled states based on magnetostrictive interaction in\na ferrimagnetic YIG sphere. The mechanism makes use of\nthe nonlinearity of such a magnomechanical interaction. The\nentanglement is in the steady state and robust against cavity\ndissipations and environmental temperature. We show strate-\ngies to detect the entanglement and a possible configuration\nthat is promising to realize the proposal. We analyse in de-\ntail various practical imperfections which would help future\nexperimental realizations. This work may find applications in\nquantum information science, quantum metrology, and quan-\ntum tasks that require entangled CV MW fields.\nAcknowledgments . We thank Junjie Liu and Yi-Pu Wang\nfor fruitful discussions on potential experimental realiza-\ntions. 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Marco,1and Juan de la Figuera1\n1Instituto de Qu´ ımica-F´ ısica “Rocasolano”, CSIC, Madrid 28006, Spain\n2Universidad Complutense de Madrid, Madrid 28040, Spain\n3Elettra Sincrotrone S.C.p.A, Trieste, Italy∗\n4Sandia National Laboratories, Livermore, CA 94550, USA\nThe oldest known magnetic material,magnetite, is of curren t interest for use inspintronics as a thinfilm. An\nopen question is how thin can magnetite films be and still reta in the robust ferrimagnetism required for many\napplications. Wehavegrownone-nanometer-thick magnetit ecrystalsandcharacterizedthem insitubyelectron\nand photoelectron microscopies including selected-area x -ray circular dichroism. Well-defined magnetic pat-\nterns are observed in individual nano-crystals up to at leas t 520 K, establishing the retention of ferrimagnetism\ninmagnetite two-unit-cells thick.\nThe trend in both magnetic data storage and spintronics is\nto reducethe thickness and/orlateral size of the device mat e-\nrials to the nanoscale. Size reductionin magnetic storage h as\nthe obvious advantage of increasing the bit density. Advan-\ntages also exist for spintronic applications. For example, the\nmagnetic layers of spin filters can be switched with smaller\nmagnetic fields as the layers become thinner. But size reduc-\ntion can also change a material’s magnetic behavior. For ex-\nample, as a ferromagnet is decreased in size, at some point\nthermalexcitationscanovercomethemagneticanisotropye n-\nergy, leading to superparamagnetism. Then the material is\nonly useful below a ”blocking” temperature, where the mag-\nnetization is stable over some relevant timescale. Several ap-\nproaches are used to delay the onset of superparamagnetism,\nincludingusinghighmagneticanisotropymaterials1orbyex-\nchange bias2. Understandinghow to stabilize magnetic order\ninlow-dimensionalstructuresisanimportantconcern.\nIron-containing oxides are a class of magnetic materials\nthatprovidegoodchemicalstabilityinoxidizingatmosphe res\nand have extremely high Curie temperatures. Strong mag-\nnetism in the iron oxide magnetite has been known since\nthe ancient Greeks3. The material, historically referred to as\nlodestone, is currently a promising candidate for spintron ic\napplications4. Bulkmagnetite(Fe 3O4)isaferrimagnetwitha\n850 K Curie temperature5and becomes multiferroic at low\ntemperatures6,7, allowing electrical control of magnetic do-\nmains. Thepredictionofhalf-metalcharacter8,whichimplies\nthattheconductionelectronsare100%spin-polarized,lea dto\nitsuse asaspininjector9.\nOxides like magnetite have much more complicated struc-\ntures and larger unit cells than metal ferromagnets, giv-\ning the possibility of tuning their properties to a larger ex -\ntent, specially given the often observed strong coupling to\nstrain effects10,11. Magnetite ultrathin nanostructures have\nbeen grown on a variety of substrates, including oxides12–18,\nsemiconductors18–22and metals23–28. Here we examine the\npresence of stable ferrimagnetic domains in ultrathin mag-\nnetite, a subject with conflicting reports in the literature .\nWhile some reports state than magnetite films close to 3 nm\nthick present a well defined magnetic structure20,22,23,29,30,\nothers indicate that clear signs of superparamagnetic beha v-\niorareobservedatthesamethickness12,31–34. Inparticularwe\nFIG. 1. ( A–D)Selected LEEM images from a sequence acquired\nduring the growth of the magnetite crystals. The first three f rames\nshowthecompletionoftheFeOlayer,whilethelastframesho wsthe\nfinal film with magnetite crystals with well-defined edges. Th e field\nof view is10 µm andthe electron beam energy is19 eV.\nnotethat,tothebestofourknowledge,therearenoreportso f\nstable magnetization domains on nanometer-thick magnetit e.\nWe grow one-nanometer-thick,micron-wide, magnetite crys -\ntals on ruthenium25using reactive molecular beam epitaxy\n(i.e., depositing iron in a backgroundof oxygen) while mon-\nitoring the growth in real time by low-energy electron mi-\ncroscopy (LEEM27). We shown by imaging their individ-\nual magnetization patterns and their x-ray magnetic circul ar\ndichroism (XMCD) spectra that they are ferrimagnetic. The\nobservationofmagnetismnearthe limitofunit-cellthickn ess\nshows that, under appropriate conditions, there is still ro om\ntodecreasethe thicknessofmagnetitenanostructureswith out\nintroducingsuperparamagnetism.\nThe experiments were performed at the Nanospectroscopy\nbeamline of the Elettra storage ring35. The beamline facil-2\nities include an Elmitec III low-energy electron microscop e\nwith an hemispherical energy analyzer. The microscope has\nthe option of selecting either an electron beam or an x-ray\nbeam to probe the specimen surface. The electron beam al-\nlowsforregularLEEMuse,includingfastreal-spaceimagin g\nof the surface during growth of the oxide films and selected-\narea diffractionmeasurements. In photoemissionmicrosco py\nmode (PEEM), the instrument is able to record selected-area\nx-ray absorption spectra using the secondary electrons emi t-\ntedsubsequenttothex-rayabsorptionprocess,orspatiall yre-\nsolved photoemissionimages. The capability of selecting t he\npolarization of the x-ray beam (plus or minus circular polar -\nization)allowsXMCDmeasurements.Thex-raybeamisfixed\nrelative to the sample at an angle of 16◦with the film plane,\nso thex-rayXMCD measurementsare mostlysensitive tothe\nin-planemagnetization.\nTheRusingle-crystalsubstratewith(0001)orientationwa s\ncleanedbyexposureto5 ×10−8mbarofmolecularoxygenat\n1000K,followedbyflashingto1500Kinvacuum. Thesam-\nplewasorientedsothattheincomingx-raybeamwasaligned\nalong a mirror plane of the Ru surface (i.e., along a [11 ¯20]\ndirection in real space). The iron oxide films were grown by\nreactive molecular beam epitaxy (MBE) in 5 ×10−7mbar of\nmolecularoxygenwiththesubstrateat900K.Ironwasevapo-\nratedfroma2-mm-diameterironrodheatedbyelectronbom-\nbardment inside a water-cooling jacket. Oxygen was intro-\nduced into the experimentalchamberby means of a capillary\nthat increased the gas flux at the sample position by about a\nfactorof2.\nIronoxidegrowthonmetalsubstratesusingmolecularoxy-\ngenastheoxidizingagentisexpectedtooccurintwostages24:\ninitially an FeO wetting layer covers the substrate, with a\nthicknessthat dependsonthe particularsubstrate. Thenma g-\nnetite nucleates and grows as 3-dimensional islands on the\nFeO film. This growth mode makes it difficult to obtain\nultra-thinmagnetitecrystalswithoutactuallyimagingth efilm\ngrowth, as we do here. In Figure 1 several frames are pre-\nsentedfromasequenceofLEEMimagesacquiredduringiron\noxide growth. In the experiment shown, the FeO initially\ngrows as islands comprised of two Fe-O layers. (Along its\n[111]direction,FeOiscomposedofalternatingplanesofir on\nand oxygen). When the FeO film is close to completely cov-\neringthe substrate,someregionshaveasingleFeO layer.\nWhen further iron is deposited on a complete FeO film,\nlarge(uptoseveralmicrometer)triangularislandsnuclea teon\ntop of the film (Figures1D and 2A)27. These crystalsand the\nFeO wetting layer are found to exhibit different low-energy\nelectron diffraction (LEED) patterns (Fig. 2A), x-ray phot o-\nelectron spectroscopy (XPS) spectra (Fig. 2B) and x-ray ab-\nsorptionspectra(notshown).\nTheLEEDpatternsofthedifferentoxidephasesareknown\nto differ24. The diffracted beams that arise from the period-\nicity of the hexagonal oxygen layers appear at very similar\npositions in iron oxides due to their similar oxygen-oxygen\ndistances, 0.297–0.320 nm. However, the different arrange -\nment of the iron atoms within the layers of each phase gives\nriseto1×1,2×2and√\n3×√\n3R30◦LEEDpatternsforbulk-\nterminatedFeO(111),magnetite(111)andhematite(0001), re-\nFIG. 2. (color online)( A) LEEM image of a magnetite crystal (the\nfield of view is 4 µm and the electron energy is 8 eV), with insets\nshowingthe low-energyelectrondiffractionpatterns (acq uiredusing\nan electron beam of 28 eV) of the crystal and its surrounding w et-\ntinglayer. ( B)Fe2pcore-level x-ray photoelectron spectra acquired\nfrom the crystal (orange, lower curve) and the wetting layer (green,\nupper curve). The solid lines are the sum of the different ind ividual\ncontributions (not shown) that are expected tobe present in the XPS\nspectra of FeO and magnetite. The inset shows reference spec tra\nfor FeO(top, green), a Langmuir-Blodgett filmcontaining Fe2+and\nFe3+(middle, purple), and hematite (bottom, gray). A non-linea r\nbackground has been substracted from the spectra.\nspectively. As seen inFigure2A, thewetting layerhasa 1 ×1\nLEEDpattern(withadditionalspotsduetoacoincidencepat -\nternwiththeunderlyingRusubstrate),suggestingaFeO(11 1)\nsurface. In contrast, the large triangular crystal has a 2 ×2\nLEED pattern, which is indicative of magnetite. The island’ s\noxygenlatticespacingobtainedfromLEEDisthesameasthe\nwetting layer, 0.32 ±0.04 nm, i.e., the magnetite crystals are\nstrainedby6%.\nAimed at identifying more precisely the chemical nature\nof the triangular crystals, the Fe 2 pcore level XPS spectra\nwere recorded from a crystal and from its surrounding iron\noxide layer (Figure 2B). For comparison, the inset shows the\nsame Fe 2 pcore level peaks recordedfrom several iron com-\npounds obtained using a conventional laboratory XPS spec-\ntrometer. (The upper spectrum corresponds to an FeO film\nproduced by vacuum evaporation of Fe metal on a Ru sub-\nstrate and subsequent oxidation; the middle spectrum corre -\nsponds to a 3-nm-thick Fe-containing film produced by the\nLangmuir-Blodgett (LB) technique that contains both Fe2+\nand Fe3+36; the bottom spectrum was obtained from pure α-\nFe2O3powder.) The spectrumfromthe wettinglayer (Figure\n2B,top)presentsthe samefeaturesasthereferenceFeO film,\nconfirmingthatthewettinglayerisFeO.Incontrast,themai n\nphotoemissionpeaksintheFe2 pspectrumfromthetriangular\ncrystal (Figure2B, bottom)appearat higherbindingenergi es\nthan those in the wetting-layer spectrum, indicating that t he\naverage Fe oxidation state in the crystal is higher than in th e\nwetting layer. The characteristic shake-up satellite of ex clu-\nsively Fe3+-containing phases (i.e., the peak at 718–719 eV)\nis not evident in the crystal’s spectrum. The spectrum thus\nresemblesthat of the mixedFe2+-Fe3+LB film shownin the\nmiddle of the inset of Figure 2B, indicating that the crystal\nis a mixed-valence Fe2+/Fe3+oxide. Consistently the octa-3\nFIG. 3. (color online)( A) XAS and ( B) XMCD image at 705.8 eV.\nThe field of view is 30 µm. (C) XMCD image recorded in rema-\nnence showing the magnetization pattern of same crystal pre sented\nin A. The field of view is 4 µm. The inset showns the experimen-\ntal geometry. ( D) Top: XAS spectrum from the magnetite crystal.\nBottom: XMCD difference spectrum.\nhedral positions in magnetite’s inverse spinel structure5are\npopulatedwith both Fe2+and Fe3+while the tetrahedral po-\nsitions are occupied only by Fe3+. This result together with\nthe LEED pattern [and the x-ray circular magnetic dichroism\n(XMCD)spectraofFig.3D]indicatethattheultrathincryst als\nare indeed magnetite. We cannot, however,evaluate their de -\ntailedstoichiometry. (Magnetiteisoftennon-stoichiome tric.)\nWeuseXMCDinPEEM37torevealin-situthemagneticor-\nderoftheindividualmagnetitecrystals. Tomeasurethex-r ay\nabsorption(XAS)spectraforthemagnetitecrystals,anima ge\nof the secondary electron emission (which is proportional t o\nthe x-ray absorption) was collected while the photon energy\nwas scanned over the Fe L 3,2x-ray absorption edges in two\ndifferent scans using opposite x-ray helicities. Such a XAS\nimage, acquiredclose to the Fe L 3absorptionedge, is shown\nin Figure 3A. The image intensity from the area correspond-\ning to the magnetite crystal of Figure 2A was integrated and\naveraged for the two x-rays helicities, giving the XAS spec-\ntrumshowninFigure3D(top). Thespectrumprovidesfurther\nsupport that the crystal is magnetite38, which has a signifi-\ncant XMCD signal39–41at the shoulder before the maximum\nof the L 3XAS spectra. Taking images at this photon energy\n(705.8eV)withdifferenthelicitiesandsubtractingthemp ixel\nbypixelgivestheXMCDimagesofFigure3B(largerfieldof\nviewof30 µm,showingthewellseparatedmagnetitecrystals,\nallofwhichpresentmagneticdomains)andFigure3C(where\nthesameislandofFigure2Aisshown). Theuniformgrayin-\ntensity of the FeO wetting layer indicates that it has no mag-\nnetic circular dichroic contrast. We thus do not find any fer-\nromagneticordersuchastheoneobservedonFeO/Fe(110)42,\nin agreement with the antiferromagnetic order expected bot h\nin bulk FeO (which is antiferromagneticwith a N´ eel temper-aturebelow roomtemperature5) andin an ultra-thinFeO film\non Pt(111)43. In contrast, the magnetite crystals show a clear\ndichroic contrast, establishing that they have non-zero lo cal\nmagnetization. The magnetic domain patterns (Fig. 3C) are\nintricate, with long straight domain walls oriented along t he\n{112}directions of the magnetite crystal. The two domains\nmarked as M+andM−in Figure 3C have the similar mag-\nnitudeofthemagnetizationcomponentalongtheilluminati on\ndirection of the x-ray beam. The magnetization pattern per-\nsists duringannealingup to 520K, where changesin the sur-\nfacetopographyare alreadydetected.\nIn order to calculate the dichroic XMCD spectra, only the\narea that corresponds to a given domain in a XAS image\nwith a given helicity is selected: a different XAS spectra ca n\nbe collected for each specific combination of domain type\n(M+,M−) and x-ray polarization ( P+,P−). To avoid\nspurious signals, the I(+M,+P)andI(−M,+P)curves\nwere subtracted together, as were the I(+M,−P)and the\nI(−M,−P)curves. Then each of the two difference spec-\ntra for+Pand−Pare subtracted from one another, after\nnormalizingbythe differenceinXAS intensityinthe wettin g\nlayertoaccountforilluminationdifferences. TheXMCDdif -\nference spectrum (Figure 3D, bottom) shows a well-defined\npeak structure at the L 3and L2edges that is characteristic\nof magnetite41. The two negative peaks at the L 3edge orig-\ninate mostly from the iron cations sitting at the octahedral\nsites, with nominal valences +2 and +3. The positive peak\ninthemiddlecorrespondsmostlytothetetrahedralFe3+ions.\nTheoppositesense oftheXMCD peaksfromthe ironcations\nat octahedral and tetrahedral sites indicates their mutual anti-\nferromagnetic coupling. Thus the magnetite crystals are fe r-\nrimagnetic, like the bulk material. The tetrahedral peak is\nsmaller and the octahedral Fe2+peak is larger than for bulk\nmagnetite41. These small differences may arise from contri-\nbutions from the underlying FeO wetting layer44or from an\nincompleteunitcell29.\nWe next accurately measure the thickness of individual\nmagnetitecrystals. We usethe40-nmreal-spaceresolution of\nPEEM to measure the attenuationof the photoelectronsemit-\nted fromthe Ru 3dcorelevel of the substrate whenemerging\nthrough individual magnetite crystals (Fig. 4). This metho d\nrequires an accurate value of the mean free path of electrons\ntraveling through the magnetite crystal at a given kinetic e n-\nergy. For 400 eV photons the electrons from the Ru 3 d5/2\ncore level have a kinetic energy of 120 eV. The attenuation\nof a single FeO layer was measured by comparing the spec-\ntral area of the Ru 3 d5/2core level from bare Ru, measured\nthrough a FeO bi-layer and through a single FeO layer (see\nFig. 4). The bilayer and single layer areas of FeO are easily\ndistinguished not only by the difference in the substrate co re\nlevelattenuationbutbytheirelectronreflectivity(Figur e4A).\nThe FeO bilayer presents an additional peak absent from the\nFeO monolayer. (Oscillationsin electronreflectivity with en-\nergyarise from interferencebetween electronsreflected fr om\nthefilm/substrateandfilm/vacuuminterfaces45,witheachad-\nditional peak indicating one additional layer.) Using the e x-\nperimentallydeterminedmean free path for 120 eV electrons\nin FeO (1.25 ±0.02 FeO layers, Figure 4D), the thickness of4\nFIG. 4. (color online)Determination of the magnetite cryst al thick-\nness. (A) Left: Electron reflectivity curves recorded from two re-\ngions of a continuous FeO film, in blue for the majority area (w hich\nappears light gray in the LEEM image on the right side and corr e-\nsponds to a FeO bilayer) and in red for the minority regions (w hich\nappear in dark gray in the LEEM image and correspond to a FeO\nsingle layer). Right: LEEM image of a complete FeO layer befo re\ngrowth of the magnetite crystals. (The electron energy is 16 .75 eV\nand the field of view is 4 µm). (B) Ru 3dXPS spectra recorded\nfrom clean Ru, through a FeO monolayer and bilayer, respecti vely.\n(C) Ru3dXPSspectra recorded from clean Ru, through the wetting\nlayer around the magnetite crystals, and through the magnet ite crys-\ntals, respectively. ( D) Semilogaritmic plot of the relative Ru XPS\n3d5/2peak area recorded fromthe different filmsversus coverage i n\nFeO layers. The black squares correspond tothe spectra inB, which\ngive a mean free pathof 1.25 ±0.03 ML FeO(line inthe semilogarit-\nmicplot). Thegreenandbluesquarescorrespondtothewetti nglayer\nand the magnetite crystal, respectively, from C (error bars from the\nwettinglayerarewithinthesymbolsize). Theadditionalth icknessof\nthe magnetite crystaldoes not depend onthe particular wett inglayer\nthickness.themagnetitecrystalwasestimated tobe 3.1 ±0.3FeO layers\n(see Fig. 4C). Given the relative density of the Fe-O layers\nin magnetite and in FeO (same oxygen density, 25% smaller\nirondensityformagnetite),andthethicknessoftheFe-Ola y-\nersin magnetite[0.242 ±0.06nm24] yieldsa thicknessforthe\nmagnetitecrystalsof1.0 ±0.4nm(thelargerrelativeerrorfor\nthe thickness in nm is due to the conservative estimate of the\ninfluence of the relative density of magnetite Fe-O layers in\nmagnetite and FeO). To put this numberinto context we note\nthat the thinnest magnetite film grown on a metal substrate\n(with a similar FeO interface layer24) reported to date with\na well defined bulk-like local magnetic structure, determin ed\nbyConversionElectronM¨ ossbauerspectroscopy,was aroun d\nthreetimesthickerthatourislands23. Incontrast,thinnerfilms\noften show a markedsuperparamagneticbehaviorsuch as for\n1.8 nm magnetite on spinel films reported by Eerenstein et\nal.33. Anti-phase boundaries (APB) are often blamed for the\nappearance of superparamagnetism34,46. Our films seem to\nhave a low density of APB (as detected from the size of the\nmagneticdomainsdetectedaswellasfrompreviousdarkfield\nimaging-seeFigure7inRef.27-),whichmightaccountforth e\nstable magnetization domains on our thinner magnetite crys -\ntals.\nSUMMARY\nIn summary, we have grown one-nanometer-thick crystals\nof iron oxide on a substrate. Electron diffraction, Fe core\nlevel photoelectron spectroscopy and x-ray absorption spe c-\ntroscopyestablishthat thecrystalsaremagnetite. X-raym ag-\nnetic circular dichroismrevealsthat individualultrathi n crys-\ntals have ferrimagnetic order up to 520 K. With thickness\nof only two unit cells, the crystals may well be the thinnest\nlodestoneeverandtheyestablishthatmagnetite’srobustm ag-\nnetismispreservedat thenanometerlimit.\nACKNOWLEDGMENTS\nThisresearchwassupportedbytheSpanishMinistryofSci-\nence and Innovation through Projects No. MAT2009-14578-\nC03-01, MAT2009-14578-C03-02 and MAT2010-21156-\nC03-02, by the Office of Basic Energy Sciences, Division of\nMaterialsandEngineeringSciences,U.S.DepartmentofEn-\nergy under Contract No. DE-AC04-94AL85000, and by the\nEuropean Union through 226716-ELISA. M. M. and B. S.\nthanktheSpanishMinistryofScienceandInnovationforsup -\nportingthemthroughFPI fellowships.\n∗Present address: Instituto Madrile˜ no de Estudios Avanzad os en\nNanociencia (IMDEA),Madrid, Spain\n1K. R. Coffey, M. A. Parker, and J. K. Howard,\nIEEETrans. Mag. 31, 2737 (1995).2J.Nogues and I.K.Schuller, J.Mag. Mag. Mat. 192, 203 (1999).\n3A.A. Mills,Ann. Sci. 61, 273(2004).\n4M. Bibes and A. Barthelemy, IEEE Trans. Elec. 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Lett. 79, 5162 (1997)."    },    {        "title": "1502.01071v1.Instability_of_a_ferrimagnetic_state_of_a_frustrated_S_1_2_Heisenberg_antiferromagnet_in_two_dimensions.pdf",        "content": "arXiv:1502.01071v1  [cond-mat.mtrl-sci]  4 Feb 2015Japanese Journal of Applied Physics RAPID COMMUNICATION\nInstability of a ferrimagnetic state of a frustrated S= 1/2\nHeisenberg antiferromagnet in two dimensions\nHiroki Nakano1∗and Toru Sakai1,2\n1Graduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n2Japan Atomic Energy Agency, SPring-8, Sayo, Hyogo 679-5148 , Japan\nTo clarify the instability of the ferrimagnetism which is th e fundamental magnetism of ferrite, numerical-\ndiagonalization study is carried out for the two-dimension alS= 1/2Heisenberg antiferromagnet with frus-\ntration. We find that the ferrimagnetic ground state has the s pontaneous magnetizationin small frustration;\ndue to a frustrating interaction above a specific strength, t he spontaneous magnetization discontinuously\nvanishes so that the ferrimagnetic state appears only under some magnetic fields. We also find that, when\nthe interaction is increased further, the ferrimagnetism d isappears even under magnetic field.\nFerrite is a magnetic material that is indispensable in modern society. It is be-\ncause this material is used in various industrial products including mo tors, generators,\nspeakers, powder for magnetic recording, and magnetic heads et c. It is widely known\nthat fundamental magnetism of the ferrite is ferrimagnetism.1–4)The ferrimagnetism is\nan important phenomenon that has both ferromagnetic nature an d antiferromagnetic\nnature at the same time. The occurrence of ferrimagnetism is unde rstood as a mathe-\nmatical issue within the Marshall-Lieb-Mattis (MLM) theorem5,6)concerning quantum\nspin systems. A typical case showing ferrimagnetism is when a syste m includes spins\nof two types that antiferromagnetically interact between two spin s of different types in\neach neighboring pair, for example, an ( S,s)=(1, 1/2) antiferromagnetic mixed spin\nchain, in which two different spins are arranged alternately in a line and coupled by the\nnearest-neighbor antiferromagnetic interaction. The ferrimagn etic state like the above\ncase, in which the spontaneous magnetization is fixed to be a simple fr action of the\nsaturated magnetization determined by the number of up spins and that of down spins\nin the state, is called the Lieb-Mattis (LM) type ferrimagnetism. Ano ther example of\nferrimagnetism is a system including single-type spins that are more t han one in a unit\ncell, although the ferrimagnetism can appear even in a frustrating s ystem including\n∗E-mail: hnakano@sci.u-hyogo.ac.jp\n1/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n\tB\n \tC\n \nβα\nα\b\nFig. 1. (Color) Network of antiferromagnetic interactions studied in this p aper. The black and red\nbonds represent J1andJ2interactions. Green squares denote finite-size clusters of 24 and 30 sites in\n(a) and (b), respectively. Note that the two-dimensional networ k composed only of the black bonds is\ncalled the Lieb lattice.\nonly a single spin within a unit cell.7,8)\nThe antiferromagnet on the Lieb lattice illustrated in Fig. 1 correspo nds the second\ncase, in which there are three spins in a unit cell. The MLM theorem hold s in the Lieb-\nlattice antiferromagnet. If antiferromagnetic interactions are a dded to this Lieb lattice\nso that magnetic frustrations occur, however, the MLM theorem no longer holds. In this\nsituation, the ferrimagnetic state is expected to become unstable . The problem of how\nthe ferrimagnetism collapses owing to such frustrating antiferrom agnetic interactions is\nan important issue to understand the ferrimagnetism well and to ma ke ferrimagnetic\nmaterials more useful in various products. This problem was studied in theS= 1/2\nHeisenberg antiferromagnet on the spatially anisotropic kagome lat tice,9,10)where the\nexistence of an intermediate phase with weak spontaneous magnet ization is clarified\nbetween the LM type ferrimagnetic phase and the nonmagnetic pha se including the\nisotropic kagome-lattice antiferromagnet. We are then faced with a question: is there\nany other different behavior of the collapse of the ferrimagnetism?\nUnder circumstances, the purpose of this study is to demonstrat e the existence of\na different behavior of collapsing ferrimagnetism in the case of an S= 1/2 Heisenberg\nantiferromagnet on the lattice shown in Fig. 1 to answer the above q uestion. When the\nantiferromagnetic interactions denoted by the red bonds vanish, the system is unfrus-\ntrated and thus it certainly shows ferrimagnetism in the ground sta te. In this study, we\nexamine the case when the red-bond interactions are switched on.\n2/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\nThe model Hamiltonian examined inthis study is given by H=H0+HZeeman, where\nH0=/summationdisplay\ni∈α,j∈βJ1Si·Sj+/summationdisplay\ni∈α′,j∈βJ1Si·Sj\n+/summationdisplay\ni∈α,j∈α′J2Si·Sj, (1)\nHZeeman=−h/summationdisplay\njSz\nj. (2)\nHereSidenotes an S= 1/2 spin operator at site i. Sublattices α,α′, andβand the\nnetwork of antiferromagnetic interactions J1andJ2are depicted in Fig. 1. Here, we\nconsider the case of isotropic interactions. The system size is deno ted byNs. Energies\nare measured in units of J1; thus, we take J1= 1 hereafter. We examine the properties\nof this model in the range of J2/J1>0. Note that, in the case of J2= 0, sublattices α\nandα′are combined into a single sublattice; the system satisfies the above conditions\nof the MLM theorem. Thus, ferrimagnetism of the LM type is exactly realized in this\ncase. In the limit of J2/J1→ ∞, on the other hand, the lattice of the system is reduced\nto a trivial system composed of isolated S= 1/2 spins and isolated dimers of two spins.\nIts ground state is clearly different from the state of the LM-type ferrimagnetism in the\ncase ofJ2= 0. One thus finds that while J2becomes larger, the ground state of this\nsystem will change from the ferrimagnetic one in the case of J2= 0 to another state,\nwhich we survey here.\nNext, we discuss the method we use here, which is numerical diagona lization based\non the Lanczos algorithm.11)It is known that this method is nonbiased beyond any\napproximations and reliable for many-body problems, which are not o nly localized spin\nsystems such as the Heisenberg model12,13)treated in th present study but also strongly\ncorrelatedelectronsystemsincludingtheHubbardmodel14–16)andthet-Jmodel.14,17,18)\nA disadvantage of this method is that the available system sizes are lim ited to being\nsmall. Actually, the available sizes in this method are much smaller than t hose of the\nquantum Monte Carlo simulation19,20)and the density matrix renormalization group\ncalculation;21)however, it is difficult to apply both methods to a two-dimensional (2D )\nfrustrated system like the present model. This disadvantage come s from the fact that\nthe dimension of the matrix grows exponentially with respect to the s ystem size. In\nthis study, we treat the finite-size clusters depicted in Fig. 1 when t he system sizes\nareNs= 24 and 30 under the periodic boundary condition. Note that each o f these\nclusters forms a regular square although cluster (b) is tilted from a ny directions along\n3/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n–2 0\nh/J 1–1 01M/M sJ2/J 1 = 0.55(b)0 5 10 \nM–10–5 E /J 1Ns = 30\nJ2/J 1 = 0.55(a)\n2\nFig. 2. (Color) Results for J2/J1= 0.55. Lowest energy in each subspace of Mfor the system of\nNs= 30 is shown in panel (a). The magnetization process is depicted in pa nel (b); red and black lines\nrepresent results for Ns= 24 and 30, respectively.\ninteraction bonds.\nWecalculatethelowestenergyof H0inthesubspacecharacterizedby/summationtext\njSz\nj=Mby\nnumerical diagonalizations based on the Lanczos algorithm and/or t he Householder al-\ngorithm. The energy is represented by E(Ns,M), whereMtakes every integer up to the\nsaturation value Ms(=SNs). We here use the normalized magnetization m=M/Ms.\nSome of Lanczos diagonalizations have been carried out using the MP I-parallelized\ncode, which was originally developed in the study of Haldane gaps.22)Note here that\nour program was effectively used in large-scale parallelized calculation s.23–25)\nTo obtain the magnetization process for a finite-size system, one fi nds the magneti-\nzation increase from MtoM+1 at the field\nh=E(Ns,M+1)−E(Ns,M), (3)\nunder the condition that the lowest-energy state with the magnet izationMand that\nwithM+1 become the ground state in specific magnetic fields. Note here th at it often\nhappens that the lowest-energy state with the magnetization Mdoes not become the\n4/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n–0.2 0\nh/J 1–0.4 –0.2 00.20.4M/M sJ2/J 1 = 0.64(b)\n0010 2 4 6\nM–13–12–11E /J 1Ns = 30\nJ2/J 1 = 0.64(a)\n0.2 3\nFig. 3. (Color) Results for J2/J1= 0.64. Lowest energy in each subspace of Mfor the system of\nNs= 30 is shown in panel (a). The magnetization process is depicted in pa nel (b); red and black lines\nrepresent results for Ns= 24 and 30, respectively. Main panel is a zoomed-in view of its inset wit h a\nwide range. The broken lines represent the results before the Max well construction is carried out.\nground state in any field. The magnetization process in this case is de termined around\nthe magnetization Mby the Maxwell construction.26,27)\nNow, we observe the case of J2/J1= 0.55; results are shown in Fig. 2. Figure 2(a)\ndepicts the lowest energy level in the subspace belonging to MforNs= 30. The levels\nforM= 0 toM= 5 are identical within the numerical accuracy. For M >5, the\nenergies increase with M. This behavior indicates that the spontaneous magnetization\nisM= 5. In Fig. 2(b), we draw the magnetization process determined by eq. (3) in the\nfull range from the negative to the positive saturations. The spon taneous magnetization\nm= 1/3 appears and the state at m= 1/3 shows the plateau with a large width. It is\nobserved that, above m= 1/3, the magnetization grows continuously. These behaviors\nare common with those of the LM ferrimagnetism at the unfrustrat ed case of J2= 0.\nNext, let us examine the case of J2/J1= 0.64; results are shown in Fig. 3. The\nMdependence of the lowest energy belonging to Mis different in M <3 from the\n5/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\ncase ofJ2/J1= 0.55. This difference affects with the disappearance of the spontane ous\nmagnetization, which is shown in Fig. 3(b). This discontinuous disappe arance occurs\natJ2/J1∼0.59 forNs= 24 and at J2/J1∼0.63 forNs= 30. An important point\nis that an intermediate state with smaller but nonzero spontaneous magnetizations is\nabsent between the m= 1/3 state and the nonmagnetic state. This behavior is clearly\ndifferent from the presence of such an intermediate state in the sp atially anisotropic\nkagome lattice.9,10)We speculate that this difference comes from the point that the\ncompeting interaction in the present model has a strong quantum n ature localized\nat pairs of dimerized spins. The discovery of the future third case o f the collapsing\nferrimagnetism would contribute to confirm our speculation. Note a lso that the plateau\natm= 1/3 shows a large width. This suggests that the ferrimagnetic state is realized\nif external magnetic fields are added.\nTo examine the properties of the m= 1/3 states in a more detailed way, we evaluate\nthe local magnetization defined as\nmξ\nLM=1\nNξ/summationdisplay\nj∈ξ/angbracketleftSz\nj/angbracketright, (4)\nwhereξtakesα,α′andβ. Here, the symbol /angbracketleftO/angbracketrightdenotes the expectation value of the\noperator Owith respect to the lowest-energy state within the subspace with a fixed\nMof interest. Recall here that the case of interest in this paper is M=Ms/3. Here\nNξdenotes the number of ξsites. Results are shown in Fig. 4. In the region of small\nJ2/J1,αandα′spins are up and βspin is down, although each of magnetizations is\nslightly deviated from the full moment due to a quantum effect. This s pin arrangement\nis a typical behavior of ferrimagnetism. On the other hand, in the re gion of large J2/J1,\nthe magnetizations at αandα′spins are vanishing and βspin shows almost a full\nmoment up. This marked change in the local magnetizations occurs a tJ2/J1∼1.38 for\nNs= 24 and at J2/J1∼1.40 forNs= 30, which suggests the occurrence of the phase\ntransition around at J2/J1∼1.4. Therefore one finds that, for J2/J1larger than this\ntransition point, the ferrimagnetic state cannot be realized even u nder magnetic fields.\nItisunfortunatelydifficulttodeterminethetransitionpointintheth ermodynamiclimit\nprecisely only from the present two samples of small clusters. For t he determination,\ncalculations of larger clusters are required in future studies. Note here that similar\nobservations of the local magnetizations were reported in Refs. 2 8 and 29, which treated\ntheHeisenberg antiferromagnet ontheCairo-pentagonlattice,30)a2Dnetwork obtained\nby the tiling of single-kind inequilateral pentagons. The same behavio r ofmξ\nLMis also\n6/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\nobserved when the kagome-lattice antiferromagnet31–37)is distorted in the√\n3×√\n3\ntype.25,38)The relationship between these models should be examined in future s tudies.\nNote also that, in the present model, the change around the trans ition point seems\ncontinuous irrespective of whether the system size is Ns= 24 or 30. This aspect is\ndifferentfromtheobservationintheCairo-pentagon-latticeantif erromagnet,28,29)where\nthe change around the transition point seems continuous for Ns= 24 but discontinuous\nforNs= 30. We speculate that whether the change is continuous or discon tinuous in\nfinite-size data is related to whether the number of unit cells in finite- size clusters is an\neven integer or an odd integer. To confirm this speculation, furthe r investigations are\nrequired in future. It will be anunresolved question whether the tr ansitionis continuous\nordiscontinuousinthethermodynamiclimit.Figure5depictsthemagn etizationprocess\natJ2/J1∼1.39. No jumps seem to appear in the process at J2/J1corresponding to\nthe transition point. It is unclear whether the width at m= 1/3 survives or vanishes\nalthough this m= 1/3 width at J2/J1∼1.39 is smaller than those in Figs. 2(b) and\n3(b). Future studies would clarify how themagnetization process b ehaves in the vicinity\nof the transition point.\nIn summary, we have investigated how the ferrimagnetic state of t heS= 1/2\nHeisenberg antiferromagnet onthe 2Dlatticecollapses owing to mag netic frustrationby\nnumerical-diagonalization method. We capture a discontinuous vanis hing of the sponta-\nneous magnetization without intermediate phase showing spontane ous magnetizations\nthat are smaller than that of the Lieb-Mattis ferrimagnetic state w hen a frustrating\ninteraction is increased. We also observe the disappearance of the ferrimagnetic state\nunder magnetic fields for even larger interaction showing frustrat ion. It is known that\norganicmolecularmagnetscanrealizeferrimagnetism.39,40)Sincevarietyoflatticestruc-\ntureleadingtoaninteractionnetworkisavailableinsuchorganicmolec ularmagnets,the\nexperimental confirmation might be done in these magnets more eas ily than metallic-\nelement compounds. Further studies concerning instability of the f errimagnetism would\ncontribute much for our development of more stable ferrimagnetic materials.\nAcknowledgments This work was partly supported by JSPS KAKENHI Grant Numbers 23 340109\nand 24540348. Nonhybrid thread-parallel calculations in numerical diagonalizations were based on\nTITPACK version 2 coded by H. Nishimori. Part of calculations in this st udy were carried out as an\nactivity of a cooperative study in Center for Cooperative Work on C omputational Science, University\nof Hyogo. Some of the computations were also performed using fac ilities of the Department of\nSimulation Science, National Institute for Fusion Science; Center f or Computational Materials\nScience, Institute for Materials Research, Tohoku University; Su percomputer Center, Institute for\n7/10Jpn. J. Appl. Phys. RAPID COMMUNICATION\n0.5 1 1.5\nJ2/J 1–0.4 –0.2 00.20.4L ocal magnetization1.38 1.4 1.4200.5\n2\nFig. 4. (Color) Behavior of local magnetizations vs. the ratio of interactio nsJ2/J1together with a\nzoomed-in view near the transition point in inset. Closed circles and clo sed diamonds denote results\nforαandβforNs= 24, respectively. Results for α′forNs= 24 are identical those for αwithin the\nnumerical accuracy because αandα′are symmetric in the Ns= 24 cluster. Open circle, open\ntriangle, and open squares represent results for α,α′, andβforNs= 30, respectively. Due to the\ntilting for Ns= 30,αandα′are not symmetric, although results of αandα′forNs= 30 are slightly\ndifferent but very similar. To avoid invisibility from overlapping of symbo ls, results of α′are shown\nonly in inset.\n0 1 2 \nh/J 100.51M/M sJ2/J 1 = 1.39\n3\nFig. 5. (Color) Magnetization process for J2/J1= 1.39. Red and black lines represent results for\nNs= 24 and 30, respectively.\nSolid State Physics, The University of Tokyo; and Supercomputing D ivision, Information Technology\nCenter, The University of Tokyo. This work was partly supported b y the Strategic Programs for\nInnovative Research; the Ministry of Education, Culture, Sports , Science and Technology of Japan;\nand the Computational Materials Science Initiative, Japan. 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China\n(Dated: October 8, 2018)\nAbstract\nThe spin ordering and spin-charge coupling in LuFe 2O4were investigated on the basis of den-\nsity functional calculations and Monte Carlo simulations. The 2:1 ferrimagnetism arises from the\nstrong antiferromagnetic intra-sheet Fe3+-Fe3+and Fe3+-Fe2+as well as some substantial antifer-\nromagnetic Fe2+-Fe3+inter-sheet spin exchange interactions. The giant magneto capacitance at\nroom temperature and the enhanced electric polarization at 240 K of LuFe 2O4are explained by\nthe strong spin-charge coupling.\nPACS numbers: 75.80.+q,71.20.-b,77.80.-e,64.60.De\n1Recently, multiferroics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have attracte d much attention be-\ncause of their potential applications in novel magnetoelectric and m agneto-optical devices.\nAmong the newly discovered multiferroics, LuFe 2O4is particularly interesting due to its\nlarge ferroelectric (FE) polarization [3] and giant magnetocapacita nce at room temperature\n[4]. In the high-temperature crystal structure of LuFe 2O4with space group R ¯3m, layers of\ncomposition Fe 2O4alternate with layers of Lu3+ions, such that there are three Fe 2O4layers\nper unit cell. Each Fe 2O4layer is made up of two triangular sheets (hereafter, T-sheets) o f\ncorner-sharing FeO 5trigonal bipyramids (Fig. 1). Below 320 K ( TCO) LuFe 2O4undergoes a\nthree-dimensional (3D) charge ordering (CO) (2Fe2.5+⇒Fe2++ Fe3+) with the√\n3×√\n3\nsuperstructure in each T-sheet; in each Fe 2O4layer, one T-sheet has the honeycomb net-\nwork of Fe2+ions with a Fe3+ion at the center of each Fe2+hexagon (hereafter, the type\nA T-sheet), while the other T-sheet has an opposite arrangement of the Fe2+and Fe3+ions\n(hereafter the type B T-sheet).\nLuFe2O4, with the novel CO-driven “electronic ferroelectricity”, [3] prese nts several fun-\ndamental questions. First, LuFe 2O4shows strong Ising behavior with the easy axis along c\n[11, 12]. The spin anisotropy of the non-CO state is understandable because the spin down\nelectron of the Fe2.5+ion partially occupies the degenerate ( dx2−y2,dxy) orbitals [5, 13]. How-\never, the Ising behavior below TCOis puzzling because the insulating√\n3×√\n3 CO breaks\nthe 3-fold rotational symmetry hence lifting the degeneracy of th e (dx2−y2,dxy) orbitals [5].\nSecond, LuFe 2O4undergoes a ferrimagnetic spin ordering below 240 K ( TN) [11, 14, 15, 16].\nA number of experimental studies found this spin ordering to be two -dimensional (2D) in\nnature [11, 14, 17]. In contrast, a recent neutron diffraction stu dy observed a finite spin\ncorrelation along cand suggested a 3D spin structure without considering CO [16]. The\nM¨ ossbauer [14] and neutron diffraction [15] studies led to a detailed ferrimagnetic structure\nof LuFe 2O4, in which the majority spin lattice consists of all Fe2+ions plus one-third of the\ntotal Fe3+ions while the minority spin sublattice consists of the remaining Fe3+ions. This\n2:1 ferrimagnetic order was suggested to originate from weak ferr omagnetic (FM) interac-\ntions between the next-nearest neighbor (NNN) Fe sites in the tria ngular antiferromagnetic\n(AFM) Ising lattice [11]. However, using the spin exchange paramete rs estimated from the\nenergy parameters of LaFeO 3, Nakaet al.[18] predicted quite a different spin structure that\nincludes some Fe sites without unique spin direction. Therefore, the detailed ferrimagnetic\nstructure and its origin remain unclear. Third, LuFe 2O4exhibits a giant magnetodielec-\n2tric response at room temperature [4], and a room-temperature d ynamic magnetoelectric\ncoupling was also reported [19]. Furthermore, the FE polarization o f LuFe 2O4was found\nto increase around TN[3]. These observations suggest the occurrence of coupling betwe en\nthe CO and magnetism. The understanding of the spin-charge coup ling is crucial for future\nmagnetodielectric applications of LuFe 2O4.\nIn this Letter, we explore these isuues on the basis of first principle s density functional\ncalculations for the first time. A large spin anisotropy is found along t hecdirection due\nmainly to the Fe2+ions of the B-sheet, the spin ground state of the√\n3×√\n3 CO state has\nthe 2:1 ferrimagnetic spin arrangement proposed by Siratori et al.[15], and there occurs\nstrong spin-charge coupling in LuFe 2O4.\nOur density functional theory calculations employed the frozen-c ore projector aug-\nmented wave method [20] encoded in the Vienna ab initio simulation package [21], and\nthe generalized-gradient approximation (GGA) [22]. To properly des cribe the strong elec-\ntron correlation in the 3d transition-metal oxide, the GGA plus on-s ite repulsion U method\n(GGA+U) [23] was employed with the effective Uvalue (Ueff=U−JwithJ= 0) of\n4.61 eV [5]. It is known experimentally [11, 14, 17] that the interlayer m agnetic interactions\nin LuFe 2O4are weak, which is understandable due to its layered structure. In this work,\ntherefore, we focus on the 2D spin ordering within a single Fe 2O4layer. For the√\n3×√\n3\nCO state of LuFe 2O4, the FE ordering of the Fe 2O4layers will be assumed.\nWe first examine the magnetic anisotropy of the Fe ions by performin g GGA+U cal-\nculations, with spin-orbit coupling (SOC) included, for the FM state o f LuFe 2O4with the\n√\n3×√\n3 CO. As shown in Fig. 1(a), there are two kinds of Fe2+ions and two kinds of\nFe3+ions in the√\n3×√\n3 CO state. We label the Fe2+and Fe3+ions of the type A T-sheet\nas 2A and 3A, respectively, and those of the type B T-sheet as 2B a nd 3B, respectively.\nIn our GGA+U+SOC calculations with spins pointing along several differ ent directions, all\nFe2+and Fe3+spins are kept in the same direction. Our calculations show that the e asy\naxis is along the cdirection, as experimentally observed [11, 12]; the /bardblc-spin orientation is\nmore stable than the ⊥c-spin orientation by 1.5 meV per formula unit (FU). The orbital\nmoments of 2A, 2B, 3A, and 3B for the /bardblc-spin orientation are 0.101, 0.156, 0.031 and 0.035,\nrespectively, which are greater than those for the ⊥c-spin orientation by 0.019, 0.062, 0.015,\nand 0.018 µB, respectively. As expected, the Fe3+(d5) ions have a very small anisotropy,\nHowever, two kinds of the Fe2+ions also have different degree of spin anisotropy. The spin\n3downelectronofthe2BFe2+ionoccupiesthe( dx2−y2,dxy)manifold[5], thereforethe2BFe2+\nion has the largest spin anisotropy along c. Our calculations indicate a non-negligible or-\nbital contribution to the total magnetization, in agreement with th e X-ray magnetic circular\ndichroism result [12].\nTo determine the magnetic ground state of LuFe 2O4in the√\n3×√\n3 CO state, we extract\nits spin exchange parameters by mapping the energy differences be tween ordered spin states\nobtainedfromGGA+Ucalculationsontothecorresponding energyd ifferences obtainedfrom\nthe Ising Hamiltonian [24]:\nH=/summationdisplay\ni,jJijSizSjz, (1)\nwhere the energy is expressed with respect to the spin disorder (p aramagnetic) state, Jijis\nthe spin exchange parameter between the spin sites iandj, andSizis the spin component\nalong the cdirection ( |Sz|= 2 and 2 .5 for Fe2+and Fe3+ions, respectively). We consider\nall 15 possible superexchange (SE) interactions and all 19 super-s uperexchange (SSE) in-\nteractions with the O...O distance less than 3.2 ˚A. The intra- and inter-sheet interactions\nwithin each Fe 2O4layer as well as the SSE interactions between adjacent Fe 2O4layers are\ntaken into account. To evaluate these 34 spin exchange paramete rs reliably, we considered\n111 different ordered spin states leading to 110 energy differences . The 34 spin exchange\nparameters were determined by performing a linear least-square fi tting analysis. The SSE\ninteractions are generally much weaker than the SE interactions wit h the magnitude of all\nSSE interactions less than 1.4 meV. The calculated SE parameters ar e reported in Table I.\nAll intra-sheet SE interactions are AFM, and the strongest intera ctions (∼7.3 meV) occurs\nbetween the 3B Fe3+ions because of the large energy gain of the AFM configuration and a l-\nmost zero FM coupling. The inter-sheet SE interactions are weaker than the the intra-sheet\nSE interactions, and are mostly AFM.\nWith the calculated spin exchange parameters, one can identify the spin ground state of\nthe CO state. The Metropolis Monte Carlo simulation of the Ising mode l is performed to\nsearch for the ground state. Simulations with supercells of severa l different sizes show that\nthe spin ground state has the magnetic structure shown in Fig. 2(a ), which has the same cell\nas the√\n3×√\n3 CO structure. In this state, all Fe2+ions contribute to the majority spin,\nand the Fe3+ions are antiferromagnetically coupled to the Fe2+ions in the type A T-sheet.\nIn the honeycomb lattice of the type B T-sheet, the Fe3+spins are antiferromagnetically\n4coupled. Thus, the spin ground state is ferrimagnetic, as experime ntally observed [11]. This\n2:1 ferrimagnetic structure is the same as the magnetic structure proposed by Siratori et al.\n[15], and differs from the structure proposed by Naka et al.[18].\nThe observed ferrimagnetic ordering can be readily explained in term s of the calculated\nexchangeparameters. Inthehoneycomb networkofthetypeBT -sheet, thenearest-neighbor\n(NN) 3B ions are antiferromagnetically coupled since their SE interac tion is strongly AFM.\nIn the type A T-sheet, the SE interactions between the 2A ions are AFM, and so are those\nbetween the 2A and 3A ions, which leads to spin frustration. As a con sequence, two possible\nspin arrangements compete with each other in the type A T-sheet; the first is the state in\nwhich the coupling between the NN 2A ions are AFM with the spin directio n of the 3A\nion undetermined, and the second is the state in which all 2A ions are a ntiferromagnetically\ncoupledtothe3Aions. Theenergiesofthesetwostates(consider ing onlytheSEinteraction)\nareE1=−4(J2A1,2A2+J2A1,2A4) per 3A ion, and E2=−10(J3A1,2A1+J3A1,2A2+J3A1,2A3)+\n4(J2A1,2A2+J2A1,2A4) per 3A ion, respectively. Due to the relatively strong AFM interact ions\nbetween the 3A and 2A ions (See Table I) and the large spin of the 3A io ns, the second\nstate has a lower energy, i.e., E2< E1. Without loss of generality, we can assume the 2A\n(3A) ions constitute the majority (minority) spin in the second stat e. Now, we examine\nthe spin orientation of the Fe2+ions in the type B T-sheet. The intra-sheet interactions\nof the 2B ion with 3B ions vanish due to the AFM ordering of the 3B ions. As for the\ninter-sheet interactions involving the 2B ions, the dominant one is th e AFM interaction of\nthe 2B ion with the 3A ion ( J3A1−2B1in Table I). Consequently, we obtain the ferrimangetic\nground state shown in Fig. 2(a), in which the spin of the 2B ion contrib utes to the majority\nspin of the Fe 2O4layer. For the stability of the ferrimangetic ground state, the inte r-sheet\ninteraction is essential. This was neglected in the model Hamiltonian st udy of Naka et al.\n[18]. The ferrimangetic state is not due to the FM interactions betwe en NNN Fe ions of the\nT-sheet because they must be vanishingly weak and mostly AFM.\nThe electronic structure of the ferrimangetic state calculated fo r the√\n3×√\n3 CO struc-\nture of LuFe 2O4is shown in Fig. 3. Also shown is the electronic structure calculated fo r\nthe FM state. Both states are semiconducting, and the highest oc cupied (HO) and the\nlowest unoccupied (LU) levels of both states come from the spin-up Fe2+and Fe3+ions,\nrespectively [5]. In addition, the band dispersion from Γ to A is rather small, indicating a\nvery weak interlayer interaction. However, there are some import ant differences. First, the\n5ferrimangetic state has a larger bandgap(1.68 eV) than doesthe F Mstate (0.77 eV). This is\nconsistent with the stability of the ferrimangetic state. Second, t he FM state has an indirect\nband gap with the HO and LU levels located at K and Γ, respectively. In the ferrimangetic\nstate, however, the LU level has the highest energy at Γ and the b and dispersions of the HO\nand LU levels are almost flat fromM to K. This difference comes fromth e orbital interaction\nbetween the spin down ( dx2−y2,dxy) levels of the spin up Fe3+and Fe2+ions.\nTo probe the presence of spin-charge coupling in LuFe 2O4, it is necessary to consider the\nspin ordering in a CO state other than the√\n3×√\n3 CO state. The previous electrostatic\ncalculations [5, 18] showed that the chain CO, in which one-dimensiona l (1D) chains of Fe2+\nions alternate with 1D chains of Fe3+ions in each T-sheet [Fig. 2(b)], is only slightly less\nstable than the√\n3×√\n3 CO, and has no FE polarization. We extract exchange parameters\nby mapping analysis as described above. It is found that the intra-s heet SE between the\nFe3+ions is the strongest ( J= 6.7 meV) as in the√\n3×√\n3 CO case. All intra-sheet SE’s are\nAFM with J(Fe3+-Fe3+)> J(Fe2+-Fe3+)> J(Fe2+-Fe2+). The inter-sheet SE between the\nFe3+ions is very weak ( |J|<0.3 meV), and that between the Fe2+and Fe3+ions is FM with\nJ=−1.4 meV. Interestingly, the inter-sheet SE between the Fe2+ions is rather strongly\nAFM(J= 6.3meV).MonteCarlosimulationsusingthesespinexchangeparamete rsindicate\nthat the spin state shown in Fig. 2(b) is the spin ground state. In th is spin ordering, the\nspins within each chain of Fe2+ions or Fe3+ions are antiferromagnetically coupled. The\nNN chains of Fe2+ions belonging to different T-sheets are coupled antiferromagnetic ally,\nwhereas the corresponding chains of Fe3+are almost decoupled.\nThe above results show that the spin ordering of the chain CO state is dramatically\ndifferent from that of the√\n3×√\n3 CO state. The most important difference is that the\ntotal spin moments are 2.33 µB/FU for the√\n3×√\n3 CO, but 0 µB/FU for the chain CO.\nThis evidences a strong spin-charge coupling in LuFe 2O4. The external magnetic field will\nhave different effects on the two CO states due to the the Zeeman e ffect. It is expected that\nthe magnetic field will further stabilize the ferrimagnetic√\n3×√\n3 CO state. Consequently,\nan external magnetic field will reduce the extent of charge fluctua tion and hence decrease\nthe dielectric constant. This supports our explanation for the gian t magnetocapacitance\neffect of LuFe 2O4at room temperature [5] .\nWithout considering the inter-sheet interactions, Naka et al.[18] suggested that the\ndegeneracy of the spin ground state of the√\n3×√\n3 CO state is of the order O(2N/3)( N is\n6the number of the spin sites), which is much larger than the spin dege neracy [O(2√\nN)] of\nthe chain CO state. Thus, they proposed that spin frustration ind uces reinforcement of the\npolar√\n3×√\n3 CO by a gain of spin entropy. However, our calculations show that t here\nare substantial inter-sheet spin exchange interactions between the 2B1 and 3A1 ions, which\nwould remove the macroscopic degeneracy of the spin ground stat e of the√\n3×√\n3 CO state.\nThe macroscopic degeneracy still persists for the chain CO state. Thus, our work provides\na picture opposite to what Naka et al.proposed. Furthermore, we find that the√\n3×√\n3\nCO state is more favorable for the spin ordering than is the chain CO s tate; with respect to\nthe paramagnetic state, the spin ground state is lower in energy by −78 meV/FU for the\n√\n3×√\n3 CO, but by −57 meV/FU for the chain CO. The model of Naka et al.[18] predicts\nthat the polar√\n3×√\n3 CO state is destabilized and the electric polarization is reduced by\nthe magnetic field, since it will lift the macroscopic spin degeneracy. I n contrast, our work\npredicts that the magnetic field stabilizes the ferrimagnetic√\n3×√\n3 CO state due to the\nZeeman effect, and provides an explanation for why the electric pola rization increases when\nthe temperature is lowered below the Neel temperature [3], becaus e the charge fluctuation\nhas an onset well below TCO[8].\nIn summary, our first principles results explain the experimentally ob served Ising ferri-\nmagnetism, and manifest the spin-charge coupling and magnetoelec tric effect in LuFe 2O4.\nWork at NREL was supported by the U.S. Department of Energy, un der Contract No.\nDE-AC36-08GO28308, and work at NCSU by the U. S. Department o f Energy, under Grant\nDE-FG02-86ER45259.\n7[1] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, an d Y. Tokura, Nature (London)\n426, 55 (2003).\n[2] N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S-W. Ch eong, Nature (London) 429,\n392 (2004).\n[3] N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kaku rai, Y. Murakami, K. Yoshii, S.\nMori, Y. Horibe, and H. Kitˆ o, Nature (London) 436, 1136 (2005).\n[4] M. A. Subramanian, T. He, J. Chen, N. S. Rogado, T. G. Calva rese, and A. W. Sleight, Adv.\nMater.18, 1737 (2006).\n[5] H. J. Xiang and M.-H. Whangbo, Phys. Rev. Lett. 98, 246403 (2007).\n[6] Y. Zhang, H. X. Yang, C. Ma, H. F. Tian, and J. Q. Li, Phys. Re v. Lett.98, 247602 (2007).\n[7] M. Angst, R. P. Hermann, A. D. Christianson, M. D. Lumsden , C. Lee, M.-H. Whangbo,\nJ.-W. Kim, P. J. Ryan, S. E. Nagler, W. Tian, R. Jin, B. C. Sales , and D. Mandrus, Phys.\nRev. Lett. 101, 227601 (2008).\n[8] X. S. Xu, M. Angst, T. V. Brinzari, R. P. Hermann, J. L. Musf eldt, A. D. Christianson, D.\nMandrus, B. C. Sales, S. McGill, J.-W. Kim, and Z. Islam, Phys . Rev. Lett. 101, 227602\n(2008).\n[9] H. J. Xiang and M.-H. Whangbo, Phys. Rev. Lett. 99, 257203 (2007).\n[10] H. J. Xiang, S.-H. Wei, M.-H. Whangbo, and J. L. F. Da Silv a, Phys. Rev. Lett. 101, 037209\n(2008).\n[11] J. Iida, M. Tanaka, Y. Nakagawa, S. Funahashi, N. Kimizu ka, and S. Takekawa, J. Phys. Soc.\nJpn.62, 1723 (1993).\n[12] W. Wu, V. Kiryukhin, H.-J. Noh, K.-T. Ko, J.-H. Park, W. R atcliff II, P. A. Sharma, N.\nHarrison, Y. J. Choi, Y. Horibe, S. Lee, S. Park, H. T. Yi, C. L. Zhang, and S.-W. Cheong,\nPhys. Rev. Lett. 101, 137203 (2008).\n[13] D. Dai and M.-H. Whangbo, Inorg. Chem. 44, 4407 (2005).\n[14] M. Tanaka, H. Iwasaki, K. Siratori, and I. Shindo, J. Phy s. Soc. Jpn. 58, 1433 (1989).\n[15] K. Siratori, S. Funahashi, J. Iida, and M. Tanaka, Proc. 6th Intern. Conf. Ferrites, Tokyo and\nKyoto, Japan, 1992, p. 703.\n[16] A. D. Christianson, M. D. Lumsden, M. Angst, Z. Yamani, W . Tian, R. Jin, E. A. Payzant,\n8S. E. Nagler, B. C. Sales, and D. Mandrus, Phys. Rev. Lett. 100, 107601 (2008).\n[17] S. Funahashi, J. Akimitsu, K. Siratori, N. Kimizuka, M. Tanaka, and H. Fujishita, J. Phys.\nSoc. Jpn. 53, 2688 (1984).\n[18] M. Naka, A. Nagano, and S. Ishihara, Phys. Rev. B 77, 224441 (2008); A. Nagano, M. Naka,\nJ. Nasu, and S. Ishihara, Phys. Rev. Lett. 99, 217202 (2007).\n[19] J. Y. Park, J. H. Park, Y. K. Jeong, and H. M. Jang, Appl. Ph ys. Lett. 91, 152903 (2007).\n[20] P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994); G. Kresse and D. Joubert, ibid59, 1758 (1999).\n[21] G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15 (1996); Phys. Rev. B 54, 11169\n(1996).\n[22] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett .77, 3865 (1996).\n[23] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, Phys. Rev. B52, R5467 (1995); S. L.\nDudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P . Sutton, Phys. Rev. B 57,\n1505 (1998).\n[24] M.-H. Whangbo, H.-J. Koo and D. Dai, J. Solid State Chem. 176, 417 (2003).\n9TABLE I: Calculated superexchange parameters (in meV) in th e√\n3×√\n3 CO state of LuFe 2O4\n(For the spin sites of the 2A, 3A, 2B and 3B ions,see Fig. 1 )\nA-AJ3A1,2A1J3A1,2A2J3A1,2A3J2A1,2A2J2A1,2A4\n3.2 4.0 4.7 1.9 3.6\nB-BJ3B1,3B2J3B1,3B4J2B1,3B1J2B1,3B2J2B1,3B3\n7.0 7.6 1.5 2.8 1.3\nA-BJ3A1,3B1J3A1,2B1J2A1,2B1J2A1,3B2J2A1,3B3\n2.0 1.9 ∼0−0.6 1.2\n10Fe\nOO3A1\n3B43B13B33B2\n2B12A42A1 2A2\n2A3\nFIG. 1: (Color online) Schematic representation of the√\n3×√\n3 CO structure. Large, medium, and\nsmall circles represent the Fe2+, Fe3+, and O2−ions, respectively. The type A (type B) T-sheet has\nthe honeycomb network of Fe2+(Fe3+) ions with a Fe3+(Fe2+) ion at the center of each hexagon.\n2A and 3A (2B and 3B) refer to the Fe2+and Fe3+ions of the type A (type B) T-sheet, respetively.\nThe region enclosed by dashed lines indicates the unit cell o f the CO structure. There is a mirror\nplane of symmetry, which is parallel to the caxis and crosses the 3A1 and 2B1 sites. The inset\nshows an isolated FeO 5trigonal bipyramid.\nFe3+Fe2+3B3Ac\n(a)\n(b)2B2A\nFIG. 2: (Color online) Schematic representations of (a) the spin ground state of the√\n3×√\n3\nCO structure and (b) one of the macroscopic spin ground state s of the chain CO structure. The\narrows denote the spin directions. The region enclosed by th e dashed lines on the bottom T-sheet\nindicates the magnetic unit cell of the spin structure.\n11-2-1012Energy (eV)\nΓ M K AΓ-2-1012Energy (eV)(a) Ferromagnetic\n(b) FerrimagneticFe3+\n3+Fe\nFe2+Fe2+\nFIG. 3: (Color online) Band structures calculated for (a) th e FM state and (b) the ferrimagnetic\nstate of the√\n3×√\n3 CO structure of LuFe 2O4. The solid and dashed lines represent the up-spin\nand down-spin bands, respectively. The√\n3×√\n3×1 hexagonal cell is used in the calculations.\n12"    },    {        "title": "1706.01925v1.Magnetic_order_and_interactions_in_ferrimagnetic_Mn3Si2Te6.pdf",        "content": "Magnetic order and interactions in ferrimagnetic Mn 3Si2Te6\nAndrew F. May,1,\u0003Yaohua Liu,2Stuart Calder,2David S. Parker,1Tribhuwan\nPandey,1Ercan Cakmak,1Huibo Cao,2Jiaqiang Yan,1and Michael A. McGuire1\n1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831\n2Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831\n(Dated: June 8, 2017)\nThe magnetism in Mn 3Si2Te6has been investigated using thermodynamic measurements, \frst\nprinciples calculations, neutron di\u000braction and di\u000buse neutron scattering on single crystals. These\ndata con\frm that Mn 3Si2Te6is a ferrimagnet below TC\u001978 K. The magnetism is anisotropic, with\nmagnetization and neutron di\u000braction demonstrating that the moments lie within the basal plane\nof the trigonal structure. The saturation magnetization of \u00191.6\u0016B/Mn at 5 K originates from the\ndi\u000berent multiplicities of the two antiferromagnetically-aligned Mn sites. First principles calculations\nreveal antiferromagnetic exchange for the three nearest Mn-Mn pairs, which leads to a competition\nbetween the ferrimagnetic ground state and three other magnetic con\fgurations. The ferrimagnetic\nstate results from the energy associated with the third-nearest neighbor interaction, and thus long-\nrange interactions are essential for the observed behavior. Di\u000buse magnetic scattering is observed\naround the 002 Bragg re\rection at 120 K, which indicates the presence of strong spin correlations well\naboveTC. These are promoted by the competing ground states that result in a relative suppression\nofTC, and may be associated with a small ferromagnetic component that produces anisotropic\nmagnetism below \u0019330 K.\nKeywords: Mn3Si2Te6, ferrimagnetism, competing interactions, magnetic frustration, di\u000buse scattering\nI. INTRODUCTION\nhttps://doi.org/10.1103/PhysRevB.95.174440\nUnderstanding and manipulating magnetic anisotropy\nis important in both basic and applied physics research.\nFor instance, anisotropic magnetism is essential for the\ndevelopment of permanent magnets, and it appears to\nbe a fundamental component in many systems that dis-\nplay unconventional superconductivity. Recently, re-\nsearch on materials that are crystallographically lay-\nered (held together by van der Waals bonds) has gained\nprominence in condensed matter physics. This is largely\ndriven by the now-realized prospect of building van der\nWaals heterostructures from materials with complemen-\ntary properties.1Two-dimensional and quasi-2D materi-\nals are of fundamental interest from a bulk perspective\nas well, in part because they exist in the limit of very\nanisotropic interactions (strong in-plane interactions and\nweak cross-plane interactions). For instance, the com-\npounds CrSiTe 3, MnPS 3, and CrI 3have layers connected\nby van der Waals bonds, and demonstrate bulk mag-\nnetic ordering with anisotropic interactions manifesting\nthemselves in anisotropic properties, suppressed 3D or-\ndering temperatures, two-dimensional order, and persis-\ntent short-range correlations above TC.2{9In this work,\nwe probe the behavior of Mn 3Si2Te6, which can be con-\nsidered as a three-dimensional analogue of CrSiTe 3.\nWhile its crystal structure is interesting, very little re-\nsearch has been reported on Mn 3Si2Te6.10,11The initial\nreport of its existence and physical properties discussed\nthe magnetism within the framework of an incorrect sto-\nichiometry and structure,10and thus the initial hypoth-\nesis of ferrimagnetism could not be fully evaluated. De-\nspite containing nominally Mn2+with no orbital mo-ment (3d5with S=5/2, L=0), the magnetism displays\na large anistropy \feld on the order of 10 T at 5 K.10\nThe crystal structure was reported several years after\nthe initial characterization,11but the magnetic proper-\nties were not revisited. Mn 3Si2Te6has a trigonal crystal\nstructure (space group No. 163) that is shown in Fig. 1.11\nFig. 1 also contains the magnetic structure obtained from\nour neutron di\u000braction data. Mn 3Si2Te6is composed of\nMnTe 6octahedra that are edge-sharing within the ab\nplane (Mn1 site), and along with Si-Si dimers this creates\nlayers of Mn 2Si2Te6. The layered framework is analogous\nto that of CrSiTe 3, which is hexagonal and has a van der\nWaals gap between the layers. In Mn 3Si2Te6, however,\nthe layers are linked by the \flling of one-third of the octa-\nhedral holes within the van der Waals gap by Mn atoms\nat the Mn2 site, yielding a composition of Mn 3Si2Te6.11\nImportantly, the multiplicity of Mn1 is twice that of Mn2.\nIn this work, Mn 3Si2Te6was characterized using mag-\nnetization, speci\fc heat and electrical resistivity mea-\nsurements, as well as via powder x-ray and single crystal\nneutron di\u000braction. We \fnd Mn 3Si2Te6to be a ferri-\nmagnet due to antiparallel alignment of moments on the\nMn1 and Mn2 atomic positions, the di\u000berent multiplicity\nof which yields a net magnetization with moments pre-\nferring to lie in the trigonal plane. The data, including\ndi\u000buse neutron scattering, reveal the existence of strong\nspin correlations well-above TC, which may be associated\nwith short range order or the persistence of correlated\nexcitations in the paramagnetic region. The experimen-\ntal probes were complemented by \frst principles calcula-\ntions that revealed a competition between antiferromag-\nnetic (AFM) exchange interactions that are frustrated\nwith respect to each other. Interestingly, the longer-\nrange, third-nearest neighbor coupling (Mn1-Mn2 pairs\nat 5.41 \u0017A) dominates over the in-plane coupling of second-arXiv:1706.01925v1  [cond-mat.str-el]  6 Jun 20172\nnearest neighbors at 4.06 \u0017A (Mn1-Mn1 pairs). The net\nresult is an antiferromagnetic con\fguration close in en-\nergy to the ferrimagnetic one, and this competition for\nthe ground state leads to a relative suppression of the\nordering temperature.\nFIG. 1: (a) Crystal structure of Mn 3Si2Te6viewed down [110].\n(b,c) Single layers of the di\u000berent MnTe 6octahedra viewed\ndown [001], where (b) is an image of Mn1 layers and (c) a\nMn2 layer. In all panels, one unit cell is outlined, MnTe 6\noctahedra are shown, and arrows indicate relative alignments\nof Mn moments within the trigonal plane.\nII. EXPERIMENTAL DETAILS\nMn3Si2Te6single crystals were obtained by melting a\nstoichiometric mixture of the elements in a vacuum sealed\nquartz ampoule at T=1125 K. High-purity Si lump (Alfa\nAesar 99.9999%), and Te shot (Alfa Aesar, 99.999%)\nwere combined with Mn granules (99.98% Alfa Aesar)\nthat were arc-melted prior to use. Single crystals of\nMn3Si2Te6were mechanically isolated from the as-grown\ningot.\nField-cooled magnetization measurements were per-\nformed in a Quantum Design Magnetic Property Mea-\nsurement System. AC magnetic susceptibility, electri-\ncal transport, and speci\fc heat capacity measurements\nwere performed in a Quantum Design Physical Property\nMeasurement System. Powder x-ray di\u000braction mea-\nsurements were performed using a PANalytical X'Pert\nPro MPD with a Cu K \u000b;1incident-beam monochroma-\ntor. An Oxford PheniX Cryostat was utilized to obtain\ndata between 20 K and 300 K. Additional powder x-ray\ndi\u000braction measurements were performed between 300\nand 400 K on a PANalytical X'Pert di\u000bractometer with\nCu K\u000bradiation and an XRK900 oven-type furnace. The\nfurnace chamber was purged with He for 12 hours prior\nto initiating the high-temperature measurements and gas\nwas continuously \rowed throughout the measurements.\nThe measurements were delayed at each temperature for\n30 minutes to allow the sample to equilibrate. At 300 K,\nRietveld re\fnement of data from the low Tstage yieldeda=7.0321(6) \u0017A andc=14.249(1) \u0017A, while re\fnement of the\ndata from the high Tstage yielded a=7.0343(2) \u0017A and\nc=14.249(2) \u0017A.\nSingle crystal neutron di\u000braction data were measured\nusing the HB3A di\u000bractometer at the High Flux Isotope\nReactor at Oak Ridge National Laboratory (ORNL). The\nneutron wavelength of 1.546 \u0017A from the bent perfect Si-\n220 monochromator was used for the data collections,12\nand 380 re\rections were measured at 4, 100, and 380 K.\nThe nuclear-only re\fnement at 380 K yielded occupan-\ncies equal to unity within the error bars, and thus the\noccupancies were \fxed at unity for all T. The magnetic\nre\fnement at 4 K required three equivalent magnetic do-\nmains. All re\fnements were performed using the pro-\ngram FullProf.13\nDi\u000buse neutron scattering measurements were per-\nformed on the CORELLI di\u000buse scattering spectrome-\nter at the Spallation Neutron Source located at ORNL.\nCORELLI is a time-of-\right spectrometer with a pseudo-\nstatistical chopper, which generates both an elastic scat-\ntering signal (with an average energy resolution \u00191 meV)\nand a total scattering signal from a single measurement.\nAn approximately 7 mg single crystal was attached to\nan Al plate in a manner to facilitate inspection of the\nHHL scattering plane. The measurements were taken\nat temperatures of 6, 120, and 350 K, where the sam-\nple was rotated in steps of 3\u000eover ranges of 120\u000e(6 K),\n120\u000e(120 K) and 90\u000e(350 K). A signi\fcant amount of\ntime was spent at 120 K measuring 30 degrees centered\naround the 002 Bragg re\rection. Data around 002 were\nalso collected on warming from 6 K to 120 K. Mantid was\nutilized to perform the Lorentz and spectrum corrections\nand merge the full volume of the scattering data.14\nFirst principles calculations were performed us-\ning the linearized augmented plane-wave (LAPW)\ncode WIEN2K,15within the generalized gradient\napproximation.16Spin-orbit coupling was only included\nduring the calculation of magnetic anisotropy within the\nobserved ground state. Sphere radii of 2.04, 2.5 and\n2.5 Bohr were chosen for Si, Mn and Te, respectively,\nwith an RK maxof 7.0 (here RK maxis the product of the\nsmallest sphere radius and the largest planewave expan-\nsion wavevector). All calculations used the experimen-\ntal, room-temperature lattice parameters of a=7.029 \u0017A\nandc=14.255 \u0017A taken from Ref. 11. For each mag-\nnetic con\fguration, the internal coordinates were relaxed\nwithin the magnetic ordering pattern until the forces\nwere less than 2 mRyd/Bohr. Optimization within the\nmagnetic state yields signi\fcantly di\u000berent atomic coor-\ndinates than optimization within the non-magnetic state\n(an energy gain of some 250 meV/Mn results), suggest-\ning a strong coupling of magnetism to the lattice. The\nstructure theoretically optimized within the ferrimag-\nnetic ground state is much closer to the experimental\nstructure than that optimized within the non-magnetic\nstate. For example, the value for Mn1 of \u0001 z \u0011zExp: -\nzCalc is -8.7\u000210\u00004for the ferrimagnetic ground state\nbut nearly 5 times as large in magnitude at 4.02 \u000210\u000033\nfor the non-magnetic state. Similarly large ratios apply\nfor the Si z and Te y and z coordinates, with the Te x co-\nordinate \u0001x roughly twice as large for the non-magnetic\nstate as for the ferrimagnetic state. Additional details\nare provided in the Supplemental Information.\nIII. RESULTS AND DISCUSSION\nA. Magnetization, neutron di\u000braction and\ntheoretical calculations\nThe magnetization Mdata for single crystalline\nMn3Si2Te6are shown in Fig. 2a. Mn 3Si2Te6is observed\nto have a Curie temperature of TC=78 K. The ordering\ntemperature was also con\frmed by AC magnetic suscep-\ntibility measurements, the results of which are shown in\nFig. 3. The anisotropy of MbelowTCsuggests the or-\ndered moments lie primarily within the ab-plane. This is\ndemonstrated by a signi\fcantly larger Mwhen the ap-\nplied \feldHlies within the ab-plane as compared to when\nHjjc; note the di\u000berent vertical axes in the main panel\nof Fig. 2a. Consistent with this, below TCthe magnetiza-\ntion saturates rapidly for Hjjaband reaches\u00191.6\u0016B/Mn\natT=5 K (inset Fig. 2a). The data imply an anisotropy\n\feld of approximately 9 T (90 kOe), and a small ferro-\nmagnetic component is observed for Hjjc. These data\nare consistent with the initial report on Mn 3Si2Te6.10\nWe note there is no remanant moment for either orien-\ntation, which is di\u000berent from the prior report and, to\nsome extent, suggests the present single crystals are of\nhigh quality.\nFig. 2b plots the magnetization data as H=M , which is\nequivalent to 1 =\u001fwhen the susceptibility can be de\fned\nas\u001f=M=H (whenMis linear inH). The data demon-\nstrate Curie-Weiss behavior between approximately 350\nand 750 K, the region where 1/ \u001fis linear in T. The\ndata above 400 K were \ft to a simple Curie-Weiss law,\n\u001f= C/(T\u0000\u0002), where C is the Curie constant and \u0002\nthe Weiss temperature. This \ftting produced an e\u000bec-\ntive moment of 5.6 \u0016B/Mn and a Weiss temperature of\n-277 K. The e\u000bective moment is consistent with the pres-\nence of Mn2+ions and the negative Weiss temperature\nindicates antiferromagnetic correlations. The saturation\nmagnetization (inset, Fig. 2a) is about one-third of that\nexpected on the basis of this e\u000bective moment, which sug-\ngests the presence of one uncompensated Mn2+local mo-\nment per formula unit in the ordered phase. In Fig. 2b,\nthe low-temperature data were collected on a large mass\nof ground crystals, while high-temperature data were col-\nlected using six small crystals sealed under vacuum in a\nthin quartz tube (the crystals are free to rotate in an\napplied \feld).\nThe AC susceptibility \u001fAC= dM/dHcan be described\nby in-phase (real) \u001f' and out-of-phase (imaginary) \u001f\"\ncomponents, which are shown in Fig. 3. The out-of-\nphase component \u001f\" relates to dissipative losses, for in-\nstance the movement of domain walls in ferromagnets;\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s48 /s50 /s48 /s52 /s48 /s54 /s48 \n/s48 /s49 /s48 /s50 /s48 /s51 /s48 /s52 /s48 /s53 /s48 /s54 /s48 /s48 /s46 /s48 /s48 /s46 /s53 /s49 /s46 /s48 /s49 /s46 /s53 \n/s48 /s49 /s48 /s48 /s50 /s48 /s48 /s51 /s48 /s48 /s52 /s48 /s48 /s53 /s48 /s48 /s54 /s48 /s48 /s55 /s48 /s48 /s56 /s48 /s48 /s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s40/s98 /s41/s32/s77 /s47/s72 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41/s72 /s32 /s61 /s32 /s50 /s48 /s32 /s79 /s101 /s40/s97 /s41\n/s48 /s46 /s48 /s48 /s46 /s50 /s48 /s46 /s52 /s48 /s46 /s54 \n/s32\n/s72 /s32 /s32 /s99 \n/s72 /s32 /s124/s124/s32 /s99 /s77 /s32 /s40\n/s66 /s47 /s77 /s110 /s41\n/s72 /s32 /s40/s107 /s79 /s101 /s41/s84 /s32 /s61 /s32 /s53 /s75 \n/s32\n/s72 /s32 /s61 /s32 /s49 /s32 /s107 /s79 /s101 \n/s72 /s32 /s61 /s32 /s49 /s48 /s32 /s107 /s79 /s101 /s72 /s47/s77 /s32/s40/s109 /s111 /s108/s45/s77 /s110 /s47/s99 /s109 /s32/s51 \n/s41\n/s84 /s32/s40/s75 /s41/s84 /s32 /s60 /s32 /s51 /s56 /s48 /s75 /s32 /s112 /s117 /s108 /s118 /s101 /s114/s105 /s122 /s101 /s100 \n/s99 /s114/s121 /s115/s116 /s97 /s108 /s115/s84 /s32 /s62 /s51 /s52 /s48 /s75 /s32 /s99 /s114/s121 /s115/s116 /s97 /s108 /s115\n/s114/s97 /s110 /s100 /s111 /s109 /s32 /s111 /s114/s105 /s101 /s110 /s116 /s97 /s116 /s105 /s111 /s110 FIG. 2: (a) Anisotropic magnetization data for\nMn3Si2Te6upon cooling in an applied \feld of 20 Oe;\nthe left axis ( H?c) and right axis ( Hkc) have the same\nunits. The inset shows isothermal magnetization data at\nT=5 K, and together these results reveal easy-plane mag-\nnetization. (b) Inverse susceptibility data combining high\nand lowTmeasurements. The dashed curve represents a\nCurie-Weiss \ft extended to lower T.\n\u001f\" is typically zero for simple antiferromagnets and non-\nzero in ferromagnets and sometimes in metals.17The AC\ndata shown in Fig. 3 were collected in zero applied DC\n\feld, with a small amplitude A=2 Oe and a frequency\nf=997 Hz. These data reveal a sharp onset at the Curie\ntemperature, with \u001f' and\u001f\" increasing below TCfor both\norientations. In addition, the data reveal anisotropy\naboveTC, which was also observed with the DC mea-\nsurements. A comparison of \u001f' and the DC analog M=H\n(H=20 Oe) is shown in the Supplemental Information.\nAs shown in Fig. 2b, the low-\feld magnetization data\non ground crystals deviate from the high TCurie-Weiss\nbehavior below\u0019330 K. Single crystal magnetization\ndata reported in Fig. 4 demonstrate that this behav-\nior is associated with a relatively abrupt onset of mag-\nnetic anisotropy below \u0019330 K, which is observed in\nboth the low-\feld DC data (Fig. 4a) and the AC data\n(Fig. 4b). The anisotropy is the same as that in the ferri-\nmagnetic phase below TC(easy-plane anisotropy). Upon4\ncooling below 330 K, \u001f\" becomes non-zero for in-plane\ndata, and it reaches a maximum near 110 K before ris-\ning sharply at TC. This may suggest the anisotropy is\nassociated with, or derived from, a ferromagnetic compo-\nnent (uncompensated moment) that is present for H?c.\nIndeed, isothermal magnetization data reveal a small\n(soft) ferromagnetic contribution for H?cin the re-\ngionTC<T < 330 K, which increases upon cooling. The\nlow-\feldM(H) data atT=110 K are shown in Fig. 5, and\nthe corresponding AC data are shown in the Supplemen-\ntal along with M(H) curves at di\u000berent T. Interestingly,\n\u001f' plateaus near 85-110 K, which would suggest that this\nferromagnetic component has saturated. That is, this\nplateau in\u001f' seems to suggest that the anisotropy does\nnot originate in precursory short-range order of the ferri-\nmagnetic phase, which would be expected to continue in-\ncreasing all the way down to TCas the correlation length\nbegins to diverge. However, the ferromagnetic-like con-\ntribution suggests the anisotropy originates from some\nform of magnetic order, as opposed to a crystal-\feld in-\nduced anisotropy of a paramagnet. As discussed below,\nour di\u000braction measurements did not reveal any signi\f-\ncant structural change near 330 K, but a structural con-\ntribution and/or origin cannot be ruled out at this time.\nThe true nature of the transition near 330 K is thus un-\nclear and warrants further investigation. As discussed in\nthe Supplemental Information, this behavior seems to be\nintrinsic to our single crystals.\nSingle crystal neutron di\u000braction and spectroscopy\nwere utilized to examine the nuclear and magnetic struc-\ntures of Mn 3Si2Te6. As shown in Fig. 6, the intensity of\nthe 002 Bragg re\rection increases upon cooling through\nTC. This suggests a strong in-plane component of the\nmoment, consistent with the magnetization data. The\nlack of additional scattering at the 110 re\rections at low\nTis consistent with the moment being in the ab-plane,\nbut this observation alone is not de\fnitive given the pres-\nence of three magnetic domains. The square root of\nthe integrated magnetic intensity is proportional to the\nordered moment, and can thus be used as a magnetic\norder parameter. Order parameter data employing in-\ntensity from the 002 re\rection were obtained using both\nHB3A and CORELLI, as shown in Fig. 6b where \fts to\npower law behavior are included. The data in Fig. 6b\nfrom CORELLI are the integrated intensity of 002, while\nthat from HB3A were constant Q data collections; nei-\nther data set are corrected for background. Additional\ntemperature-dependent data from HB3A are shown in\nthe Supplemental Information.\nThe order parameter \fts shown as solid curves in\nFig. 6b yield a critical exponent \f\u00190.25 andTCvalues\nconsistent with that obtained from magnetization mea-\nsurements, though some minor discrepancies are observed\nbecause the data were collected while ramping temper-\nature in di\u000berent conditions. The \fts employ a sim-\nple power law, ( I\u0000I0)0:5=A(TC-T)\f, whereI0repre-\nsents the non-magnetic intensity (background and nu-\nclear, assumed temperature independent). Data from\n/s48 /s46 /s48 /s49 /s46 /s48 /s50 /s46 /s48 /s51 /s46 /s48 /s52 /s46 /s48 /s53 /s46 /s48 /s54 /s46 /s48 \n/s56 /s48 /s49 /s48 /s48 /s49 /s50 /s48 /s49 /s52 /s48 /s49 /s48 /s45/s53 /s49 /s48 /s45/s52 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 \n/s55 /s48 /s56 /s48 /s57 /s48 /s49 /s48 /s48 /s49 /s49 /s48 /s49 /s50 /s48 /s49 /s51 /s48 /s49 /s52 /s48 /s49 /s53 /s48 /s48 /s46 /s48 /s48 /s48 /s46 /s48 /s49 /s48 /s46 /s48 /s50 /s48 /s46 /s48 /s51 /s48 /s46 /s48 /s52 /s56 /s48 /s49 /s48 /s48 /s49 /s50 /s48 /s49 /s52 /s48 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 /s49 /s48 /s49 /s72 \n/s68 /s67 /s61 /s48 \n/s40/s98 /s41/s32\n/s72 \n/s65 /s67 /s32/s99 \n/s72 \n/s65 /s67 /s32/s124/s124/s32 /s99 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41/s40/s97 /s41\n/s65 /s67 /s32/s109 /s97 /s103 /s110 /s101 /s116/s105/s99 /s32/s115 /s117 /s115 /s99 /s101 /s112 /s116/s105/s98 /s105/s108/s105/s116/s121 \n/s39/s39\n/s84 /s32/s40/s75 /s41/s32/s39 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41/s39\n/s84 /s32/s40/s75 /s41FIG. 3: (a) The real component \u001f' and (b) the imaginary\ncomponent \u001f\" of the AC magnetic susceptibility data near\nthe ferrimagnetic ordering temperature. The insets show the\ndata on log-scale using the same units as the primary panels.\nData collected upon cooling using A=2 Oe and f=997 Hz.\njust below TCdown to\u001940 K are \ft, and the result-\ning curves are extended to lower Tin Fig. 6b for com-\nparison. For the CORELLI data we obtain \f=0.26(1)\nandTC=81.7(2) K while for the HB3A data we obtain\n\f=0.23(1) and TC=78.5(1) K. The standard deviations\noriginate from the \ftting procedures, but the absolute\nerrors on critical exponents estimated in this way are\nexpected to be larger. The inset contains these same\nresults in log-log representation to emphasize the simi-\nlar temperature-dependence for the two data sets, and\nto highlight the good quality of the \fts. The criti-\ncal exponent of \f\u00190.25 for Mn 3Si2Te6can be com-\npared to those obtained for simple mean \feld mod-\nels. For instance, a 2D Ising system is expected to\nhave\f=0.125, while the 3D Ising and Heisenberg mod-\nels have\f=0.326 and 0.367, respectively.18In CrSiTe 3,\na critical exponent of \f=0.12 was reported, suggesting\n2D character of the magnetism in that van der Waals\nbonded ferromagnet.2,4Mn3Si2Te6is a crystallographi-\ncally three-dimensional material. The deviation of \ffrom\nthe simple 3D models is not surprising, however, given\nthe existence of multiple Mn sites and ferrimagnetic or-5\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s49 /s48 /s45/s52 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 \n/s53 /s46 /s48 /s54 /s46 /s48 /s55 /s46 /s48 /s56 /s46 /s48 \n/s51 /s48 /s48 /s51 /s49 /s48 /s51 /s50 /s48 /s51 /s51 /s48 /s51 /s52 /s48 /s51 /s53 /s48 /s45/s48 /s46 /s49 /s48 /s46 /s48 /s48 /s46 /s49 /s48 /s46 /s50 /s48 /s46 /s51 /s40/s97 /s41\n/s40/s98 /s41\n/s32/s77 \n/s68 /s67 /s32/s40/s101 /s109 /s117 /s47/s103 /s41\n/s84 /s32/s40/s75 /s41/s72 \n/s68 /s67 /s32 /s61 /s32 /s50 /s48 /s32 /s79 /s101 /s72 \n/s68 /s67 /s32 /s32 /s99 \n/s72 \n/s68 /s67 /s32 /s124/s124/s32 /s99 \n/s72 \n/s68 /s67 /s32 /s61 /s32 /s48 /s72 \n/s65 /s67 /s32 /s32 /s99 \n/s72 \n/s65 /s67 /s32 /s124/s124/s32 /s99 \n/s39 /s39 /s39 \n/s32/s32/s39 /s32/s97 /s110 /s100 /s32 /s39 /s39 /s32/s40/s49 /s48 /s45 /s51 \n/s32/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41\nFIG. 4: (a) Anisotropic DC magnetization data collected\nupon cooling in an applied \feld of 20 Oe. (b) Anisotropic\nAC susceptibility data showing the real \u001f' and imaginary \u001f\"\ncomponents. The AC data were collected using A=14 Oe and\nf=997 Hz with zero applied DC \feld.\n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s54 \n/s72 \n/s68 /s67 /s32 /s32/s99 \n/s72 \n/s68 /s67 /s32 /s32/s99 \n/s72 /s32/s40/s79 /s101 /s41\n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41/s84 /s32 /s61 /s32 /s49 /s49 /s48 /s75 \nFIG. 5: Isothermal DC magnetization data at 110 K demon-\nstrating the ferromagnetic contribution for H?cin\nMn3Si2Te6.\nder.\nRe\fnement of the 4 K neutron di\u000braction data from\n/s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51 /s49/s52 /s49/s53/s48/s49/s50/s51/s52/s53/s54\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s49/s50/s51\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s48/s48/s50/s72/s66/s51/s65/s48/s48/s50\n/s49/s49/s48/s84 /s32/s61/s32/s52/s32/s75\n/s49/s48/s48/s32/s75/s49/s49/s50\n/s49/s49/s50/s99/s111/s117/s110/s116/s115/s47/s116/s105/s109/s101/s32/s40/s49/s48/s51\n/s47/s51/s115/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41\n/s40/s98/s41/s32/s73/s32/s48/s46/s53\n/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s32\n/s32/s72/s66/s51/s65\n/s32/s67/s79/s82/s69/s76/s76/s73/s76/s111/s103/s91/s40 /s73 /s45/s73\n/s48/s41/s48/s46/s53\n/s93\n/s76/s111/s103/s91 /s84/s45/s84\n/s67/s93FIG. 6: (a) Rocking curves of various Bragg re\rections at 4\nand 100 K. The increase of intensity upon cooling into the\nmagnetically-ordered state is strongest for re\rections with\nnon-zero L. (b) Order parameter for the ferrimagnetic struc-\nture (002 intensity), with data from the single crystal neutron\ndi\u000bractometer (HB3A) and the elastic di\u000buse neutron scatter-\ning spectrometer (CORELLI). The solid curves are \fts to a\nsimple power law, and the inset shows the data and \fts in a\nlog-log plot.\nHB3A yielded the magnetic structure shown in Fig. 1\n(Rf= 4:6 and\u001f2= 3:6). This spin structure is de-\n\fned by parallel alignment of the moments on Mn1 com-\nbined with anti-parallel alignment to the moments on\nMn2. Within the limits of these data, the re\fned mo-\nments are the same on both Mn sites, consistent with\nthe expectation of similar oxidation states (both posi-\ntions are octahedrally-coordinated by Te). At 4 K, the\nre\fned moment is 4.2(1) \u0016B/Mn when the moments are\nconstrained to be equal and to lie within the trigonal\nplane. When permitted, a small moment of 0.2(2) \u0016B/Mn\nalong thecaxis is observed, but the quality of the re\fne-\nment does not improve. However, the isothermal mag-\nnetization data M(H) at 5 K suggest a soft ferromag-\nnetic component for Hjjc(inset Fig. 2a), and the low-\n\feldM(T) data are anisotropic below TC(Fig. 2a). It is\npossible that an applied \feld causes alignment of canted\nmoments or domains, and the lack of an applied \feld dur-\ning neutron di\u000braction leads to an average canting that is6\nnegligible. Note that in Fig. 1, the moments are shown to\nbe along the aaxis, but their speci\fc orientation within\ntheab-plane cannot be determined due to symmetry.\nIn trying to understand this magnetic con\fguration,\nwe \frst consider the crystal structure and apparent ex-\nchange interactions. Mn1-Mn2 have an interatomic dis-\ntance of 3.55 \u0017A at room temperature,11and are linked\nthrough face-sharing octahedra along the caxis (Fig.1),\nwhich would lead to a direct exchange interaction that\nwe expect to be antiferromagnetic (AFM) for these Mn2+\nions. The Mn1-Mn1 interaction is more complicated be-\ncause it occurs via edge sharing octahedra, which re-\nsults in a competition of direct interaction (AFM) and\n\u001990\u000eMn1-Te-Mn1 interactions that can be either ferro-\nmagnetic (FM) or AFM depending on the panddor-\nbitals involved.19Since the ground state is ferrimagnetic\nwith parallel alignment of Mn1 moments, it is tempt-\ning to suggest that FM superexchange is the dominant\nin-plane interaction. Such an explanation would seem\nconsistent with the observation of ferromagnetism in the\nstructurally-related material CrSiTe 3, in which the anal-\nogous edge-sharing octahedra have ferromagnetic cou-\npling and orovide the shortest exchange pathway (there is\nno inter-plane occupancy).4Theoretical calculations were\nperformed to get a better understanding of the coupling\nmechanisms.\nFirst principles calculations were performed to under-\nstand the ferrimagnetic ground state and the associated\nTC. Energies of the following six states were calculated:\nnon-magnetic, ferromagnetic (FM), ferrimagnetic states\nFI1 and FI2, and antiferromagnetic states AF1 an AF2.\nFI1 is the ferrimagnetic ground state, as also observed\nexperimentally, and its magnetic symmetry is shown in\nFig. 1. The \frst competing state is AF1, in which all\nMn1-Mn1 planar neighbors are anti-aligned, while all\nMn1-Mn2 nearest neighbors are aligned. Additional de-\ntails regarding the various magnetic con\fgurations are in\nthe Supplemental Information. The energies associated\nwith the magnetic states studied fall more than an eV per\nMn below the non-magnetic state, and the Mn moments\ncan accurately be considered local moments.\nTABLE I: The calculated relative energies for several mag-\nnetic ordering patterns, along with net magnetic moments\nreported as \u0016B/Mn. In all cases, each Mn is calculated to\nhave a moment consistent with Mn2+.\nOrdering \u0001E (meV/Mn) M (\u0016B/Mn)\nFI1 (ground state) 0 1.67\nAF1 19.1 0\nFI2 31.5 1.67\nAF2 32.2 0\nFM 105.4 4.82\nThe calculations predict that the ferrimagnetic con-\n\fguration is the ground state, being substantially lower\nin energy than the paramagnetic or ferromagnetic states\n(105 meV/Mn below the ferromagnetic state). As sum-marized in Table I, AF1 is only 21 meV/Mn above the\nground state, while AF2 and FI2 are slightly more than\n30 meV/Mn above FI1. The energy of the ground state\nrelative to competing states determines TC, which is to\nsay that the existence of competing magnetic con\fgura-\ntions acts to suppress TC. Using a mean \feld approach,\nthe Curie temperature is one-third of the energy di\u000ber-\nence between the ground state and the competing state,\nwhich in this case gives 19.1 meV/3k B= 73.9 K, in good\nagreement with the experimentally observed TC=78 K.\nUltimately, the similar energies of the competing states\nreveal a competition between di\u000berent coupling mecha-\nnisms and/or pathways.\nThe theoretical analysis was extended by mapping the\nobserved energetics to a Heisenberg model assuming an\nIsing spin S=\u00061, which yielded the \frst three e\u000bective\nexchange constants. The de\fnitions and relative values\nof these J iare given in Table II. Compared to the third-\nnearest neighbor at 5.42 \u0017A, the other Mn-Mn distances\nare\u00197\u0017A or greater and have been neglected. The mean\n\feld approach to a Weiss temperature is one-third of the\naverage interaction energy per magnetic ion, which in\nthis case is \u0002\u0019-757 K/3 = -252 K (accounting for the\nmultiplicities), which is in good agreement with the ex-\nperimental value of -277 K. To give a sense for the energy\nscales, the calculations yielded J 1=-35(4) meV/Mn.\nThe calculations revealed that the \frst three exchange\nconstants are all antiferromagnetic in nature, which re-\nsults in a frustrated system of competing interactions.\nAs shown in Table II, J 1provides the strongest exchange,\nand its AFM nature is consistent with our expectations\nbased on the direct exchange pathway. Interestingly, the\nsecond-nearest neighbor interaction (J 2) is AFM and is\nweaker than the third-nearest neighbor interaction (J 3).\nThus, the longer-range J 3dominates the in-plane J 2and\ntogether with the strong J 1produces FM alignment of the\nMn1 moments within the ab-plane despite the energy cost\ndue to an AFM J 2. The occupancy of the Mn2 position is\ntherefore essential for generating FM alignment of Mn1\nmoments in the ground state of Mn 3Si2Te6. Thus, this\nseems to contrast the behavior observed in ferromagnetic\nCrSiTe 3where the hypothetical Cr2 position is vacant\nbut the in-plane coupling appears to be ferromagnetic.4\nThe multiplicity of each exchange interaction is criti-\ncal in determining the order of competing ground states.\nFor instance, in ferromagnetic MnBi the nearest-neighbor\nexchanges are AFM in nature, but a larger number of\nlonger-range, ferromagnetic interactions results in a high\nCurie temperature of 630 K.20In Mn 3Si2Te6, the multi-\nplicity of J 2and J 3are equal despite a di\u000berent number\nof pertinent neighbors (due to crystallographic site mul-\ntiplicities), and are a factor of three larger than that of\nJ1(see Table II). The energy minimization in the ground\nstate comes from satisfying both J 1and J 3. However, due\nto the multiplicity di\u000berence, the \frst competing state\n(AF1) is realized by satisfying both J 3and J 2. Thus, due\nto the Mn-Mn coordinations, several competing states\nexist in Mn 3Si2Te6.7\nSince under some circumstances strong correlations\ncan be present in 3 d-based chalcogenides, we have also\nperformed GGA+U calculations of the ferrimagnetic\nground state and the ferromagnetic state, applying a\nHubbard U of 3 eV to the Mn dorbitals. There are two\nmain e\u000bects of adding this U: the ferrimagnetic band\ngap is increased slightly, and the energetics of the sys-\ntem change signi\fcantly. In particular, the ferromagnetic\nstate is now just 43 meV/Mn above the ferrimagnetic\nstate, to be compared with 105 meV/Mn in the GGA-\nonly calculations. It is likely that similar e\u000bects would\nbe obtained for the other excited states. Since the GGA\nitself provides excellent agreement with both the mag-\nnitude of the Curie temperature and the Curie-Weiss \u0002\nobtained from the exchange constants, it appears that the\nregular GGA is indeed an appropriate tool for studying\nthis system.\nSince Mn 3Si2Te6shows a surprisingly large anisotropy\n\feld - approximately 9 T - we have also performed \frst\nprinciples calculations21{23of the magnetic anisotropy in\nthe ferrimagnetic state in this system, which necessitates\nthe use of the spin-orbit interaction. Here the moments\nare constrained to lie either along the caxis, or within the\nabplane. We \fnd that the energy is some 0.65 meV/Mn\nlower in the planar con\fguration, leading to a calcu-\nlated anisotropy \feld HAof 13 T, in reasonable agree-\nment with the experimental observations. It is interest-\ning to note that this anisotropy appears to arise from an\nanisotropy in the Mn orbital moments, which are of mag-\nnitude 0.023 \u0016B/Mn in the axial con\fguration (all orbital\nmoments are parallel to the corresponding spin moments\nand hence are of opposite sign for Mn1 and Mn2), but\nare found to be 0.037 \u0016B/Mn1 and -0.048 \u0016B/Mn2 in the\nplanar con\fguration. The existence of an orbital moment\nimplies a deviation from the 3 d5, S=5/2 electronic con-\n\fguration, though it appears that only a small orbital\ncontribution exists.\nIt is interesting to compare the magnetic exchange\nconstants in Mn 3Si2Te6to those in MnTe. MnTe has\nthe hexagonal NiAs structure-type in its bulk form, and\nit contains the same exchange pathways as de\fned in\nTable II, but with slightly di\u000berent bond angles, in-\nteratomic distances, and multiplicities.24MnTe has an\nantiferromagnetic structure composed of ferromagnetic\nplanes that stack antiferromagnetially along the caxis.25\nThe planar J 2is ferromagnetic in MnTe,24and thus\nthe interactions in MnTe are not frustrated. There-\nfore, while the magnetic structures of Mn 3Si2Te6and\nMnTe are similar (ferromagnetic planes stacked antifer-\nromagnetically), the origins are rather di\u000berent. In both\ncompounds, J 2is associated with edge-sharing octahe-\ndra (see Table II). The octahedra are nearly ideal in\nMnTe (bond angles of 90.0 \u00060.1\u000e), while they are rather\ndistorted in Mn 3Si2Te6(for instance, one bond angle\ndrops to 85.1\u000e). In relation to J 2, the distortions in\nMn3Si2Te6lead to a signi\fcant deviation from the ideal\nangle of 90\u000e(87.2\u000e) associated with Mn-Te-Mn superex-\nchange, as well as a shorter interatomic distance for thecoupling (4.06 \u0017A compared to 4.13 \u0017A in MnTe). This\ncomparison further highlights the importance of frus-\ntrated interactions in Mn 3Si2Te6, which produce com-\npeting magnetic ground states. As discussed above, the\nenergy di\u000berence between the lowest energy (magnetic)\ncon\fgurations determines the magnetic ordering tem-\nperature, and thus the existence of competing interac-\ntions/states leads to a suppressed Curie temperature in\nferrimagnetic Mn 3Si2Te6. However, since the dominant\nexchange interactions in MnTe are not frustrated, they\ndo not result in a competing ground state that lies close\nin energy. The bulk ordering temperature of MnTe is\nthus rather large, TN=310 K.25\nTABLE II: Schematic de\fning the three nearest neighbor\nexchange constants. All are found to be antiferromagnetic\n(J<0), and the values shown were obtained for an Ising spin\nof\u00061 in the Hamiltonian. The relevant multiplicities and\ndistances in the relaxed cell are provided; see Fig. 1 for more\ncrystallographic details. The interaction multiplicity takes\ninto account the number of neighbors at a given site and the\nassociated crystallographic multiplicity (four Mn1 and two\nMn2 per unit cell).\nExchange Multiplicity Distance\n(K/Mn) (per unit cell) ( \u0017A)\nJ1= -402(50) 4 3.54\nJ2= -73(9) 12 4.06\nJ3= -172(22) 12 5.42\nB. Lattice behavior and electrical resistivity\nAs shown in Fig. 7, the evolution of the aandclattice\nparameters appears smooth across TCand near 330 K.\nThe thermal expansion along the aaxis is fairly typical\nfor allT. Thecparameter, however, shows a change in\ntemperature dependence across TCand an unusual curva-\nture up to near room temperature. This seems to bolster\nthe idea that an ordering transition occurs near 330 K\nand has some coupling to the lattice, though additional\nmeasurements would be useful. The neutron di\u000braction\ndata at 100 and 380 K re\fned well within the published\ncrystal structure.\nThe speci\fc heat capacity of Mn 3Si2Te6is shown in\nFig. 8. An anomaly can be observed near TC, and as\nshown in the inset an applied \feld ( Hjjc) causes a broad-\nening of this feature to higher temperature. This behav-\nior is consistent with ferromagnetic-like ordering, where\nan applied \feld suppresses \ructuations and broadens the\nspeci\fc heat anomaly by removing more of the magnetic\nentropy above TC. While we do not have a good lattice\n(phonon) standard for Mn 3Si2Te6, an overestimation of\nthe entropy change that occurs near TCcan be made by\nintegrating CP=Twith a straight line for a background\n(see Supplemental Information). When \ftting between\n60 and 95 K, this procedure yields an entropy change of8\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s46/s57/s57/s48/s48/s46/s57/s57/s50/s48/s46/s57/s57/s52/s48/s46/s57/s57/s54/s48/s46/s57/s57/s56/s49/s46/s48/s48/s48/s49/s46/s48/s48/s50\n/s97\n/s32/s84 /s32/s40/s75/s41/s99/s97 /s47/s97\n/s51/s48/s48/s75/s32/s38/s32 /s99 /s47/s99\n/s51/s48/s48/s75\nFIG. 7: Temperature dependence of the lattice parameters\nnormalized to values at 300 K, with data for co\u000bset for clarity.\nError bars are smaller than the size of the data markers, and\nthe dashed line indicates TC.\nonly 8% of that expected for ordering of S=5/2 local\nmoments on Mn2+. The magnetic entropy per mole of\nMn in the paramagnetic state is Smag= RLn[2S+1] =\nR Ln(6) = 14.9 J/mol-Mn/K, and the integration from\n60 to 95 K only yields \u0001 Smag\u00191.2 J/mol-Mn/K. Again,\nwe emphasize that this is likely to be an overestimation\nof \u0001Smagdue to the use of a linear background; this\napproach is utilized to emphasize how much entropy is\nreleased away from TC(both above and below). A sim-\nilarly small \u0001 Smag is observed in MnPS 3, where two-\ndimensional order is observed above TC,5,26as well as\nin CrSiTe 3where short range correlations are observed\naboveTCand the magnetism is strongly coupled to the\nlattice.3,4\nAs shown in Fig. 8b, there is a strong contribution to\nthe speci\fc heat capacity at low temperature, and this\ndecreases with increasing applied \feld. At the lowest\ntemperatures, the data can be well-described by a Debye\nterm (\fT3) plus a linear term ( \rT), the latter of which is\nthe typical way to include an electronic contribution. In\nthis case, however, Mn 3Si2Te6is a semiconductor and the\nstrong magnetic-\feld dependence also points to a mag-\nnetic origin; \rshould not be associated with the typical\nelectronic Sommerfeld coe\u000ecient. As shown in the Sup-\nplemental Information, the low Tmagnetic contribution\nin Mn 3Si2Te6(and CrSiTe 3) can also be described using\na term proportional to T1:5. Data below 2 K would likely\nhelp in considering the nature of the low Tmagnetic con-\ntribution in Mn 3Si2Te6.\nThe in-plane electrical resistivity \u001aabof Mn 3Si2Te6is\nshown in Fig. 9 for two crystals. The value at 300 K is\nconsistent with that reported by Ref. 10, and similar to\ntheir data we observe activated-conduction and a strong\n/s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s49/s46/s56/s50/s46/s48/s50/s46/s50/s50/s46/s52\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s50 /s52 /s54 /s56 /s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s40/s98/s41/s32\n/s48/s72 /s61/s48/s84\n/s52/s84\n/s56/s84/s67\n/s80/s32/s40/s82/s47/s109/s111/s108/s45/s97/s116/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s84 /s32/s40/s75/s41\n/s32/s67\n/s80/s32/s40/s82/s47/s109/s111/s108/s45/s97/s116/s41\n/s56/s84/s52/s84\n/s32/s32/s67\n/s80/s47/s84 /s32/s40/s109/s74/s47/s109/s111/s108/s45/s77/s110/s47/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41/s48/s72 /s61/s48FIG. 8: (a) Speci\fc heat capacity of Mn 3Si2Te6, with inset\nshowing the \feld dependence of the anomaly observed near\nTC. (b) Field dependence of the the low-temperature speci\fc\nheat data demonstrating a strong magnetic contribution.\nanomaly at the ferrimagnetic ordering temperature. An\nincrease in the carrier mobility is expected below TCdue\nto the suppression of spin \ructuations, and this may ex-\nplain the sharp dip in \u001aabbelowTC. The feature at TCis\nsomewhat dramatic, though, especially when considering\nthe implications of a small \u0001 SmagnearTC. The present\ndata, which extends to higher temperatures than pre-\nviously reported, also shows a broad feature near 330 K.\nThis appears to demonstrate the bulk-nature of magnetic\nordering associated with the susceptibility anomaly ob-\nserved at the same temperature in Fig. 2, and the asso-\nciated behavior in Fig. 9 could be related to carrier scat-\ntering or electronic structure e\u000bects.\nTo inspect the changes in \u001a(T) above and below\nthe magnetic transitions, we \ft the data to \u001a=\n\u001a0Exp[EA=kBT] whereEAandkBare the activation en-\nergy and Boltzmann constant, respectively. The \ftted\ncurves are shown in red in Fig. 9. Firstly, we note that\nthese \fts su\u000ber from inadequacy of the simple model\n(particularly at the lowest T), as well as the limited\namount of data utilized. However, the trends and magni-\ntudes are worth noting. The activation energy obtained\nby \ftting\u001afrom 350-380 K is roughly 0.3 eV; this is the9\n/s48 /s49 /s48 /s48 /s50 /s48 /s48 /s51 /s48 /s48 /s52 /s48 /s48 /s49 /s69 /s45/s51 /s48 /s46 /s48 /s49 /s48 /s46 /s49 /s49 /s49 /s48 \n/s32/s32/s97 /s98 /s32/s40 /s45/s109 /s41\n/s84 /s32/s40/s75 /s41\nFIG. 9: The in-plane electrical resistivity of\nMn3Si2Te6generally increases with decreasing tempera-\nture, with a notable feature near TC. The decrease of \u001a\nbecomes more rapid above \u0019330 K. The black and green\ncurves are data collected on di\u000berent crystals, and the red\nlines are \ftted curves using \u001a=\u001a0Exp[EA/kBT], which only\n\fts the data over small temperature ranges but does reveal\nan apparent increase in activation energy upon warming\nacross the two magnetic anomalies.\nregion where all of the material appears to be in a para-\nmagnetic state. Below the magnetic anomaly observed\nat 330 K, the activation energy decreases substantially,\nEA=0.04 eV for 160-300 K. Below TC, the simple Arrhe-\nnius behavior is not a good model for the data. For\ncomparison sake, a small temperature range allows for a\nreasonable \ft to be obtained, with EA=0.007 eV for 25-\n40 K. Interestingly, these results show the decrease of the\nactivation energy with increasing magnetic order, and if\nEais related to an energy gap then this is rather unusual\nbehavior. Note that the reduction of the activation en-\nergy at low Tcould be due to the activation of low-level\ndefects.\nC. Di\u000buse neutron scattering\nMotivated by the bulk measurements that revealed\nanomalous behavior below 330 K, neutron scattering was\nperformed on CORELLI to search for di\u000buse scattering.\nThese measurements revealed di\u000buse scattering near the\n002 re\rection at 120 K (Fig. 10), and di\u000buse scattering\nalong L in the (H-HL) plane at 6 K (Fig. 11). The scatter-\ning data were binned into 3D bins with three orthogonal\naxes along [H H 0], [H -H 0] and [0 0 L] directions, respec-\ntively. The bin sizes are 1%, 2% and 5% of the reciprocal\nvectors [1 1 0], [1 -1 0] and [0 0 1], respectively.\nFigure 10a is a 2D map of the elastic neutron scat-\ntering intensity for Mn 3Si2Te6at 120 K, showing di\u000busescattering around the 002 re\rection. The 2D map shows\nthe integrated intensity from [-0.035 -0.035 0] to [0.035\n0.035 0], i.e., 7 bins, along the [H H 0] out-of-plane direc-\ntion. The strong scattering intensity reaching out from\n000 in Fig. 10 was strongly angle dependent and did not\nappear consistent with the usual instrument background,\nbut it was essentially independent of temperature. We\nthus speculate that this feature may have been caused\nby the glue utilized to orient the small crystal.\nThe di\u000buse scattering is emphasized in Fig. 10c, which\nis a map of the intensity at 120 K with the intensity at\n350 K subtracted. By contrast to the 002 re\rection, the\nstrong nuclear-only 004 re\rection is sharp and is not sur-\nrounded by di\u000buse scattering. This di\u000buse cloud around\n002 is not observed at 6 or 350 K, suggesting it is from\nshort-range magnetic scattering. It is interesting to spec-\nulate whether this di\u000buse scattering may be related to the\nweak ferromagnetic component of the magnetization seen\nbelow 330 K. This hypothesis could be tested with addi-\ntional temperature-dependent INS measurements. The\nabsence of this di\u000buse scattering at 6 K suggests that it\ncoalesces into the ferrimagnetic order. This is consistent\nwith the observation of di\u000buse scattering around the 002\nre\rection, which has a strong magnetic contribution in\nthe ferrimagnetic phase. The strong contribution from\nthe long-range magnetic order to the intensity of the 002\nre\rection would also hinder detection of di\u000buse scattering\nat 6 K.\nIn Fig. 10b, line scans along L are shown around the\n002 and 004 re\rections using di\u000berent integration areas\ndH (resolution is inversely proportional to dH). The data\nfor 004 demonstrate the behavior expected for a nuclear-\nonly Bragg peak; the 004 re\rection has a sharp peak with\nlittle dependence on the temperature or integration area.\nBy contrast, the nuclear component of the 002 re\rection\nis very weak (see Fig. 10d) and the data at 120 K demon-\nstrate a very broad region of increased intensity centered\naround 002. The sharpening of the intensity at 002 with\ndecreasing dH (higher resolution) suggests the presence\nof a small, sharper Bragg re\rection underneath a di\u000buse\ncloud of scattering.\nThe di\u000buse scattering in Fig. 10 demonstrate the ex-\nistence of short-range spin correlations well-above TCin\nMn3Si2Te6, though the nature of these correlations is\nunclear. The scattering could be from short range order,\npotentially relating to the magnetization anomaly near\n330 K. Alternatively, the di\u000buse scattering could be asso-\nciated with excited states that persist above TCand cause\ninelastic scattering that cannot be entirely excluded due\nto the nature of the experiment. Above TC, the param-\nagnons could be broad, un-gapped and without a de\fned\ndispersion, which would promote isotropic, di\u000buse scat-\ntering in these nearly-elastic measurements. Both short-\nrange order and excited states will exist above a second\norder magnetic transition, but in some cases these short-\nrange correlations persist to much higher temperatures\nthan is typically observed (alternatively, their correla-\ntions lengths are relatively longer above TC).10\nElastic scattering 120K \nTotal Scattering 120K Total Scattering 350K (a) \n(b) (d) (c) (120K minus 350K) Total Scattering, difference map \nFIG. 10: Data from the neutron elastic di\u000buse scattering spec-\ntrometer CORELLI. (a) Estimate of the elastic scattering at\n120 K showing di\u000buse scattering near the 002 Bragg re\rec-\ntions. Line scans of the total scattering at (b) 120 K and (d)\n350 K across 002 and 004 re\rections with di\u000berent integration\nwidths along H indicated in the legend. (c) Di\u000berence map of\ntotal scattering (120 K with 350 K baseline) emphasizing the\ndi\u000buse scattering near the 002 re\rection.\nThe existence of short-range correlations above TCis\ngenerally enhanced when TCis suppressed relative to the\nstrongest exchange interaction. Geometric frustration,\ncompeting interactions, or strong anisotropy in the ex-\nchange constants can all lead to such behavior. In these\ncases, strong magnetic exchange interactions can pro-\nmote short-range correlations (of the ground or excited\nstates) well-above TCbecause thermal energy has less in-\n\ruence near TC. As noted in Ref. 27, the relevant ratio\nto consider when thinking about short-range correlations\nnearTCis notT=T Cbut ratherkBT/JS. In many quasi-\n2D systems, anisotropic interactions lead to a relative\nsuppression of kBTC, and a strong JS can enhance cor-\nrelation lengths above TC. For Mn 3Si2Te6, competing\nantiferromagnetic interactions appear to provide the un-\nderlying mechanism that ultimately promotes the persis-\ntent, short-range correlations above TC. A counter ex-\nample is MnF 2, where the thermal energy is large relative\nto the exchange energy at the antiferromagnetic ordering\nTN,27and as a result the majority of the magnetic en-\ntropy in MnF 2is released below TN\u001967 K.27,28\nAt 6 K, di\u000buse scattering was detected along L in the\n(H-HL) plane, as shown in Fig. 11. The anisotropy of\nthis low-temperature di\u000buse scattering suggests it is as-\nsociated with a short correlation length along the caxis.\nThe 6 K di\u000buse scattering is centered around L=2, and\n(a) \n(b) 6K \n350K FIG. 11: Total neutron scattering in the (H-HL) plane at\n(a) 6 K and (b) 350 K. Di\u000buse scattering is observed along L,\nnotably at 6 K near 1-12. These 2 D maps show the intensity\nfrom [-0.005 -0.005, 0] to [0.005, 0.005 0], i.e., a single bin,\nalong the [H H 0] out-of-plane direction.\nis only observed around particular re\rections (H-K =\n3n+2) that are only contributed to by Mn scattering\n(both sites contribute). This enhances the relative contri-\nbution of the magnetic intensity in these re\rections and\nfacilitates detection of this di\u000buse scattering. Despite ap-\npearing to have a magnetic origin, there is perhaps a mi-\nnor indication of this di\u000buse scattering at 350 K (though\nthis is within the noise of the background scattering).\nTherefore, this feature may be associated with defects\non the Mn positions. From a magnetic perspective, the\ndi\u000buse scattering seems to suggest that there is a short\ncorrelation length of spin canting along the caxis. While\nthis seems plausible and consistent with our other mea-\nsurements, the current data cannot determine the origin\nof this di\u000buse scattering. An additional scattering ex-\nperiment with an applied \feld would be the next step in\ndetermining the origin of this anisotropic di\u000buse scatter-\ning. We note that the tails extending toward L = 0 (the\nasymmetry of 004 for instance) are an artifact of the time\nof \right measurement technique and are not observed in\nthe elastic data (see Supplemental Information).\nIV. SUMMARY\nThis work has characterized the magnetism of\nMn3Si2Te6and found it to be a ferrimagnet due to anti-\nparallel alignment of moments on di\u000berent Mn atomic po-11\nsitions. If the potential canting of the moments along the\ncaxis is ignored, then the magnetic structure is fairly sim-\nple. By contrast, the mechanisms that lead to this ground\nstate are relatively complex. A competition between an-\ntiferromagnetic exchange interactions exists, and it is\na third-nearest neighbor interaction that ultimately de-\ntermines the ground state. This frustration suppresses\nTCand likely enhances the importance of \ructuations,\nleading to persistent short-range correlations above TC.\nMagnetization measurements also suggest that some ad-\nditional ordering may exist below \u0019330 K. Di\u000buse scat-\ntering was observed at 6 K that may indicate short-range\ncorrelations of spin canting along the caxis. Due to the\ncompeting antiferromagnetic interactions, the manipula-\ntion of bond distances or the targeted substitution of theMn2 atoms would likely yield new ground states with\npotentially higher ordering temperatures.\nV. ACKNOWLEDGMENTS\nThis work was supported by the U. S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences, Materi-\nals Sciences and Engineering Division. Work at ORNLs\nHigh Flux Isotope reactor and Spallation Neutron Source\nwere supported by the Scienti\fc User Facilities Division,\nO\u000ece of Basic Energy Sciences, U.S. Department of En-\nergy (DOE).\n\u0003Electronic address: mayaf@ornl.gov\n1A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013).\n2V. Carteaux, F. Moussa, and M. Spiesser, EPL (Euro-\nphysics Letters) 29, 251 (1995).\n3L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M.-W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, et al., APL Materials 3, 041515 (2015),\nURL http://scitation.aip.org/content/aip/journal/\naplmater/3/4/10.1063/1.4914134 .\n4T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler,\nM. B. Stone, J.-Q. Yan, and D. Mandrus, Phys. Rev. B 92,\n144404 (2015), URL http://link.aps.org/doi/10.1103/\nPhysRevB.92.144404 .\n5P. A. Joy and S. Vasudevan, Phys. Rev. B 46,\n5425 (1992), URL http://link.aps.org/doi/10.1103/\nPhysRevB.46.5425 .\n6A. Wildes, S. Kennedy, and T. 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Hen-\nnion, Phys. Rev. B 73, 104403 (2006).\n25K. Adachi, J. Phys. Soc. Jpn. 16, 2187 (1961).\n26Y. Takano, N. Arai, A. Arai, Y. Takahashi, K. Takase, and\nK. Sekizawa, J. Magn. Magn. Mater. 272, E593 (2004).\n27L. J. de Jongh and A. R. Miedema, Adv. Phys. 23, 1\n(1974).\n28W. O. J. Boo and J. W. Stout, J. Chem. Phys. 65, 3929\n(1976).12\nSupplementary Information\nThis Supplemental Materials contains additional infor-\nmation regarding magnetization measurements and \frst\nprinciples calculations, as well as neutron di\u000braction and\nscattering data for Mn 3Si2Te6. It also includes addi-\ntional details for the analysis of speci\fc heat capacity\ndata, and a comparison of the low-temperature speci\fc\nheat for Mn 3Si2Te6, CrSiTe 3and CrI 3.\nMagnetization measurements\nThe bulk ordering temperature of TC=78 K was char-\nacterized using AC and DC magnetization measure-\nments. These measurements also revealed that the mag-\nnetization is anisotropic below 330 K, which is well-above\nthe bulk ordering temperature. The anisotropy of the\nmagnetization in this intermediate temperature region is\nconsistent with that in the ordered region - the moment\nis largest for \felds applied perpendicular to the c-axis.\nThis is emphasized in Fig. S1, where isothermal magne-\ntization data are shown for a variety of temperatures. A\nsmall, ferromagnetic-like contribution for H ?ccan be\nseen above TC=78 K. This is best observed at 110 and\n150 K, though anisotropy is easily observed in the M(H)\ndata at higher T.\nA small ferromagnetic component can sometimes\nbe attributed to an impurity, but the observed\nanisotropy strongly suggests that this is intrinsic to our\nMn3Si2Te6single crystals. Also, there was a change\nin the in-plane resistivity at approximately this same\ntemperature. None of the known binaries in the Mn-\nSi and Mn-Te phase diagrams order ferromagnetically\nas bulk materials and no other Mn-Si-Te ternary com-\npounds are known. Furthermore, we do not detect ob-\nvious signs of these binary impurities in powder x-ray\ndi\u000braction or the 3D full volume single crystal scattering\ndata from CORELLI. We note that the previous work on\nMn3Si2Te6did not report data above 300 K, nor did they\nreport anisotropic susceptibility measurements, and the\nCurie Weiss \ftting was performed using data collected in\na large applied \feld.1It is worth mentioning that some of\nthe potential impurities, such as MnSi 1:7,2have complex\nmagnetism that can be tuned to ferromagnetism using\n\fnite size e\u000bects.\nThe AC magnetization measurement naturally probes\nthe ferromagnetic-like contribution observed at low \felds.\nTo demonstrate this, data at T=110 K are examined in\nmore detail in Fig. S2. The (soft) ferromagnetic con-\ntribution for H?cis manifested as a larger \u001f' than\nforHjjc(especially at HDC=0). When the DC \feld\nis increased from zero, the ferromagnetic component is\nrapidly quenched (or saturated), and thus \u001f' decreases\nwith increasing applied \feld for H?c. Similarly, the\nout-of-phase component of the AC susceptibility ( \u001f\") de-\ncreases with applied \feld as the ferromagnetic portion is\nquenched (likely due to the loss of domain movement). In\n/s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s46 /s50 /s48 /s46 /s52 /s48 /s46 /s54 \n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s49 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s49 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s51 /s84 /s32 /s61 /s32 /s49 /s49 /s48 /s32 /s75 /s72 /s32 /s32 /s99 \n/s72 /s32 /s32 /s124/s124/s32 /s99 \n/s32/s84 /s32 /s61 /s32 /s55 /s48 /s32 /s75 \n/s72 \n/s68 /s67 /s32/s40/s79 /s101 /s41\n/s32\n/s84 /s32 /s61 /s32 /s50 /s48 /s48 /s32 /s75 \n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41\n/s84 /s32 /s61 /s32 /s49 /s53 /s48 /s32 /s75 FIG. S1: Isothermal magnetization curves showing the\nanisotropy of the induced moment as a function of applied\n\feld, with these plots highlighting the data at small applied\n\felds.\nthe other orientation, Hjjc,\u001f' is essentially independent\nof applied \feld because MDCis very linear with Hfor this\norientation (the ferromagnetic contribution is negligible\naboveTC=78 K forHjjc, see Fig. S1). The anisotropy\nof\u001f' and\u001f\" are greatly reduced by H=1000 Oe.\nThe data in Figures S1 and S2 thus reveal a ferromag-\nnetic contribution that is related to the anisotropy in the\nmagnetization below 330 K. This is consistent with the\nanisotropy observed in the magnetization measured dur-\ning DC measurements, which is greatest at small applied\n\felds.\nThe small ferromagnetic contribution leads to a dis-\ncrepancy between \u001fACand the DC analog MDC=HDC.\nMDC=HDC=\u001fwhenMis linear inH(as for a paramag-\nnet at high T). For comparison sake, we plot these two\nquantities together in Fig. S3. The di\u000berence between\nthese two quantities starts to grow below the 330 K fea-\nture, and then increases again below TC. Note that for\nthis orientation H?c, the demagnetization e\u000bects are13\n/s48 /s46 /s48 /s48 /s48 /s46 /s48 /s50 /s48 /s46 /s48 /s52 /s48 /s46 /s48 /s54 /s48 /s46 /s48 /s56 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s54 \n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s56 /s48 /s46 /s48 /s49 /s50 \n/s32/s32/s72 \n/s68 /s67 /s32 /s32 /s99 \n/s72 \n/s68 /s67 /s32 /s124/s124/s32 /s99 \n/s40/s99 /s41/s40/s98 /s41\n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41/s84 /s32 /s61 /s32 /s49 /s49 /s48 /s75 /s40/s97 /s41\n/s39 /s39 /s39 \n/s32/s32/s72 \n/s65 /s67 /s32 /s32 /s99 \n/s72 \n/s65 /s67 /s32 /s124/s124/s32 /s99 /s39 /s32/s97 /s110 /s100 /s32 /s39 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s72 \n/s68 /s67 /s32/s40/s79 /s101 /s41\nFIG. S2: Isothermal magnetization for T=110 K showing (a)\nDC magnetization and (b,c) AC susceptibility data. The AC\ndata were collected using an amplitude of A=14 Oe and a\nfrequency of f=997 Hz.\nexpected to be small (\feld applied in the plane of a thin\nplate).\nFirst principles calculations\nFig. S4 shows the magnetic con\fgurations explored\nwith \frst principles calculations (the non-magnetic and\nferromagnetic states are not shown). Our \frst princi-\nples calculations included a non-magnetic state, a ferro-\nmagnetic state, the ferrimagnetic states FI1 (Mn1 and\nMn2 spins opposed, the calculated ground state) and\nFI2 (Mn1-Mn1 planar neighbors anti-aligned and half of\nMn1-Mn2 neighbors aligned and half antialigned), and\ntwo antiferromagnetic states AF1 and AF2 (both these\nstates areq= 0 states). In state AF1, all Mn1-Mn1 pla-\nnar neighbors are anti-aligned, while all Mn1-Mn2 near-\nest neighbors are ferromagnetically aligned. In AF2 the\nMn1-Mn1 planar neighbors are antialigned and Mn1-Mn2\nnearest neighbors are anti-aligned.\nThe calculated energies were mapped to a Heisenberg\nmodel with J 1the nearest-neighbor exchange (between\nMn1-Mn2), J 2the second-nearest neighbor exchange that\nis between planar Mn1-Mn1, and J 3that couples Mn1\nand a third-nearest spin at Mn2. These calculations as-\nsumed an e\u000bective spin S of \u00061 in the Hamiltonian. The\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 /s49 /s48 /s49 /s49 /s48 /s50 \n/s77 \n/s68 /s67 /s47/s72 \n/s68 /s67 \n/s39 \n/s72 /s32 /s32/s99 \n/s32/s32\n/s65 \n/s65 /s67 /s32/s61 /s32/s49 /s52 /s32/s79 /s101 \n/s72 \n/s68 /s67 /s32/s61 /s32/s50 /s48 /s32/s79 /s101 /s39 /s32/s32/s97 /s110 /s100 /s32 /s77 \n/s68 /s67 /s47/s72 \n/s68 /s67 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41FIG. S3: Comparison of AC susceptibility and quantity M=H\nfrom DC measurements. The DC measurements were con-\nducted upon cooling in an applied \feld of 20 Oe. The AC\ndata were obtained using HDC=0,A=14 Oe and a frequency\noff=997 Hz. The data are for magnetization with \felds lying\nin the basal plane of the trigonal crystal.\nvalues obtained are all antiferromagnetic: -34.6 , -6.3,\nand -14.8 meV/Mn for J 1, J2, and J 3, respectively. Ef-\nfective uncertainties of approximately 10-15 percent are\nobtained for each of these constants. The uncertainty\narises due to an over-determined set of equations deter-\nmining the exchange constants (one more equation than\nvariable, so that the constants are \\best-\ft\" constants).\nRemoval of one of the equations allows an exact set, for\nthis equation subset, to be determined and thereby the\nuncertainty within the \\best-\ft\" constants assessed.\nAs mentioned, the \frst principles calculations depict\na strong relationship of physical structure to the mag-\nnetism (speci\fcally, the non-symmetry-dictated internal\npositions). In Table S3, we compare the atomic coor-\ndinates extracted from the relaxations to the published\nstructure. Note that all calculations used the experimen-\ntal lattice parameters of a= 7.029 \u0017A andc= 14.255 \u0017A,\nwhich were determined at room-temperature and re-\nported in Ref. 3. LAPW sphere radii of 2.04 Bohr for\nSi, and 2.5 for Mn and Te were employed and su\u000ecient\nnumbers of k-points - at least 600 in the full Brillouin\nzone - were used to determine energy di\u000berences. An\nRKmaxof 7.0, where RK maxis the product of the small-\nest LAPW sphere radius (in this case Si) and the largest\nplanewave expansion wavevector, was employed.\nIn a certain sense, it is natural that signatures of the\nferrimagnetic ground state would remain well above the\nCurie temperature of 78 K. The DFT results \fnd the en-\nergy of the ferrimagnetic state to be well more than an\neV per Mn below that of the non-magnetic state. This\nis more than two orders of magnitude larger than the\n7 meV/Mn scale of TC, indicating a strong likelihood of14\nFIG. S4: The magnetic con\fgurations examined by \frst prin-\nciples calculations in Mn 3Si2Te6are shown (the non-magnetic\nand ferromagnetic states are not shown). FI1 is the ground\nstate, and relative to that energy we obtained AF1=20.8,\nFI2=32.6, and AF2=33.8 meV/Mn.\n\\disordered local moments\"4well aboveTC. Supporting\nthis interpretation is the behavior of the resistivity, which\nshows semiconducting behavior well above TC. This is\nconsistent with the semiconducting gap we \fnd theoret-\nically in the ferrimagnetic state, while the non-magnetic\nstate from theory has a metallic behavior.\nTABLE S3: The internal coordinates of the non-symmetry\ndictated atomic positions. We include the symmetry-dictated\nplanar coordinates of Mn1 and Si for reference. \\Exp.\" refers\nto the experimental published structure,3NM to the non-\nmagnetic calculations, and FI to the ferrimagnetic calcula-\ntions.\nAtom Site Symmetry Origin x y z\nMn1 4f Exp. 1/3 2/3 0.00068\nFI-Calc. 1/3 2/3 0.00155\nNM-Calc. 1/3 2/3 0.00470\nSi 4e Exp 0 0 0.08153\nFI-Calc. 0 0 0.08186\nNM-Calc. 0 0 0.08298\nTe 12i Exp. 0.65869 0.00361 0.12788\nFI-Calc. 0.65040 0.00081 0.12697\nNM-Calc. 0.64213 0.02033 0.11615Single crystal neutron di\u000braction data - HB3A data\nRocking curves from the four-circle di\u000bractometer\nHB3A at the High Flux Isotope Reactor (ORNL) are\nshown in Fig. S5. These demonstrate a strong temper-\nature dependence for the 002 re\rection and not the 004\nre\rection. The re\fned magnetic structure prohibits mag-\nnetic scattering contributions to (004), which is shown\n(Fig. S6) to increase only slightly upon cooling across the\nmagnetic ordering temperature of TC\u001978 K. The inten-\nsities of other nuclear-dominated re\rections are shown in\nFig. S6.\n/s53 /s54 /s55/s45/s49/s48/s49/s50/s51/s52/s53/s54/s55\n/s49/s50 /s49/s51/s48/s48/s52/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s49/s48/s51\n/s32/s99/s111/s117/s110/s116/s115/s41\n/s32/s40/s100/s101/s103/s46/s41/s32/s48/s48/s50\n/s52/s75\n/s49/s48/s48/s75\n/s51/s56/s48/s75\n/s32/s40/s100/s101/s103/s46/s41\nFIG. S5: Rocking curves for single crystal neutron di\u000braction\ndata of (a) the 002 Bragg re\rection and (b) the 004 Bragg\nre\rection at 4, 100, and 350 K.\nDi\u000buse Scattering - CORELLI data\nDi\u000buse magnetic scattering was investigated using the\nsame crystal as the neutron di\u000braction measurements.\nThe crystal is approximately 7 mg in mass, which is small\nwhen searching for di\u000buse scattering intensity. As such,\nlong counting times were employed. In addition, the mea-\nsurement is facilitated by the large moment on the Mn\natoms. The main text discussed di\u000buse scattering along\nL in the (H-HL) plane, which was strongest at 6 K. This\ndi\u000buse scattering is centered around L=2, and is essen-\ntially absent for L <1 and L>3. As discussed in the\nmain text, the source for this scattering is not entirely\nclear. Given that this occurs for non-zero H, and is not\npresent for large L, it potentially relates to a canting of\nthe moments out of the abplane. Additional data for this\nscattering plane, including the estimated elastic contri-\nbution, are shown in Fig. S7\nThe di\u000buse scattering at 6 K is very anisotropic, occur-15\n/s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53\n/s32\n/s49/s48/s48 /s49/s49/s48\n/s48/s48/s52 /s50/s50/s48/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121\n/s84 /s32/s40/s75/s41\nFIG. S6: Temperature dependence of di\u000braction intensity at\nseveral Bragg re\rections as obtained from the single crys-\ntal neutron di\u000braction beamline HB3A. Based on the re\fned\nmagnetic structure, the 004 re\rection should not possess any\nmagnetic contribution and thus the relative change with tem-\nperature is just a re\rection of the evolution of the lattice or\npeak position (constant Q measurements). The other re\rec-\ntions, with H6= 0, could contain some magnetic contribution\nif the moments have a component along the caxis. However,\nthe relative change with Tfor these peaks is similar to that\nfor 004, suggesting that magnetic scattering is minor for the\npeaks shown here. The data have been normalized and then\no\u000bset from one another.\n350K  \n120K  \n6K Total scattering  Elastic estimate  \n(a) (d) \n(b) \n(c) (e) \n(f) \n120K  \n6K 350K  \n120K  \n6K 350K  \nFIG. S7: Neutron scattering intensity maps from the 2D de-\ntectors of CORELLI showing scattering in the H-HL plane\n(a,d) 6 K, (b,e) 120 K and (c,f) 350 K. The two columns are the\ntotal scattering and the estimated elastic scattering. These\n2D maps show the intensity from [-0.005 -0.005, 0] to [0.005,\n0.005 0], i.e., a single bin, along the [H H 0] out-of-plane di-\nrection.ring only along L and not along H. Such one-dimensional\ndi\u000buse scattering is sometimes observed in quasi-2D crys-\ntal structures due to the presence of stacking faults. An\nimportant feature observed here is the lack of di\u000buse scat-\ntering for H = 0. From a magnetic scattering perspective,\nonly non-zero H re\rections in this plane would possess\nscattering from a component of the moment along the c\naxis. Therefore, these results would be consistent with\nthe presence of some canting along the caxis that only\nhas short coherence lengths. This would be consistent\nwith the average structure being that of a non-canted\nmoment, and we recall that only a very small canted mo-\nment is observed when allowed during re\fnement of the\nsingle crystal neutron di\u000braction data at 4 K. Short co-\nherence lengths of a canted moment could perhaps be\ncaused by the competing exchange interactions. The ex-\nistence of a large J 1promotes ferrimagnetic Mn1-Mn2-\nMn1 units along the caxis, and the coupling of these\nvia competing interactions produces the long-range fer-\nrimagnetic structure. Thus, it may be possible that each\nof these units (or clusters of these units) has a certain\ncanting angle (or lack thereof), but the coherence be-\ntween them is somewhat weak. It is also possible that\na crystallographic defect on the Mn positions causes the\ndi\u000buse scattering, which becomes observable through a\nstrong magnetic contribution at low T.\nAnalysis of speci\fc heat capacity data\nAs mentioned in the main text, we performed a \ft of\nthe speci\fc heat capacity data to estimate the magnetic\nentropy change at TC. Importantly, this was done in a\nmanner that would generally maximize any contribution\n(overestimation). The reason for this is to show that even\nwhen overestimated, the entropy change across TCis very\nsmall (about 8% of anticipated, for the data and baseline\nshown). The data utilized are shown in Figure S8, and\nthe inset shows the data after subtracting the baseline\n(same units as primary y-axis).\nTo further examine the magnetic contribution to the\nspeci\fc heat, we have compared data for Mn 3Si2Te6with\nthat of CrSiTe 3and CrI 3in Fig. S9. Both of these\nCr-based compounds are ferromagnetic semiconductors\nand are quasi-2D with van der Waals gaps; CrSiTe 3\nhasTC=33 K and CrI 3hasTC=61 K.5,6While the mag-\nnetic order of these materials is di\u000berent from that of\nMn3Si2Te6, they all possess relatively large magnetic\ncontributions to the speci\fc heat capacity at low T. In\nFig.S9a, the data are plotted in the conventional CP=T\nversusT2manner. This appears valid for CrI 3and does\nan adequate job of describing the data for Mn 3Si2Te6,\nthough the data for Mn 3Si2Te6have some curvature over\nthe wide temperature range considered. A more extreme\ncurvature is found for CrSiTe 3, where non-linearity of\nCP=TversusT2is clearly evident. Instead, CrSiTe 3is\nbetter modeled using a magnetic contribution that is pro-\nportional to T1:5(Fig.S9b). This treatment also seems16\n/s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53\n/s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s32/s32/s100/s97/s116/s97\n/s32/s98/s97/s115/s101/s108/s105/s110/s101/s67\n/s80/s47/s84 /s32/s40/s74/s47/s109/s111/s108/s45/s77/s110/s47/s75/s50\n/s41\n/s84 /s32/s40/s75/s41/s67\n/s80/s47/s84/s32/s45/s32/s98/s97/s115/s101/s108/s105/s110/s101\n/s84 /s32/s40/s75/s41\nFIG. S8: Speci\fc heat data as CP=TversusT, the integra-\ntion of which provides an entropy change. To integrate over\nthe anomaly near TCwithout a phonon reference material, we\nhave taken a straight line for the baseline and this will typ-\nically cause an overestimation of the entropy change across\nTC. The inset shows the data after removal of this baseline,\nwhich would essentially correspond to the magnetic contribu-\ntion if an appropriate baseline were utilizedto work for Mn 3Si2Te6, but clearly fails for CrI 3. At the\nlowestT, the data for Mn 3Si2Te6are well-described by a\nlinear term, as shown in the main text.\n\u0003Electronic address: mayaf@ornl.gov\n1R. Rimet, C. Schlenker, and H. Vincent, J. Magn. Magn.\nMater. 25, 7 (1981), URL http://dx.doi.org/10.1016/\n0304-8853(81)90141-4 .\n2S. Zhou, K. Potzger, G. Zhang, A. M ucklich, F. Eichhorn,\nN. Schell, R. Gr otzschel, B. Schmidt, W. Skorupa, M. Helm,\net al., Phys. Rev. B 75, 085203 (2007), URL http://link.\naps.org/doi/10.1103/PhysRevB.75.085203 .\n3H. Vincent, D. Leroux, D. Bijaoui, R. Rimet, and\nC. Schlenker, J. Solid State Chem. 63, 349 (1986).\n4A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, J.Phys. F 13, 979 (1983).\n5J. Dillon Jr and C. Olson, J. Appl. Phys. 36, 1259 (1965).\n6M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales,\nChem. Mater. 27, 612 (2015), URL http://pubs.acs.org/\ndoi/abs/10.1021/cm504242t .\n7L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M.-W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, et al., APL Materials 3, 041515 (2015),\nURL http://scitation.aip.org/content/aip/journal/\naplmater/3/4/10.1063/1.4914134 .17\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s77/s110\n/s51/s83/s105\n/s50/s84/s101\n/s54\n/s67/s114/s83/s105/s84/s101\n/s51\n/s67/s114/s73\n/s51\n/s32/s77/s110\n/s51/s83/s105\n/s50/s84/s101\n/s54\n/s67/s114/s83/s105/s84/s101\n/s51\n/s67/s114/s73\n/s51/s67\n/s80/s47/s84 /s32/s40/s109/s74/s47/s109/s111/s108/s45/s84/s77/s47/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41\n/s40/s98/s41\n/s32/s67\n/s80/s47/s84/s49/s46/s53\n/s32/s40/s109/s74/s47/s109/s111/s108/s45/s84/s77/s47/s75/s50/s46/s53\n/s41\n/s84/s49/s46/s53\n/s32/s40/s75/s49/s46/s53\n/s41/s40/s97/s41\nFIG. S9: Low temperature speci\fc heat capacity of\nMn3Si2Te6, CrSiTe 3and CrI 3are plotted assuming (a) CP\n=\rT+\fT3formulation and (b) using a CP=\r0T1:5+\fT3\nrepresentation. In the axis labels, the term TM = transition\nmetal. Here, \fis assumed to be related to the Debye tem-\nperature as usual, but \ris not meant to represent the usual\nSommerfeld coe\u000ecient because these materials are semicon-\nducting and should not possess a signi\fcant electronic contri-\nbution toCPat lowT. The data for CrSiTe 3are taken from\nRef. 7 and data for CrI 3are from Ref. 6."    },    {        "title": "2401.13617v1.Current_Driven_Domain_Wall_Motion_in_Curved_Ferrimagnetic_Strips_Above_and_Below_the_Angular_Momentum_Compensation.pdf",        "content": "Current-Driven Domain Wall Motion in Curved Ferrimagnetic Strips Above and\nBelow the Angular Momentum Compensation\nD. Osuna Ruiz *1,∗O. Alejos2, V. Raposo1, and E. Mart´ ınez1\n1Department of Applied Physics, University of Salamanca, Salamanca 37008, Spain and\n2Department of Electricity and Electronics, University of Valladolid, Valladolid, Spain.\n(Dated: January 25, 2024)\nCurrent driven domain wall motion in curved Heavy Metal/Ferrimagnetic/Oxide multilayer strips\nis investigated using systematic micromagnetic simulations which account for spin-orbit coupling\nphenomena. Domain wall velocity and characteristic relaxation times are studied as functions of\nthe geometry, curvature and width of the strip, at and out of the angular momentum compensation.\nResults show that domain walls can propagate faster and without a significant distortion in such\nstrips in contrast to their ferromagnetic counterparts. Using an artificial system based on a straight\nstrip with an equivalent current density distribution, we can discern its influence on the wall terminal\nvelocity, as part of a more general geometrical influence due to the curved shape. Curved and narrow\nferrimagnetic strips are promising candidates for designing high speed and fast response spintronic\ncircuitry based on current-driven domain wall motion.\nI. INTRODUCTION\nA magnetic domain wall (DW) is the transition re-\ngion that separates two uniformly magnetized domains\n[1]. These magnetic configurations are interesting due to\nfundamental physics, but also due to potential technolog-\nical applications [2, 3]. In fact, during the last decades\nDWs have been at the core of theoretical and experimen-\ntal studies which have provided with a deep understand-\ning of different spin-orbit coupling phenomena [4–8]. For\ninstance, straight stacks where an ultra-thin ferromag-\nnetic (FM) layer is sandwiched between a heavy metal\n(HM) and an oxide (Ox), present perpendicular mag-\nnetic anisotropy (PMA) and therefore, the domains are\nmagnetized along the out-of-plane direction of the stacks:\nup(+⃗ uz) or down (−⃗ uz). DWs in these HM/FM/Ox\nstacks adopt an homochiral configuration due to the\nDzyaloshinskii-Moriya interaction (DMI) [5, 7, 9]. Adja-\ncent DWs have internal magnetic moments along the lon-\ngitudinal direction ( ⃗ mDW=±⃗ ux), and the sense is im-\nposed by the sign of the DMI, which in turns depends on\nthe HM [5]. For left-handed stacks such as Pt/Co/AlO,\nup-down (UD) and down-up (DU) DWs have internal mo-\nments with ⃗ mDW=−⃗ uxand⃗ mDW= +⃗ ux, respectively\n[5]. These DWs are driven with high efficiency by inject-\ning electrical currents along the longitudinal direction of\nthe HM/FM/Ox stack [4]. Due to the spin-Hall effect [5],\nthe electrical current in the HM generates a spin polar-\nized current which exerts spin-orbit torques (SOTs) on\nthe magnetization of the FM layer, and drives series of\nhomochiral DWs which are displaced along the longitudi-\nnal direction ( x-axis). DW velocities of VDW∼500 m/s\nhave been reported upon injection of current densities of\nJHM∼1 TA/m2in Pt/Co/AlO [4]. Consequently, these\nstacks have been proposed to develop highly-packed mag-\nnetic recording devices, where the information coded in\n∗osunaruiz.david@usal.esthe domains between DWs can be efficiently driven by\npure electrical means. Both UD and DU DWs move\nwith the same velocity along straight stacks, but some\nimplementations of these memory or logic devices would\nrequire to design 2D circuits, where straight parts of\nHM/FM/Ox stack are connected each other with curved\nor semi-rings sections. However, recent experimental ob-\nservations [10] and theoretical studies [11] have pointed\nout that adjacent UD and DU DWs move with differ-\nent velocity along curved HM/FM/Ox stacks, which is\ndetrimental for applications because the size of the do-\nmain between adjacent DWs changes during the motion,\nwith the perturbation of the information coded therein.\nTherefore, other systems must be proposed in order to\ndesign reliable 2D circuits for DW-based memory and\nlogic devices.\nOther stacks with materials and/or layers with anti-\nferromagnetic coupling, such as synthetic antiferromag-\nnets (SAF) and ferrimagnetic (FiM), have proven to out-\nperform FM in terms of current-driven DW dynamics\n[11–15]. Ultrafast magnetization dynamics in the THz\nregime, marginal stray field effects and insensitivity to\nexternal magnetic fields are other significant advantages\nof materials with antiferromagnetic coupling with respect\nto their FM counterparts. As conventional antiferromag-\nnets (AFs), FiM alloys are also constituted by two spec-\nimens, typically a rare earth (RE) and transition metal\n(TM), that form two ferromagnetic sublattices antiferro-\nmagnetically coupled to each other. GdFeCo, GdFe or\nTbCo are archetypal FiM alloys, with the RE being Gd\nor Tb and the TM being FeCo or Co. In contrast to AFs\nwith zero net magnetization, the magnetic properties of\nFiMs, such as magnetization and coercivity, are largely\ninfluenced by the relative RE and TM composition (or\nequivalently, temperature). This fact offers additional\ndegrees of freedom to control the current-driven DW ve-\nlocity. The spontaneous magnetization of each sublattice\nMS,ican be tuned by changing the composition of the\nFiM and/or the temperature of the ambient ( T) [13, 14].\nFor a given composition of the FiM (RE xTM 1−x), therearXiv:2401.13617v1  [cond-mat.mtrl-sci]  24 Jan 20242\nare two relevant temperatures below the Curie thresh-\nold. One is the magnetization compensation temperature\n(TM) at which the saturation magnetization of the two\nsublattices are equal ( Ms1(TM) =Ms2(TM)), so the FiM\nbehaves as a perfect antiferromagnetic material, with\nzero net magnetization and diverging coercive field. The\nother is the temperature at which the angular momentum\ncompensates, TA, at which Ms1(TA)/γ1=Ms2(TA)/γ2,\nwhere γiis the gyromagnetic ratio of each sublattice\n(i:1,2 for 1:TM and 2:RE). As the gyromagnetic ratio\ndepends on the Land´ e factors ( gi) which are different for\neach sublattice, the angular compensation temperature\nTAis in general different from the magnetization compen-\nsation temperature ( TM). Consequently, the FiM have\na net magnetization at TA, so conventional techniques\nused for FMs can be also adopted to detect the magnetic\nstate of FiM samples [16]. Moreover, recent experimen-\ntal observations have evidenced that the current-driven\nDW velocity along straight HM/FiM stacks can be sig-\nnificantly optimized at the angular momentum compen-\nsation temperature ( T=TA), with velocities reaching\nVDW∼2000 m/s for typical injected density current of\nJHM∼1 TA/m2along the HM underneath [14]. The\nDW velocity drops either below ( T < T A) and above\n(T > T A) angular momentum compensation. Note that\nalternatively to tuning the temperature for a fixed com-\nposition xof the FiM alloy RE xTM 1−x, even working at\nroom temperature ( T= 300 K) the DW velocity peaks\nat a given composition where angular momentum com-\npensates [13]. Therefore, both studies, either fixing the\ncomposition ( x) and changing temperature of the ambi-\nent (T), or fixing the ambient temperature and modifying\nthe FiM composition are equivalent for our purposes of\nDW dynamic. Although the current-driven DW motion\n(CDDWM) along HM/FiM stacks suggests their poten-\ntial for memory and logic applications, previous studies\nhave been mainly focused on straight FiM strips [13–15].\nThe further develop of novel DW-based devices also re-\nquires to analyze the dynamics of DWs along HM/FiM\nwith curved parts which would connect straight paths to\ndesign any 2D circuit. Such investigation of the dynam-\nics along curved is still missing, and it is the aim of the\npresent study.\nHere we theoretically explore the CDDWM along\ncurved HM/FiM stacks by means of micromagnetic ( µm)\nsimulations. Our modeling allows us to account for the\nmagnetization dynamics in the two sublattices indepen-\ndently. We explore the CDDWM below, at and above\nthe angular momentum compensation (AMC) for differ-\nent curved samples, with different widths and curvatures,\nand considering the realistic spatial distribution of the\ninjected current along the HM. In particular, we will in-\nfer and isolate the relevance of different aspects govern-\ning such dynamics, as the role of the non-uniform cur-\nrent and other purely geometrical aspects of the curved\nshape. This work completes previous studies on straight\nsamples [11, 13–15], and will be practical for designing\nmore compact and efficient DW-based devices. The restof the paper is organized as follows. In Sec. II we de-\nscribe the numerical details of the micromagnetic model\nalong with the material parameters and the geometri-\ncal details of the evaluated samples. Sec. III presents\nthe micromagnetic results of the CDDWM in different\nscenarios. Firstly, exploring the role of the FiM sam-\nple width ( w) for a fixed the curvature ( ρ, given by the\ninverse of the average radius, ρ= 1/re), and secondly\nfixing the width and varying the curvature. After that,\nwe present results which allow us to infer the role of non-\nuniform current and geometrical aspect ( w, ρ) comparing\ncurved and straight samples. The main conclusions are\nsummarized in Sec. IV.\nII. MATERIALS AND METHODS\nCDDWM is numerically studied here along curved\nHM/FiM stacks as schematically shown in Fig. 1, where\nriandroare the inner and outer radius rorespectively,\nandre= (ro+ri)/2 is the mean effective radius. wand\ntFiMare the width and the thickness of the FiM respec-\ntively. The relaxed magnetization configuration of the\nsublattice i= 1, shown in Fig. 1 (opposite configuration\nin sublattice i= 2), serves as the initial state to study\nthe CDDWM upon of current injection along the HM un-\nderneath. The temporal evolution of the magnetization\nof each sublattice is given by the Landau-Lifshitz-Gilbert\nequation (LLG) [17],\nd⃗ mi(t)\ndt=−γ0,i⃗ mi(t)×⃗Heff,i+αi⃗ mi×d⃗ mi(t)\ndt+⃗ τSOT,i ,\n(1)\nwhere here the sub-index istands for i: 1 and 2 sub-\nlattices respectively. γi=giµB/ℏandαiare the gy-\nromagnetic ratios and the Gilbert damping constants,\nrespectively. giis the Land´ e factor of each layer, and\n⃗ mi(⃗ r, t) =⃗Mi/Ms,iis the normalized local magnetization\nto its saturation value ( Ms,i), defined differently for each\nsublattice: Ms,i(i: 1,2). In our micromagnetic model the\nFiM strip is formed by computational elementary cells,\nand within each cell we have two magnetic moments, one\nfor each component of the FiM. The respective effective\nfield ( ⃗Heff,i) acts on the local magnetization of each sub-\nlattice ( ⃗ mi(⃗ r, t)), and it is the sum of the magnetostatic,\nthe anisotropy (PMA), the DMI and the exchange fields\n[11, 15]. The magnetostatic field on each local moment\nin the sublattice is numerically computed from the av-\nerage magnetization of each elementary cell using simi-\nlar numerical techniques as for the single FM case (see\n[11, 15]). We checked that the demagnetising field has a\nmarginal influence in the simulation results compared to\nother contributions to the effective field. For the PMA\nfield, the easy axis is along the out-of-plane direction ( z-\naxis), and the anisotropy constants for each sublattice are\nKu,i(PMA constant). Diis the DMI parameter for each\nsublattice i: 1,2 [11, 15]. The exchange field of each sub-\nlattice includes the interaction with itself (intra-lattice3\nexchange interaction, ⃗Hexch,i ) and with the other sublat-\ntice (inter-lattice exchange interaction, ⃗Hexch, 12). The\ninter-lattice exchange effective field is computed as for\na single FM sample, Hexch,i =2Ai\nµ0Ms,i∇2⃗ mi, where Aiis\nthe intralattice exchange parameter. The inter-lattice ex-\nchange contribution ⃗Hexch, 12to the effective field ⃗Heff,i,\nacting on each sublattice is computed from the corre-\nsponding energy density, ωexch,i =−Bij⃗ mi·⃗ mj, where\nBij(in [J m−3]) is a parameter describing the inter-lattice\nexchange coupling between sublattices (here, we used the\nnotation i: 1 and j: 2).\nIn Eq.(1), ⃗ τSOT,i are the SOTs acting on each sub-\nlattice, which are related to the electrical current along\nthe HM ( ⃗JHM). Based on preliminary studies [18], here\nwe assume that ⃗ τSOT,i is dominated by the spin Hall ef-\nfect (SHE), so ⃗ τSOT,i =−γ0HSL⃗ mi×(⃗ mi×⃗ σ) where\nHSL=ℏθSH,iJHM\n2|e|µ0MstFiM[19]. ℏis the Planck constant, and\nθSH,i is the spin Hall angle, which determines the ra-\ntio between the electric current and the spin current\n(Js=θSHJHM) for each sublattice. ⃗ σ=⃗ uJ×⃗ uzis\nthe unit vector along the polarization direction of the\nspin current generated by the SHE in the HM, being\northogonal to both the direction of the electric current\n⃗ uJand the vector ⃗ uzstanding for the normal to the\nHM/FiM interface. For a longitudinal current ( ⃗ uJ=⃗ ux),\nthe spin current is polarized along the transverse di-\nrection, ⃗ σ=−⃗ uy. For curved samples where the cur-\nrent density ⃗JHM=JHM(r)⃗ uJhas azimuthal direction\n(⃗ uJ=−⃗ uϕ), the direction of the polarization is radial,\n⃗ σ=⃗ uJ×⃗ uz=⃗ ur, as shown in Fig 1. A potential dif-\nference is applied between the ends of the curved track\nto inject current in the right circulation. Therefore, a\ngap of 25 nm is also modelled, leading to a split ring\nshape for the strip (see inset in Fig. 1). The spatial\ndistribution of current as a function of the radial coor-\ndinate ( ri< r < r o) is taken from [11, 20], and it de-\npends on the width ( w) and the radial distance ( r) as\nJHM(r) =J0w/(rlog (1 + w/ri)), where J0is the nom-\ninal, uniform current density, in an equivalent straight\nstrip of same cross-section ( w×tHM, where tHMis the\nthickness of the HM strip).\nIn order to illustrate the current-driven DW dynam-\nics along curved HM/FiM stacks we fix tFiM = 6 nm,\nand samples with different widths ( w) and radii ( re)\nwere evaluated. The following common material pa-\nrameters were adopted for the two sublattices i: 1,2:\nAi= 70 pJ/m, Ku,i= 1.4×106J/m3,αi= 0.02,\nDi= 0.12 J/m2,θSH,i = 0.155. The strength of the\nantiferromagnetic coupling between the sublattices was\nfixed to Bij≡B12=−0.9×107J/m3. The gyro-\nmagnetic ratios ( γi=giµB/ℏ) are different due to the\ndifferent Land´ e factor: g1= 2.05 and g2= 2.0. The\nsaturation magnetization of each sublattice Ms,ican be\ntuned with the composition of the FiM and/or with the\ntemperature of the ambient ( T). Here, we assume the\nfollowing temperature dependences for each sublattice:Ms,i(T) =Ms,i(0)\u0010\n1−T\nTC\u0011ai\n, where TC= 450 K is the\nCurie temperature of the FiM, Ms,1(0) = 1 .4×106A/m\nandMs,2(0) = 1 .71×106A/m are the saturation magne-\ntization at zero temperature, and a1= 0.5 and a2= 0.76\nare the exponents describing the temperature depen-\ndence of the saturation magnetization of each sublattice.\nThe temperature at which the net saturation magneti-\nzation vanishes ( Ms,1(TM) =Ms,2(TM)) is TM= 241 .5\nK, and the angular momentum compensation temper-\nature corresponding to Ms,1(TA)/g1=Ms,2(TA)/g2, is\nTA= 260 K. We evaluate the CDDWM below, at and\nabove the angular momentum compensation adopting\nthree representative temperatures: T= 220 K < T A,\nT= 260 K = TAandT= 300 K > TA. Samples were\ndiscretized using a 2D finite difference scheme using com-\nputational cells with ∆ x= ∆y= 0.2 nm and ∆ z=tFiM.\nSeveral tests were carried to certify that the presented re-\nsults are free of discretization errors.\nFIG. 1. Scheme showing the relaxed states of spins in sub-\nlattice i= 1, in the positive z-direction (white domain), in\nthe negative z-direction (black domain) and in the plane of\nthe strip for an ‘Up to Down’ (UD) domain wall (purple)\naccording to the current direction, for an exemplary curved\nstrip. The direction of the applied electric current (red arrow)\nin the Heavy Metal beneath the magnetic strip, generated\nfrom a potential difference ∆ V(see inset), is shown as well\nas the geometrical parameters of the strip.\nIII. MICROMAGNETIC RESULTS\nDue to the several combination of parameters to con-\nsider in our study, we divided this section in three sub-\nsections: (A) The study on the influence of the strip\nwidth ( w); (B) the study on curvature ( ρ=r−1\ne); and (C)\nsame studies for a straight strip with identical material\nparameters, to explore by comparison the effects of curva-\nture on the DW dynamics. In parts (A) and (B), scenar-\nios for three different temperatures, T1= 220 K , T2= 2604\nK, T3= 300 K are considered, to study the DW motion\nbelow the AMC ( T1), at the AMC ( T2) and above the\nAMC ( T3). We also define and refer to T3= 300 K as for\n‘room temperature’ in our study. Note that a change in\ntemperature only affects MSin our model, therefore it\nhas equivalent effects to changing material composition\n[13]. In addition to DW velocity, we also characterize the\ninertial motion of the DW as a function of current den-\nsity. As an example, Fig. 2 shows typical results of the\nDW position and its velocity in a ring-like strip ( w= 256\nnm and re= 384 nm) under a density current JHM= 2\nTA/m2and at T= 260 K. Qualitatively similar results\nare obtained at T= 220 K and T= 300 K (not shown).\nInsets show the (clockwise) DW displacement as a func-\ntion of time for one sublattice ( i= 1).\nA. Influence of width for a fixed curvature\nIn this study, the curvature is fixed ( re= 384 nm) and\nwidth ( w) is varied from 56 nm to 296 nm in steps of 40\nnm. Fig. 3 shows the results for the terminal DW veloc-\nity (|VDW,i|) as a function of the nominal density current\nJHM=j0, equivalent to the homogeneous density cur-\nrent in a straight strip with the same cross-section. In the\nnext sections, we use the notation JHM= J for simplic-\nity. Fig. 3 shows that temperature has a noticeable effect\non the terminal velocity on the DW type equally, Up to\nDown domain (UD) or Down to Up domain (DU). As it\nwas expected from previous work on straight FiM strips\n[11, 13], at TAthe DW velocities are greater. Also, the\nDW velocities increase for narrower strips (red symbols).\nIn fact, the observed trend is very similar to straight\nstrips: the terminal velocity is maximum at the temper-\nature of AMC, TA= 260 K and significantly increased,\nexceeding 2000 m/s for the narrowest strip as compared\nto∼1800 m/s for the widest. These results also suggest\nthat the DWs velocities are equal for both types of DWs\n(UD and DU), which would lead to no distortion of the\nsize of a domain between two adjacent DWs travelling\nalong the curved strip. This result is significantly differ-\nent from FM systems [11].\nAtT̸=TA, the DW velocity is reduced either increas-\ning or reducing temperature with respect to TA, leading\nto velocities around 1100 m/s, generally regardless of the\nwidth and DW type. For a given Jvalue, as the strip\ngets wider, however, the velocity is slightly smaller but\nthese differences are negligible (see Fig. 3(a), (c), (d),\n(f)). This result contrasts with that of a FM strip, where\na greater difference of velocities between a DU and a UD\nalong a curved strip was shown [11].\nTo characterize the inertial motion of the DW, we eval-\nuate the temporal evolution of the DW velocity ( V(t),\ncomputed from the spatial averaging of m1,z(t)) as a\nfunction of time (or instant velocity) under a current\nsquare pulse of duration 0.1 ns and start at t= 0. The\nµm results can be fitted to the following exponentials:\nV∞(1−e−t/τr) during the duration of the pulse ( t≤0.1\nxf\n0.00 ns\n0.50 ns\n0.13 ns\n0.25 ns\n0.38 ns\n(a)\n(b)\n(c)(d)FIG. 2. Micromagnetic results of applying a uniform JHM=\nJ0= 2 TA/m2(a) showing the relative and final position xf\n(b) and velocity VDW(c), as a function of time t, of an UD\nDW in a curved strip ( w= 256 nm and re= 384 nm), at\nT= 260 K. Insets (d) are snapshots of the magnetic config-\nuration in sublattice i= 1 at different times (highlighted by\nthe vertical dotted lines in (a-c)). Green solid lines are for\nguiding the eye and are co-parallel with the strip radius.\nns), and V∞e−t/τf, after the pulse ends ( t > 0.1 ns),\nwhere V∞is the DW terminal velocity (see Fig. 2(a)-(f)).\nThe characteristic relaxation times τr(or rising time) and\nτf(or fall time) represent the duration of such transients\nand characterize the inertial motion of the DW. These\nparameters can be extracted by fitting the µm results to\nthe exponentials (see solid curves in Fig. 4(a)).\nAlthough simulations were performed for both types of\nDWs, note that we only present here results for the DU\ntype wall, for sake of simplicity. Identical results (not\nshown) were obtained for the UD DW. Fig. 4(a) show\nthe ‘instantaneous’ DW velocity V(t) and the relaxation\ntimes τfor three selected values of J (see solid symbols) at\nT=TAfor the widest strip ( w= 296 nm). Solid lines are5\n𝑤\n260 K 260 K220 K 220 K\n300 K 300 K(a)\n(b)\n(c)(d)\n(e)\n(f)\nT=220 K\nT=260 K\nT=300 K\nDUUD\n(g)\nw=296 nm\nww (nm)\n56 \n176 \n296 \nw=56 nmw (nm)\n56 \n176 \n296 UD\nDU\nT=220 K\nT=260 K\nT=300 K\nFIG. 3. Terminal velocities as a function of J for a UD (a-c)\nand a DU (d-f) DW obtained for sub-lattice i= 1 and for\nthree different temperatures: below, above and at the AMC\ntemperature (220 K, 300 K and 260 K, respectively). Strips\nfor the two limiting cases are shown in the red and blue con-\ntour insets at the top. (g) Terminal velocities as a function\nofwfor a UD (full symbols) and a DU (open symbols) type\nwall for J = 2 .35 TA/m2and the three chosen temperatures.\nInset in (g) shows J(r) for two values of w. Red dashed line\nindicates J = 2 .35 TA/m2.\nthe exponential curves to which the obtained simulated\ndata is fit. For each current, the minimal τis expected\nforT=TA= 260 K. Fig. 4(b) shows that τrandτffor\nthe two limiting cases ( w= 56 nm and w= 296 nm) are\nquantitatively similar, in the order of 0.02 ns, since they\nfall within the 95% confidence interval, set by the largest\nerror bars obtained for τfrom the fitted results, among\nall J. Also, all values are similar in order to the step-size\nJ = 2.35 TA/m2\nJ = 1.60 TA/m2\nJ = 0.85 TA/m2T = 260 K\n(a) (b)FIG. 4. Instantaneous DW velocities V(t) for three values of\nJ and for a curved strip of re= 384 nm and w= 296 nm at\nT=TA(a). Symbols are the µm data, from which τ(rise and\nfall times) are extracted for the narrowest and widest strips\n(b). Dashed lines in (b) are the upper and lower bounds of a\n95% confidence interval.\nused in simulations, 0.01 ns (see Fig. 4(d)).\nSimilar values of τwere obtained for strips of other\nwidths. Relaxation times are not noticeably influenced\nby temperature, and they generally remain within the\nrange of 0.01 ∼0.03 ns for T= 200 K and T= 300 K.\nThis is more than one order of magnitude smaller than\nin FM strips, the latter being about ∼1 ns according to\nRef.[21]. Besides, the relaxation times of current-driven\nDWs in curved strips found here are in good agreement\nwith those from field-driven or thermally driven DWs in\nantiferromagnetic straight strips, in the order of picosec-\nonds [22, 23].\nB. Influence of curvature for a fixed width\nIn this study, the strip width is fixed to w= 256 nm\nand the curvature parameter ρis varied. In other words,\nthe equivalent radii re(re=ρ−1) is varied from 134\nnm to 534 nm in steps of 50 nm. Fig. 5 shows the re-\nsults for the terminal velocity ( |VDW, 1|) of DU and UD\nDWs, for several values of rein nanometers, where the\nred (blue) curve corresponds to the smaller (greater) val-\nues, for three different temperatures.\nFig. 5(a)-(f) shows that, at T=TAand for a given\ncurvature, the DW velocities of DU and UD types are\nvery similar for the whole range of currents explored.\nDW velocity reduces as the curvature increases (see Fig.\n5(g)). It is worth noting that the latter cannot be a con-\nsequence of only a nonuniform J(r) as defined in [11]. In\nfact, for curved-most strips ( re= 134 nm), the spatial-\ndependent density current J(r) varies with rsimilarly as\nit does for changing w(see inset in Fig. 5(g) and in Fig.\n3(g)), which would suggest similar variations to DW ve-\nlocities as those found in Fig. 3(g). In other words, the\nimpact of the non-uniform J(r) is not so relevant to be\nthe only source of the big differences between the DW\nvelocities for large and small curvatures (orange symbols\nin Fig. 5(g) for re= 134 nm and re= 484 nm, respec-\ntively). Fig. 5(g) also shows that as the strip curvature6\n260 K 260 K220 K 220 K\n300 K 300 K(a)\n(b)\n(c)(d)\n(e)\n(f)\n𝑟𝑒\n(g)re (nm)\n134 \n384 \n584 \nre (nm)\n134 \n384 \n584 UD\nDU\nT=220 K\nT=260 K\nT=300 K\nDUUD\nT=220 K\nT=260 K\nT=300 Kre = 134 nm\nre = 534 nm\nFIG. 5. Terminal velocities as a function of J for a UD (a-c)\nand a DU (d-f) DW in the curved strip obtained for sub-\nlattice i= 1 and for three different temperatures: below,\nabove and at the AMC temperature (220 K, 300 K and 260\nK, respectively). Strips for the two limiting cases are shown\nin the red and blue contour insets at the top. (g) Terminal\nvelocities as a function of refor a UD (full symbols) and a DU\n(open symbols) type wall for J = 2 .35 TA/m2and the three\nchosen temperatures. Inset in (g) shows J(r) for two values\nofre. Red dashed line indicates J = 2 .35 TA/m2.\nis reduced, DW velocity converges to the straight strip\ncase ( re→ ∞ ).\nForT̸=TA, the dependence of the DW velocity with\ntemperature is minimal regardless of the DW type. As\nthe strip curvature increases there is a prominent change\nin the maximal terminal DW velocity for both DW types.\nHowever, the relative difference of velocities is almost\nJ = 2.35 TA/m2\nJ = 1.60 TA/m2\nJ = 0.85 TA/m2\n(a) (b)FIG. 6. Instantaneous DW velocities V(t) for three values of\nJ and for a curved strip of re= 584 nm and w= 256 nm at\nT=TA(a). Symbols are the µm data, from which τ(rise and\nfall times) are extracted for the narrowest and widest strips\n(b). Dashed lines in (b) are the upper and lower bounds of a\n95% confidence interval.\nnegligible. Therefore, results suggest that the strip cur-\nvature affects in a similar way to width, and equally, to\nboth DWs. In other words, the terminal velocity is signif-\nicantly reduced as curvature (or width) increases, while\nthe differences between DWs remain negligible (see Fig.\n5(g)). This behavior is even more pronounced at T=TA.\nAs discussed in section III.A, the latter would imply that\nthe robustness of a transmitted bit, encoded in a domain\nbetween two DWs, can be optimised in such curved-most\nstrips and reaches larger velocities in the strip.\nFig. 6(a) show the DW velocity as a function of time\nand for three selected values of J for an effective radius of\nre= 534 nm (least curved strip) and intermediate width\nw= 256 nm, at T=TA. Results look quantitatively sim-\nilar to those shown in Fig. 4(a), where rewas fixed to an\nintermediate value of 384 nm. As in Fig. 4(b), Fig. 6(b)\nshows that τrandτffor the two limiting cases ( re= 134\nnm and re= 534 nm) are quantitatively similar, in the\norder of 0.02 ns. For all the FiM curved strips explored\nat, above and below AMC, τrandτfremain within the\nrange of 0.01 ∼0.03 ns, approximately one order of mag-\nnitude less than their FM counterparts. This is in good\nagreement with results presented in the previous section\nand other work in straight strips [8], which further sup-\nports the negligible inertia of DWs in such FiM systems.\nC. Discussion on the effective influence of a curved\nshape on the wall velocity\nA non-uniform current distribution is expected to in-\nfluence the terminal velocity of the DW for a given cur-\nvature, specially for wide curved strips [11]. In this sec-\ntion, to explore further the degree of influence of the\nnon-uniform current, equivalent studies on wandρon\na straight strip with an artificially implemented non-\nuniform J( r=y) atT=TAare performed. A straight\nstrip is a bounding case for a curved strip that shows no\neffective curvature ( re→ ∞ , ρ→0) and an homogeneous\ndensity current J=J0. Therefore, we explore whether7\n(c)(a)\n(d)w = 256 nm re = 384 nm(b)\nre = 384 nm w = 256 nm\nJ = Jx(y, re)\nJx = J 0J = Jx(y, w)\nJx = J 0\nJ = Jφ(r, re)J = Jφ(r, w)\nJφ = J 0\nFIG. 7. DW terminal velocities in a straight strip for an in-\nhomogeneous J(y, w, r e) (blue circles) and for a uniform J0\n(black crosses) as a function of radius re(a) and width w(b)\nat same temperature ( T= 260 K). Insets in (a) show the mag-\nnetic configuration of sublattice i= 1 at t= 0 and schematics\nof the current spatial distribution in the strips as examples.\n(b) DW terminal velocities in a curved strips with the same\nparameters as a function of radius re(c) and width w(d). As\nan example, inset in (c) shows the radial dependence of an in-\nhomogenous current distribution in such a strip. Dotted lines\nhighlight the cases where the two geometrical parameters ( w\nandre) are coincident among all the studies.\n‘curvature ( ρ) effects’ are mainly dominated by the in-\ntrinsic inhomogenous current, or whether they can also\narise from the curved shape itself [24]. We aim to discern\nthe actual influence of an inhomogeneous current, as part\nof an more global effect due to the curved shape. For the\nfollowing study, and since the straight shape must be re-\ntained, re(or equivalently ρ) is artificially modified in the\nnon-uniform current distribution expression: ⃗J(y, w, r e)\n[11] in the x-direction, as if the strip was curved. Note\nthat ρrepresents the inverse of the averaged or effective\ncurvature radius of the strip ( ρ=r−1\ne) and not the cylin-\ndrical radial coordinate ( r) in the system.\nFig. 7(a-b) show the velocity of an UD wall in the\nstraight strip for J=J0= 2.35 TA/m2(black crosses)\nand for an inhomogeneous J(y) (blue circles), varying re\ninJ(y, re) (see insets) for a fixed width w= 256 nm, and\nvarying width ( w) for a fixed re,J(y, re= 384). While it\nis expected that an inhomogeneous J(y, w, r e) will influ-\nence the DW velocity (especially for curved-most strips,\nseere= 134 nm in (a)), the differences with the case of\nJ=J0are almost negligible. Fig. 7(c-d) show resultsfor curved strips. For these cases, wandreare naturally\nvaried in J(r, w, r e) by modifying the shape itself. From\nthe standpoint of the applied current, this is expected\nto be equivalent to doing it by changing the shape itself.\nResults from an inhomogeneous current (blue circles, re-\nproduced from Fig. 3(g) and 5(g)) consistently tend to\nconverge to the straight strip as ρis reduced. The strip\nis straight when ρ= 0 ( re→ ∞ ).\nWhen the strip is either straight (a-b) or curved (c-\nd), for both studies (fixing wand varying reor vice-\nversa), the DW velocity is found to be the same when\nthe geometrical parameters are coincident, i.e., w= 256\nnm and re= 384 nm, as expected (see horizontal dot-\nted lines in (a-b) or (c-d)). However, when the shape\nis different, even in the cases when wandreare coinci-\ndent (and therefore, J(w, re) is expected to also be the\nsame), different DW velocities are obtained (see vertical\ndotted lines in (a-c) and (b-d)) below and slightly above\n2000 m/s, respectively. Moreover, by modifying either re\norwin the curved strip by directly changing its shape\n(see (c-d) and previous sections), there is a clear larger\nimpact on DW velocity, than by artificially (but equiva-\nlently) modifying reorwin the straight strip (see (a-c)).\nIn the curved strips, the trend when varying reorwis\nqualitatively similar regardless to the homogeneity of the\napplied J(see black crosses and blue circles in either (c)\nor (d)). Since J( r) is modeled in an equivalent way in all\nstudies by simply changing J( r) = J( y) for the straight\nstrip, marked differences between the wall velocity in (a-\nb) (straight strip) and (c-d) (curved strip), specially for\nvery curved strips, suggest that not only the non-uniform\nJ(r) is influencing the UD DW motion.\nOur results suggest that the curved shape may have\nan intrinsic influence on the wall velocity, manifested as\na more marked dependence with wandre, regardless of\nthe inhomogeneity of the current density (see Fig.(c-d)),\nas the shape becomes more curved. An influence due to\nthe inhomogeneous current, still appears naturally in the\ncurved strip, but may have a lesser impact compared to\nother geometrical factors (see differences between black\ncrosses and blue circles).\nIV. DISCUSSION AND SUMMARY\nWe have provided a study on DW motion in curved\nFiM strips, particularised to one of the two strongly cou-\npled sub-lattices, for three different temperatures, and as\na function of geometrical parameters for a HM/FiM/Ox\nmultilayer structure. We observe an absence of tilting of\nthe DW and domain distortion at different temperatures,\n40 K above and below the angular momentum compen-\nsation temperature.\nWidth and curvature effects on the DW velocity are\ndiscussed. Besides contributions from a non-uniform\nJ(r), there is an overall significant influence from the\nshape of the strip itself on DW velocity. This implies\nthat, for a fixed temperature (or composition), DW ve-8\nlocity can be optimised by optimising the geometrical\nparameters of the curved strip. The relative differences\nbetween a DU and a UD walls are marginal in general.\nIn other words, geometrical factors affect them almost\nequally, which is positive for a robust transmission of a\nbit encoded in an Up or Down domain between two adja-\ncent DWs. With reducing current, differences in veloci-\nties between curved and straight strips are still minimised\nat the expense of slower DWs. Similar effect is observed\nas width or curvature is increased. This is beneficial for\ndesigning intricate 2D circuit tracks combining curved\nand straight sections, while preserving DW velocities still\nlarger than those found in their FM counterparts. Also,\nDWs in a curved FiM strip show a negligible inertia in\ncontrast to their FM counterparts ( τFiM << τ FM) for\nall the explored scenarios. The DWs start to move and\nstop almost immediately ( τFiM∼0.02 ns) after the ap-\nplication or removal of current.\nConsidering the obtained results altogether and assum-\ningT̸=TA, which will be most of the experimental cases\nat room temperature ( T= 300 K), our study allows us\nto conclude that narrow enough FiM strips are ideal can-\ndidates for designing curved tracks for 2D spintronic cir-\ncuits of an arbitrary shape based on CDDWM, where bits\nare encoded in domains separated by walls. This is dueto very fast rise and fall times ( τr∼τf<<0.1 ns), high\nvelocities ( VDW>1000 m/s) and negligible distortion of\nthe two types of DWs (UD and DU) in all the scenarios\nexplored in this work. Greater DW terminal velocities\nand smaller time responses in curved FiM strips than\nthose in their FM counterparts are obtained. These re-\nsults can help in the further research, development and\nimprovement of FiM-based spintronic circuitry that may\nrequire compactness and high-speed functionality with\nhigh robustness to DW (and/or domain) distortion.\nV. ACKNOWLEDGEMENTS\nThis work was supported by project SA114P20 from\nJunta de Castilla y Leon (JCyL), and partially supported\nby projects SA299P18 from JCyL, MAT2017-87072-C4-\n1-P and PID2020-117024GB-C41 from the Ministry of\nEconomy, Spanish government, and MAGNEFI, from\nthe European Commission (European Union). All data\ncreated during this research are openly available from\nthe University of Salamanca’s institutional repository at\nhttps://gredos.usal.es/handle/10366/138189\n[1] A. Hubert and R. Sch¨ afer, in Magnetic Domains: The\nAnalysis of Magnetic Microstructures (Springer-Verlag\nBerlin Heidelberg, 1998).\n[2] S. S. P. Parkin, “Shiftable magnetic shift register and\nmethod of using the same,” (US6834005B1, 2004).\n[3] S. Parkin and S.-H. 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Mart´ ınez, “Domain wall\nmotion in magnetic nanostrips,” in Materials Science and\nTechnology (American Cancer Society, 2020) pp. 1–49.\n[12] S.-H. Yang, K.-S. Ryu, and S. Parkin, Nature nanotech-\nnology 10(2015), 10.1038/nnano.2014.324.[13] S. A. Siddiqui, J. Han, J. T. Finley, C. A. Ross, and\nL. Liu, Phys. Rev. Lett. 121, 057701 (2018).\n[14] L. Caretta, M. Mann, F. B¨ uttner, K. Ueda, B. Pfau,\nC. G¨ unther, P. Hessing, A. Churikova, C. Klose,\nM. Schneider, D. Engel, C. Marcus, D. Bono,\nK. Bagschik, S. Eisebitt, and G. Beach, Nature Nan-\notechnology 13, 1154 (2018).\n[15] E. Mart´ ınez, V. Raposo, and ´Oscar Alejos, Journal of\nMagnetism and Magnetic Materials 491, 165545 (2019).\n[16] S. Arpaci, V. Lopez-Dominguez, J. Shi, L. S´ anchez-\nTejerina, F. Garesci, C. Wang, X. Yan, V. K. Sang-\nwan, M. A. Grayson, M. C. Hersam, G. Finocchio and\nP. Khalili Amiri, Nature Communications 12(2021),\n10.1038/s41467-021-24237-y.\n[17] L. Landau and E. Lifshitz, Phys. 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Yang, East Asian Journal on Applied Mathematics 7,\n837–851 (2017)."    },    {        "title": "1812.00714v1.Mn2V0_5Co0_5Z__Z__Ga__Al__Heusler_alloys__Fully_compensated_ferrimagnets_with_high_Tc_and_compensation_temperature.pdf",        "content": "1 \n Mn2V0.5Co0.5Z (Z= Ga, Al) Heusler alloys: Fully compensated \nferrimagnets with high T c and compensation temperature  \n \nP V Midhunlal1, J Arout Chelvane2, D Prabhu3, Raghavan Gopalan3, and Harish Kumar N 1* \n1. Department of Physics, Indian Institute of Technology-Madras, Chennai-600036, India.  \n2. Defense Metallurgical Research Laboratory, Kanchanbagh (PO), Hyderabad-500058, India. \n3. International Advanced Research Centre for Powder Metallurgy and New Materials (ARCI),    \n   Chennai-600113, India. \n*nhk@iitm.ac.in \n \nAbstract:   High T C fully compensated ferrimagnets are potential candidates for spin \ntransfer torque based spintronic devices. We report the  structural and magnetic properties of \nhigh T C fully compensated ferrimagnets Mn 2V0.5Co0.5Z (Z=Ga, Al) in the melt-spun ribbon \nand arc melted bulk form. While the parent alloys Mn 2YZ (Y=V, Co; Z= Ga, Al) exhibits a \nmagnetic moment value around 2 f.u., the Mn 2V0.5Co0.5Ga alloy exhibits room \ntemperature nearly fully compensated moment value of 0.09 and 0.13 f.u. in the bulk and \nribbon form respectively. For Mn 2V0.5Co0.5Al this turned out to be 0.04 and 0.08 f.u. In \nContrast to the bulk sample’s Néel P-type ferrimagnetic behaviour, ribbon samples exhibit \nNéel N-type ferrimagnetic characteristic with a high compensation temperature of 420 K for \nZ=Ga and 275 K for Z=Al. The observed T C values are more than 640 K for all samples. The \ndifferences in the magnetic properties of arc melted and melt-spun alloys indicates that even a \nslight variation in stoichiometry and sample preparation method can influence the physical \nproperties of a compensated system. \n1. Introduction \n     Half-metallic antiferromagnets (HMAFs) or more precisely half-metallic fully \ncompensated ferrimagnets (HMFCFis) are a new class of magnetic materials which exhibits \nzero macroscopic moment with high spin polarization at the Fermi level. These materials \ndiffer from the conventional antiferromagnets in such a way that the different magnetic sub-\nlattices are chemically or crystallographically inequivalent and the compensation occurs in a \nwide range of temperature. Also, the characteristic temperature is the Curie temperature (T C) \nand not the Néel temperature as in the case of conventional antiferromagnets 1. This 2 \n interesting class of materials are expected to be utilized as tips in Spin-polarized Scanning \nTunneling Microscopes (SP-STM) and electrode material in Spin Torque Transfer (STT) \nbased Magnetic Tunnel Junctions (MTJs) which would be the building blocks of future \nMagnetic Random Access Memory (MRAM)2,3. Among the various HMAFs which includes \ncertain double perovskites, dilute magnetic semiconductors and superlattices 4,5, Heusler \nalloys have got much interest due to their magnetic moment tunability and high T C. The \nSlater-Pauling relation which expresses the total magnetic moment (M t) to the total number \nof valence electron per unit cell (Z t) can be utilized to design HMFCFis by varying the \nvalence electron number. As per the rule, M t= Zt-24 for full-Heusler alloys (general formula \nX2YZ, X & Y are transition metal and Z is a main group element) and M t= Zt-18 for half-\nHeusler alloys (chemical formula XYZ), zero magnetic moment state is expected for full-\nHeusler alloys with 24 valence electrons and half-Heusler alloys with 18 valence electrons6. \nThe ab initio studies carried out by I.Galanakis et al. have shown a way of achieving half-\nmetallic fully compensated ferrimagnetism in Mn 2VAl and Mn 2VSi Heusler alloys by \nsubstituting Co at the Mn site7. Motivated by this interesting theoretical observation, \nresearchers have carried out experimental investigation on the structural and magnetic \nproperties of (Mn 2-xCox)VZ (Z = Ga, Al) Heusler alloys8,9. Even though the compensation \nwas achieved in these alloys, the T C has drastically come down from more than 700 K to \nbelow the room temperature as the compensation point approaches (x=1) which is a \ndrawback as far as the applications are concerned. At the same time, our recent investigation \non the Co substitution at the V site of Mn 2VZ (Z= Ga, Al) alloys could sort out the issue of \ndecreasing T C. Mn2V1-xCoxZ (Z = Ga, Al) alloys exhibited near zero moment state when \nx=0.5 by preserving high T C (more than 650 K)10. Here in this report we investigate the \nstructural and magnetic properties of high T C fully compensated ferrimagnetic Mn 2V0.5Co0.5Z \n(Z = Ga, Al) ribbons with high compensation temperature. The properties are compared with \nthat of the bulk samples and the interesting deviations in the compensation behaviour are \ndiscussed. \n2. Experimental details \n    Mn2V0.5Co0.5Z (Z = Ga, Al) bulk samples were prepared by arc melting the individual \nelements with high purity. Initially, titanium was melted several times to absorb any leftover \noxygen in the chamber. Samples were melted several times after flipping to ensure \nhomogeneity. Samples were sealed in an evacuated quartz tube for annealing. \nMn2V0.5Co0.5Ga was annealed at 1073 K and Mn 2V0.5Co0.5Al was annealed at 673 K for three 3 \n \ndays followed by furnace cooling. Mn 2V0.5Co0.5Z (Z = Ga, Al) ribbon samples were prepared \nfrom the arc melted ingots by vacuum induction melting followed by melt spinning method. \nThe molten alloys were ejected over rotating copper wheel rotating at 1000 rpm. Flakes with \n30 - 40 µm thickness and approximate length of 10 mm and breadth of 2 mm were obtained. \nThe structural characterization was carried out using Rigaku SmartLab high-resolution X-ray \ndiffractometer with Cu-K α radiation. The compositions and microscopic images of the \nribbons were recorded using FEI-InspectF Scanning Electron Microscope (SEM). The \nmagnetic measurements in the temperature range 5- 300 K were carried out using Quantum \nDesign MPMS 3 SQUID VSM and high-temperature magnetic measurements (300–850 K) \nwere carried out using Microsense Vibrating Sample Magnetometer, Model EZ9. \n3. Results & Discussion \n3.1 Structural properties \n       Fig. 1(a) - (f) shows the cross-sectional and surface SEM images of the ribbon samples. \nSurfaces morphology appeared to be different for the two sides for both the alloys (one side is \nsmooth and the other side is rough). This is expected as one of the surfaces touches the wheel \nand the other surface exposed to the inert atmosphere during the melt spinning technique.  \n \n \n \n \n \n \n \n \n Fig. 1(a) Cross-sectional SEM images of Mn 2V0.5Co0.5Ga and (b) Mn 2V0.5Co0.5Al ribbons. \n(c) & (d) surface images (both sides) of Mn 2V0.5Co0.5Ga and (e) & (f) Mn 2V0.5Co0.5Al \nribbons. 4 \n \nFig. 2(a) and (b) shows the X-ray diffraction (XRD) patterns of Mn 2V0.5Co0.5Z ribbon and \nbulk samples. There was not much difference in the XRD pattern of bulk and corresponding \nribbon samples. The (111) and (200) Superlattice reflections were absent in the case of Z=Ga \nsamples while Z=Al samples exhibited these reflections with low intensity. The absence/low \nintensity could be due to the similar atomic scattering factors of x-ray for the atoms which are \nin the same period in the periodic table. Here it is to be noted that as per the earlier reports, \nthe parent Mn 2VGa alloy has not exhibited any superlattice reflection in the XRD pattern and \nthe neutron diffraction pattern has shown the superlattice peaks with huge intensity11,12. It is \nreported that Mn 2VZ alloys crystallize in the cubic L21 structure (space group:225, Fm𝟑ഥm) \nand Mn 2CoZ alloys crystallize in the cubic Xa structure (space group:216, F𝟒ഥ3m)12–14 . As the \nparent alloys/end members possess different crystal structure, it is difficult to predict the \ncrystal structure of Mn 2V0.5Co0.5Z samples. Bearing this in mind, Rietveld refinement of the \nXRD patterns were carried out by using  FullProf software15. assuming both L21 and Xa \nstructures for all samples. Interestingly both the structures have given a similar fit with nearly \nthe same lattice parameter and χ2 values. To have a better understanding, the XRD patterns \nwere simulated for both the structures and it was evident that the patterns are \nindistinguishable. Since the lattice parameters are same for both the structures, refined pattern \nassuming Xa structure is shown in Fig. 2 (a) and (b). The obtained lattice parameters are  \n5.872 and 5.857 Å for Z=Ga and Al ribbon respectively which are close to the bulk values. \nThe estimated composition, lattice parameter and other magnetic parameters are shown in \ntable I. \n Fig. 2(a) Refined X-ray diffraction patterns of Mn 2V0.5Co0.5Z ribbon samples and (b) Bulk \nsamples. 5 \n  \n3.2 Magnetic properties \n       Our earlier report has investigated the detailed magnetic properties of the Mn 2V1-xCoxZ \n(Z = Ga, Al) bulk alloys. The obtained moment values for the end members in the series are \n1.80, 2.05, 1.83 and 2.06 f.u. for Mn 2VGa, Mn 2CoGa, Mn 2VAl and Mn 2CoAl \nrespectively10. Here we focus on the x=0.5 member in the bulk and ribbon forms, which \nexhibits total magnetic moment compensation with high T C. The isothermal magnetization \ncurves at 5 K and 300 K for the ribbons and bulk samples with x=0.5 are shown in Fig. 3 (a)-\n(d). The nearly compensated magnetic moment values obtained through Honda plots of M-H \ncurves at 5 K are 0.23 and 0.10 f.u. for Z= Ga and Al ribbon respectively. This is slightly \nhigher than the magnetization values of the corresponding bulk samples which are 0.1 and \n0.06 f.u. The magnetic parameters are shown in table I. The moment was found to \ndecrease with increase in temperature for the ribbon samples and an increment was observed \nfor the bulk samples as shown in Fig. 3. Moment values at 300 K are 0.09 and 0.04 f.u. for \nZ= Ga and Al ribbons respectively. This indicates that even though magnetic moment \ncompensation happens in both ribbon and bulk samples, the effect of temperature on the \nmagnetic properties are different. This is more evident from the temperature variation of \nmagnetization (M-T) measured in the range 5- 850 K as shown in Fig. 4 (a)-(d). Since the \nlow-temperature M-T curves (5-300 K) and high-temperature M-T curves (300-850 K) are \nrecorded using different instruments, combined normalized curves are shown in the figures. Alloy Type EDS Composition Lattice \nparameter  \n( Å  ) M (f.u.) \nTC \n(K) 5 K 300 K \nMn2V0.5Co0.5Ga Ribbon Mn 2.09V0.51Co0.52Ga0.87 5.872 0.23 0.09 672 \nBulk Mn 2.08V0.52Co0.47Ga0.91 5.878 0.10 0.13 706 \nMn2V0.5Co0.5Al Ribbon Mn 2.03V0.51Co0.52Al0.91 5.857 0.10 0.04 641 \nBulk Mn 2.02V0.48Co0.50Al1.00 5.825 0.06 0.08 659 \nTable I: Structural and magnetic parameters of Mn 2V0.5Co0.5Z alloys in the ribbon and bulk forms   6 \n \nThe low temperature M-T curves measured at 100 Oe field and in the range 5-300 K are also \nshown in the insets of Fig. 4 (a)-(d).  \n \n \n \n \n \n \n Fig. 3(a) isothermal magnetization curves of Mn 2V0.5Co0.5Ga and (b) \nMn2V0.5Co0.5Al ribbons measured at 5 and 300 K. (c) & (d) isothermal \nmagnetization curves of Mn 2V0.5Co0.5Ga and Mn 2V0.5Co0.5Al bulk samples. 7 \n \n \n \nThe M-T curves for the ribbons show different behaviour compared to the bulk samples. For \nthe ribbons, magnetic moment was found to decrease with increase in temperature from a \nnon-zero value and nearly full compensation occurs around 420 K for Z=Ga and 275 K  for \nZ=Al. This behaviour is in agreement with the decrease in moment observed in the M-H \ncurves recorded at different temperatures. By increasing the temperature further, \nferrimagnetic to paramagnetic transition has been occurred and the T C values were found to \nbe 672 K and 641 K for Z=Ga and Z= Al ribbons respectively. This is slightly less compared \nto the bulk values of 706 K and 659 K for Z=Ga and Z=Al respectively. The M-T curves \nclearly show the presence of a full compensation temperature which was absent in the case of \nbulk samples. A similar behaviour is observed in the compensated ferrimagnet Mn 1.5FeV0.5Al Fig. 4(a) M-T curves of Mn 2V0.5Co0.5Ga and (b) Mn 2V0.5Co0.5Al ribbons \nmeasured at 100 Oe field (c) & (d) M-T curves of Mn 2V0.5Co0.5Ga and \nMn2V0.5Co0.5Al bulk samples. Insets shows the low temperature ZFC and FC \nM-T curves.  8 \n (bulk arc melted sample)16,17. It is to be noted that the compensation temperature for \nMn1.5FeV0.5Al bulk alloy was 127 K (which has been tuned up to 226 K by varying the \nstoichiometry) which is much lower than the compensation temperature of Mn 2V0.5Co0.5Ga \nribbon reported in this paper (420 K). In the earlier reported Mn 1.5FeV0.5Al system, three \ndifferent kinds of M-T behaviour was observed, when the stoichiometry was varied. The first \ncase is the fully compensated ferrimagnetic state where a zero moment is observed near 0 K \nand it is maintained up to a certain temperature (50 K in the case of Mn 1.5FeV0.5Al) followed \nby an increase in magnetization and then transition from ferrimagnetic state to paramagnetic \nstate. The authors could also simulate this M-T curve using a molecular field model for a two \nsub-lattice ferrimagnet. The second case is the overcompensated ferrimagnetic state where \nthe compensation temperature was shifted to a certain temperature (308 K) by choosing an \nappropriate stoichiometry for the parent alloy. Here the magnetization is non-zero near 0 K \nand then decreases with increase in temperature, reaching a fully compensated state and then \nfollow similar behaviour as in the first case. This is Néel N-type ferrimagnetic behaviour. In \nthe third case which is known as the Néel P-type ferrimagnetic behaviour, full compensation \ndoes not occur in the entire temperature range and the moment keeps on increasing with the \nincrease in temperature18. Now comparing these three M-T characteristics, it is clear that the \nMn2V0.5Co0.5Z ribbon samples fall in the second category which is Néel N-type ferrimagnet. \nBut the decrease in the magnetization up to the full compensation temperature is not as fast as \nthat of Mn 1.5FeV0.5Al alloy indicating different exchange coupling strength for the different \nmagnetic sublattices. As far as the bulk samples are concerned, a full compensation point was \nnot observed as in the case of ribbon samples. An increase in the magnetic moment was \nobserved till the T C indicating that the bulk samples could be Néel P-type ferrimagnets. The \nTC was found to be 706 K and 659 K for Z=Ga and Al respectively. It is to be noted that even \nthough the moment is varying with the temperature, the magnitudes show that the samples \nare in almost a near compensated state at least up to 420 K for the ribbon and 300 K for the \nbulk samples. It is expected that a slight variation in the composition could highly affect the \nmagnetic sub-lattices ordering of a fully compensated system which would give a different \nmagnetic response with the temperature as in the case of Mn 1.5FeV0.5Al 17. In addition to this, \nthe sudden quenching of molten liquid would have affected the magnetic sub-lattices ordering \nof ribbon samples (quenching rate is around 104 K/s). Mn 2V0.5Co0.5Ga ribbon and bulk \nsamples exhibit a negligibly small transition around 50 K. Unlike the Z=Ga samples, Z=Al \nsamples (ribbon and bulk) has not shown any transition in the low-temperature regime 9 \n indicating that the presence of small second magnetic phase is not influencing the \ntemperature dependent compensation in Z=Ga sample.  \n4 Conclusions \n    The magnetic properties of fully compensated ferrimagnets Mn 2V0.5Co0.5Z (Z=Ga, Al) \nshow distinctly different magnetic properties in their ribbon and bulk form. While the ribbon \nsamples exhibit Néel N-type ferrimagnetic behaviour with high compensation temperature, \nthe bulk samples exhibit Néel P-type ferrimagnetic behaviour without any full compensation \ntemperature. Even though there exists a temperature dependent moment variation in the \nsamples, an overall nearly fully compensated state is preserved at least up to 420 K for the \nribbons and 300 K for the bulk samples. These materials having zero moment, high T C and \nwide temperature range of compensation would be attractive for the future spintronic devices \nutilizing fully compensated ferrimagnets.       \n \nReferences \n1 S. Wurmehl, H.C. Kandpal, G.H. Fecher, and C. Felser, J. Phys. Condens. Matter 18, 6171 \n(2006). \n2 R.A. de Groot, Phys. B Phys. Condens. Matter 172, 45 (1991). \n3 B. Balke, G.H. Fecher, J. Winterlik, and C. Felser, Appl. Phys. Lett. 90, 1 (2007). \n4 W.E. Pickett, Phys. Rev. B - Condens. Matter Mater. Phys. 57, 10613 (1998). \n5 X. Hu, Adv. Mater. 24, 294 (2012). \n6 I. Galanakis, P.H. Dederichs, and N. Papanikolaou, Phys. Rev. B - Condens. Matter Mater. \nPhys. 66, 1 (2002). \n7 I. Galanakis, K. Özdoǧan, E. Şaşioǧlu, and B. Aktaş, Phys. Rev. B - Condens. Matter Mater. \nPhys. 75, 3 (2007). \n8 K. Ramesh Kumar, J. Arout Chelvane, G. Markandeyulu, S.K. Malik, and N. Harish \nKumar, Solid State Commun. 150, 70 (2010). \n9 B. Deka, A. Srinivasan, R.K. Singh, B.S.D.C.S. Varaprasad, Y.K. Takahashi, and K. Hono, \nJ. Alloys Compd. 662, 510 (2016). \n10 P. V Midhunlal, J.A. Chelvane, U.M.A. Krishnan, D. Prabhu, R. Gopalan, and N.H. \nKumar, (2018). \n11 K. Ramesh Kumar, N. Harish Kumar, G. Markandeyulu, J.A. Chelvane, V. Neu, and P.D. 10 \n Babu, J. Magn. Magn. Mater. 320, 2737 (2008). \n12 K.R. Kumar, N.H. Kumar, P.D. Babu, S. Venkatesh, and S. Ramakrishnan, J. Phys. \nCondens. Matter 24, (2012). \n13 C. Jiang, M. Venkatesan, and J.M.D. Coey, Solid State Commun. 118, 513 (2001). \n14 G.D. Liu, X.F. Dai, H.Y. Liu, J.L. Chen, Y.X. Li, G. Xiao, and G.H. Wu, 1 (2008). \n15 J. Rodríguez-Carvajal, Phys. B Condens. Matter 192, 55 (1993). \n16 R. Stinshoff, A.K. Nayak, G.H. Fecher, B. Balke, S. Ouardi, Y. Skourski, T. Nakamura, \nand C. Felser, Phys. Rev. B 95, (2017). \n17 R. Stinshoff, G.H. Fecher, S. Chadov, A.K. Nayak, B. Balke, S. Ouardi, T. Nakamura, and \nC. Felser, AIP Adv. 7, (2017). \n18 L. Neel, Science, 174, 985 (1971).  \n \n \n \n "    },    {        "title": "2402.04719v2.Quantum_Theory_of_Spin_Transfer_and_Spin_Pumping_in_Collinear_Antiferromagnets_and_Ferrimagnets.pdf",        "content": "Quantum Theory of Spin-Transfer and Spin-Pumping in Collinear Antiferromagnets\nand Ferrimagnets\nHans Gløckner Giil and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: April 12, 2024)\nAntiferromagnets are promising candidates as active components in spintronic applications. They\nshare features with ferrimagnets in that opposing spin orientations exist in two or more sublattices.\nSpin transfer torque and spin pumping are essential ingredients in antiferromagnetic and ferrimag-\nnet spintronics. This paper develops an out-of-equilibrium quantum theory of the spin dynamics of\ncollinear magnets containing many spins coupled to normal metal reservoirs. At equilibrium, the\nspins are parallel or antiparallel to the easy axis. The theory, therefore, covers collinear antiferro-\nmagnets and ferrimagnets. We focus on the resulting semi-classical spin dynamics. The dissipation\nin the spin dynamics is enhanced due to spin-pumping. Spin accumulations in the normal metals\ninduce deterministic spin-transfer torques on the magnet. Additionally, each electron’s discrete spin\nangular momentum causes stochastic fluctuating torques on the antiferromagnet or ferrimagnet. We\nderive these fluctuating torques. The fluctuation-dissipation theorem holds at high temperatures,\nincluding the effects of spin-pumping. At low temperatures, we derive shot noise contributions to\nthe fluctuations.\nI. INTRODUCTION\nSpin transfer torque (STT) and spin pumping (SP)\nare essential ingredients in the generation and detection\nof spin currents and are central components in modern\nspintronics research and devices [1]. The use of mag-\nnetic insulators enables signal propagation without mov-\ning charges and could provide low-dissipation and ultra-\nfast memory devices [2]. Initially, much of spintronic\nresearch focused on the study of STT [3–5] and SP [6–\n8] in ferromagnets (FMs). Subsequently, this included\nalso works on fluctuations [9–11] and pumped magnon\ncondensates [12–14].\nUnlike FMs, whose macroscopically apparent magnetic\nproperties have been known for thousands of years, anti-\nferromagnets (AFMs) carry zero net magnetic moments\nand were elusive for some time. Even after their dis-\ncovery, AFMs were believed to have few potential ap-\nplications [15] and were disregarded in the early days\nof spintronics research. Recent theoretical and experi-\nmental findings have highlighted the potential of using\nAFMs in spintronics applications, thus starting the field\nof antiferromagnetic spintronics. Key discoveries were\nthe robustness of AFMs to external magnetic perturba-\ntions and the high resonance frequency of antiferromag-\nnetic material [16, 17]. The prediction [18] and subse-\nquent experimental detection [19] of an STT in AFMs\nsparked a massive interest in using AFMs as the active\ncomponent in spintronics devices [16, 20]. Moreover, it\nwas predicted that contrary to what was believed, anti-\nferromagnets are as efficient in pumping spin currents as\nFMs [21]. This effect was later experimentally detected in\nthe easy-axis AFM MnF 2[22]. These discoveries opened\nup the possibility of utilizing AFMs in spintronic applica-\ntions, enabling the possible fabrication of stray-field-free\ndevices operating in the THz-regime [16, 23], allowing for\nmuch faster device operation than in FMs.In recent years, the spin dynamics in AFMs have been\nexplored extensively, including the effects of disorder [24],\ngeneration of spin-Hall voltages [25], and the proper-\nties of antiferromagnetic skyrmions [26]. The spin dy-\nnamics of ferrimagnetic materials have also been stud-\nied [27]. Phenomenological models of intra- and cross-\nlattice torques were introduced in [28]. Ref. 29 fur-\nther discusses the competition between intra and cross-\nsublattice spin pumping in specific models of antiferro-\nmagnets.\nAs in antiferromagnets, ferrimagnets have opposing\nmagnetic moments. However, these moments have differ-\nent magnitudes, resulting in a net magnetization. These\nfeatures result in rich spin dynamics ranging from be-\nhavior reminiscent of antiferromagnets to ferromagnets.\nA prime example of a ferrimagnet is yttrium-iron-garnet\n(YIG). The low-energy magnon modes in YIG resemble\nmodes in ferromagnets.\nIn the study of non-equilibrium effects, the Keldysh\npath integral approach to non-equilibrium quantum field\ntheory is a powerful tool in the study of non-equilibrium\nsystems beyond linear response [30, 31]. Although most\nof the research on STT and SP utilized a semiclassical\napproach, some works have used the Keldysh framework\nin the study of spin dynamics in FMs out of equilib-\nrium [10, 11, 32–34]. Moreover, the Keldysh method was\nrecently used to formalize a fully quantum mechanical\ntheory of STT and SP, including the effects of quan-\ntum fluctuations [35]. These fluctuations have become\nincreasingly relevant with the development of new devices\noperating in the low-temperature regime. Nevertheless,\napplying the Keldysh method to derive microscopic re-\nlations for SP, STT, and fluctuating torques in an AFM\nsystem is lacking.\nIn this paper, we extend the approach of Ref. 35,\nwhich examined a ferromagnet in the macrospin approx-\nimation coupled to normal metals featuring spin andarXiv:2402.04719v2  [cond-mat.mes-hall]  11 Apr 20242\ncharge accumulation, to a similar system but instead\nfeaturing a collinear magnet with many individual spins\ncoupled at different sublattices to normal metals. Our\nstudy thus covers antiferromagnets, ferrimagnets, and\nferromagnets. We derive the spin dynamics using a\nfully quantum mechanical Keldysh non-equilibrium ap-\nproach. We find expressions for the spin transfer torque,\nspin-pumping-induced Gilbert damping, and fluctuat-\ning fields, including low-temperature shot-noise contri-\nbutions. The Gilbert damping and fluctuations contain\nboth inter-lattice and intra-lattice terms. Using Onsager\nreciprocal relations, we relate the spin pumping and spin\ntransfer coefficients. Our results enhance the knowledge\nof the microscopic expressions of STT and SP and fluctu-\nating torques in antiferromagnets and ferrimagnets cou-\npled to normal metals in the low-energy regime, where\nquantum fluctuations become essential.\nThe subsequent sections of this paper are structured\nas follows. In Sec. II, we introduce the model employed\nfor the itinerant electrons in the normal metals, the lo-\ncalized magnetic moments in the antiferromagnetic or\nferrimagnet, and the electron-magnon coupling between\nthem. We then present the key findings of this paper\nin Sec. III, including microscopic definitions of the spin\ntransfer torque, spin pumping, and fluctuating torques\nin many spin magnets, being an antiferromagnetic, fer-\nrimagnet, or ferromagnet. The derivation of an effective\nmagnon action, achieved by integrating fermionic degrees\nof freedom resulting from the interaction with normal\nmetals, is detailed in Sec. IV. The evaluation of this ef-\nfective action is then provided in Sec. V. Finally, Sec. VI\nconcludes the paper.\nII. MODEL\nWe consider a bipartite collinear magnet coupled to\nan arbitrary number of normal metal reservoirs. The\nmagnet can represent an antiferromagnet, a ferrimagnet,\nor a ferromagnet. The total Hamiltonian is\nˆH=ˆHe+ˆHem+ˆHm (1)\nin terms of the Hamiltonian describing the electrons in\nthe normal metal ˆHe, the Hamiltonian describing the in-\nteraction between the electrons and the magnet ˆHem, and\nthe Hamiltonian of the magnet ˆHm.\nThe Hamiltonian of the electrons combined with the\nHamiltonian representing the interaction between the\nelectrons and the magnet is\nˆHe+ˆHem=Z\ndrˆψ†\"\nHe+ℏ−1X\niuiσ·ˆSi#\nˆψ , (2)\nwhere ˆψ†= ( ˆψ†\n↑,ˆψ†\n↓) is the spatially dependent 2-\ncomponent itinerant electron field operator, and σis the\nvector of Pauli matrices in the 2 ×2 spin space. In the\nHamiltonian (2), ui(r) represents the spatially depen-\ndent exchange interaction between the localized spin atsiteiand the itinerant electrons. This interaction is lo-\ncalized around spin iinside the magnet. The sum over\nthe localized spins iconsists of a sum over sites in sublat-\nticeAand sublattice B, i.e.,P\ni. . .→P\na. . .+P\nb. . ..\nThe localized spin operator ˆSihas a total spin angular\nmomentum Si=ℏp\nsi(si+ 1) where siis the (unitless)\nspin quantum number of the localized spin, such that\nˆSi2=ℏ2si(si+ 1). For large si, the difference between\nSi/ℏandsiis a first-order correction, and we can ap-\nproximate Si≈ℏsi.\nThe spin-independent part of the single-particle elec-\ntron Hamiltonian is\nHe=−ℏ2\n2m∇2+Vc, (3)\nwhere Vcis the spatially dependent charge potential.\nIn the classical limit of the magnet, the spins at sublat-\nticeAare along a certain direction and the spins at sub-\nlattice Bare along the opposite direction in the ground\nstate. We will consider the semiclassical spin dynam-\nics near the instantaneous classical direction of the spins\nthat we let be along the zdirection and adiabatically\nadjust the evolution of the small deviation [10, 11, 35].\nIn the following, it is constructive to expand the in-\nteraction term to the second order in the magnet cre-\nation/annihilation operators using a Holstein-Primakoff\ntransformation,\nˆHem=ˆH0+ˆH1+ˆH2, (4)\nwhere ˆH0is the interaction with the classical magnetic\nground state and ˆH1(ˆH2) is the interaction term to the\nfirst (second) order. The classical ground state contribu-\ntion to the interaction is then\nˆH0=Z\ndrˆψ†Vsσzˆψ , (5)\nwhere the magnitude of the spatially dependent spin po-\ntential experienced by the itinerant electrons is\nVs(r) =X\nasaua(r)−X\nbsbub(r), (6)\nand oscillates rapidly with the staggered field.\nIn the macrospin approximation,P\niSican be treated\nas a giant spin in ferromagnets. Then, ui(r) becomes\nthe effective exchange interaction. Ref. 35 shows how the\nelectronic Hamiltonian ˆHecombined with the electron-\nmagnon Hamiltonian to zeroth order ˆH0become partic-\nularly transparent in ferromagnet-normal metal systems\nin terms of the scattering states of the itinerant electrons\nfor the macrospin dynamics. We generalize this approach\nto magnet-normal metal systems with individual local-\nized spins. In this picture, the electronic Hamiltonian\nremains simple, as in Ref. 35:\nˆHe+ˆH0=X\nsαϵαˆc†\nsαˆcsα, (7)3\nwhere ˆ csαannihilates an electron with spin s(s=↑or\ns=↓). The quantum number α=κnϵcaptures the\nleadκ, the transverse waveguide mode n, and the elec-\ntron energy ϵ. The electron energy consists of a trans-\nverse contribution ϵnand a longitudinal contribution\nϵ(k) =k2/2m, where kis the longitudinal momentum,\nsuch that ϵ=ϵn+ϵ(k). The eigenenergy is spin degener-\nate, since the leads are paramagnetic. Furthermore, we\nconsider identical leads such that the eigenenergy is in-\ndependent of the lead index. The system setup is shown\nL R N N AF\nFIG. 1. An antiferromagnet (AF) with conductors (N) on\neither side connected to a right (R) and a left (L) lead.\nin Fig. 1 for the case of two leads. In Eq. (7) and similar\nexpressions to follow, the sum over the scattering states\nimplies thatP\nαXsα=P\nκnR∞\nϵndϵXsκn(ϵ). In the scat-\ntering approach, the field operator is\nˆψs=X\nαˆcsαψsα, (8)\nwhere ψsα(r) is the wave function of a scattering state\nof spin sand quantum number α.\nThe Hamiltonian of the antiferromagnet is [ idenotes\na site at sublattice A(i=a) orB(i=b)]\nHm=ℏ−2X\nijJijˆSi·ˆSj−Kℏ−2X\ni\u0010\nˆSi·z\u00112\n+γµ0X\naHA\na·ˆSa+γµ0X\nbHB\nb·ˆSb, (9)\nwhere Jijis the symmetric exchange interaction, K > 0\nis the easy-axis anisotropy energy, and γ=g∗µB/ℏis\nthe (absolute value of) the effective gyromagnetic ratio,\nwhere g∗is the effective Land´ e g-factor and µBis the\nBohr magneton. In Eq. (9), HA,B\niis the external mag-\nnetic field in units of Am−1at lattice site i={a, b}, and\nµ0is the vacuum permeability, which appears because\nwe are employing SI units. In reality, HA=HBin the\npresence of a uniform external magnetic field. However,\nto illustrate and understand the physics, we allow the\nexternal fields at sublattices AandBto differ, and to\ndepend on the lattice site.\nWe consider the low-energy excitations from the semi-\nclassical ground state of the staggered spin orientation.\nTo this end, we carry out a Holstein-Primakoff expansion\nto the second order in magnon excitations at each sub-\nlattice AandBdescribed via the annihilation operators\nˆaaandˆbbas detailed in Appendix A. Introducing theraising/lowering fields as H±=Hx±iHy, the magnon\nHamiltonian becomes\nHm=E0+X\naEA\naˆa†\naˆaa+X\nbEB\nbˆb†\nbˆbb\n+ 2X\naa′Jaa′√sasa′ˆa†\naˆaa′+ 2X\nbb′Jbb′√sbsb′ˆb†\nbˆbb′\n+ 2X\nabJab√sasb[ˆaaˆbb+ ˆa†\naˆb†\nb]\n+γµ0ℏX\narsa\n2[HA\na−ˆaa+HA\na+ˆa†\na]\n+γµ0ℏX\nbrsb\n2[HB\nb−ˆb†\nb+HB\nb+ˆbb], (10)\nwhere the classical ground state energy E0is\nE0=X\naa′sasa′Jaa′+X\nbb′sbsb′Jbb′−2X\nabsasbJab\n−2KX\nis2\ni+ℏµ0X\nasaHA\naz−ℏµ0X\nbsbHB\nbz,(11)\nand is disregarded in the following.\nEA(B)\na(b)= 2X\nb(a)sb(a)Jab−X\na′(b′)sa′(b′)Ja(b)a′(b′)\n+ 2sa(b)K∓ℏγµ0HA(B)\na(b)z(12)\nis the energy of a local excitation, where the upper sign\nholds for sites on sublattice Aand the lower sign holds\nfor sites on sublattice B.\nIn the scattering basis of the electronic states, the cor-\nrections to the antiferromagnetic ground state electron-\nmagnon interaction to quadratic order in the magnet op-\nerators becomes ˆHem−ˆH0=ˆH1+ˆH2. The first-order\ncontribution of electron-magnon interaction is\nˆH1=X\naαβr\n2\nsah\nˆaaˆc†\n↓αWαβ\na↓↑ˆc↑β+ ˆa†\naˆc†\n↑αWαβ\na↑↓ˆc↓βi\n+X\nbαβr\n2\nsbh\nˆb†\nbˆc†\n↓αWαβ\nb↓↑ˆc↑β+ˆbbˆc†\n↑αWαβ\nb↑↓ˆc↓βi\n,(13)\nand describes the spin-flip scattering of the itinerant elec-\ntrons associated with creating or annihilating localized\nmagnons. The dimensionless matrix Wiis governed by\nthe exchange potential ui(r) and the scattering states\nwave functions ψsα:\nWαβ\niss‘=Z\ndrψ∗\nsα(r)siui(r)ψs‘β(r), (14)\nand is Hermitian, Wαβ\ni↑↓= [Wβα\ni↓↑]∗. The electron-magnon\ninteraction that is second order in the magnon operators4\nis\nˆH2=−X\naαβˆa†\naˆaa\nsah\nˆc†\n↑αWαβ\na↑↑ˆc↑β−ˆc†\n↓αWαβ\na↓↓ˆc↓βi\n+X\nbαβˆb†\nbˆbb\nsbh\nˆc†\n↑αWαβ\nb↑↑ˆc↑β−ˆc†\n↓αWαβ\nb↓↓ˆc↓βi\n,(15)\nwhere the matrix elements are defined in Eq. (14). We\nnote that our electron-magnon-interaction is isotropic in\nspin space and will give rise to zeroth-, first-, and second-\norder magnon terms in the Hamiltonian, i.e. ˆH0,ˆH1and\nˆH2, respectively. This is in contrast to the model used\nin Refs. 12 and 32, where only the first-order term ˆH1is\nconsidered.\nFinally, in normal metal reservoirs, the occupation of\nthe state is\n⟨c†\ns′αcsβ⟩=δαβnss′α, (16)\nwhere the 2 ×2 out-of-equilibrium distribution is\nnss′α=1\n2[fκ↑(ϵα)) +fκ↓(ϵα))]δss′\n+1\n2[fκ↑(ϵα))−fκ↓(ϵα))]uκ·σss′, (17)\nallowing for a (lead-dependent) spin accumulation in the\ndirection of the unit vector uk.f↑andf↓are general\ndistribution functions for spin-up and spin-down parti-\ncles, which generally differ for elastic or inelastic trans-\nport [35]. In equilibrium, the distribution function only\ndepends on energy,\nfeq\nκ↑(ϵ) =feq\nκ↓(ϵ) =f(ϵ−µ0), (18)\nwhere fis the equilibrium Fermi-Dirac distribution and\nµ0is the equilibrium chemical potential.\nIn inelastic transport, the spin- and charge accumula-\ntions µCandµScorrespond to chemical potential in a\n(spin-dependent) Fermi-Dirac function,\nfin\nκ↑(ϵ) =f(ϵ−µ0−µC\nκ−µS\nκ/2) (19a)\nfin\nκ↓(ϵ) =f(ϵ−µ0−µC\nκ+µS\nκ/2). (19b)\nFor notational simplicity, we define the chemical poten-\ntials\nµκ↑=µ0+µC\nκ+µS\nκ/2 (20a)\nµκ↓=µ0+µC\nκ−µS\nκ/2. (20b)\nIn the limit of small charge and spin accumulations com-\npared to the Fermi level, it can be derived that\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfin\nκ↑(ϵ)−f(ϵ)\u0003\n. (21)\nIn the elastic regime, the distribution function cannot\ngenerally be described as a Fermi-Dirac function. Thedistribution function is instead given as a linear combi-\nnation of Fermi-Dirac functions in the connected reser-\nvoirs [35],\nfel\nsκ(ϵ) =X\nlRsκlf(ϵ−µl), (22)\nwhere the index lruns over the reservoirs, and Rsκlis the\nlead and spin-dependent transport coefficient for reser-\nvoirl. The transport coefficients satisfy\nX\nlRsκl= 1. (23)\nIn the elastic transport regime, it is advantageous to de-\nfine the effective charge and spin accumulations through\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfel\nκ↑(ϵ)−f(ϵ)\u0003\n. (24)\nThe elastic and inelastic transport regime results in dif-\nferent results for the fluctuations in the magnetization\ndynamics of the magnet.\nHaving specified the model for the system in consider-\nation, we proceed by presenting the main results of the\npaper.\nIII. MAIN RESULTS: EQUATIONS OF MOTION\nThis section presents the main results of our work.\nOur primary result is the derivation of a Lan-\ndau–Lifshitz–Gilbert–Slonczewski (LLGS) equation for\nthe localized (normalized) spins mi=Si/Siin a gen-\neral magnet coupled to normal metal reservoirs,\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni (25)\nvalid for low-energy excitations when the equilibrium\nmagnetization is parallel (antiparallel) to the z-axis. The\nbulk antiferromagnet torque τb\nifor a site i={a, b}arises\nfrom contributions of anisotropy, exchange coupling, and\nexternal fields, and reads\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n. (26)\nwhere Eiis the energy of a local excitation and Hiis\nthe applied field. Hence, the bulk torque remains unaf-\nfected by the presence of normal metal reservoirs and the\nassociated spin- and charge accumulations.\nThe spin transfer torque τstt\niis induced by spin accu-\nmulation in the normal metals, and can be expressed as\nfollows:\nτstt\ni=ℏ−1X\nκ\u0002\nβI\niκz×µS\nκ−βR\niκz×(z×µS\nκ)\u0003\n.(27)\nIn Eq. (27), the superscripts ” R,I” denote the real and\nimaginary part. The site and lead-dependent coefficients5\nβiκare expressed in terms of the microscopic scatter-\ning matrix elements defined in Eq. (14) evaluated at the\nFermi energy:\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (28)\nand can be calculated numerically for any particular sys-\ntem configuration.\nThe spin pumping torque τsp\nicontains contributions\nfrom both sublattices and is given by\nτsp\ni=X\nj\u0002\nαR\nijz×∂tmj+αI\nijz×(z×∂tmj)\u0003\n,(29)\nwhere jruns over all sites and αijis expressed in the low-\nenergy limit using the scattering matrix elements evalu-\nated at the Fermi energy,\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (30)\nandαR(I)denotes the real (imaginary) part of the matrix.\nUsing the Onsager reciprocal relations in Appendix B, we\nfind that the spin transfer torque and spin pumping are\nrelated in the case of the most relevant case of a single\nreservoir,\nX\njαij=βi. (31)\nFinally, the fluctuating torque τf\niis expressed in terms\nof a fluctuating transverse field Hf\ni,\nτf\ni=−γµ0z×Hf\ni. (32)\nThe fluctuating field exhibit interlattice and intralattice\ncorrelators ⟨HµiHνj⟩, where µ, ν={x, y}:\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (33a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (33b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (33c)\nwhere the time arguments tandt′of the fields and the\nrelative time argument ( t−t′) of the self energies are\nomitted for simplicity. The self-energy Σ is due to charge\nand longitudinal spin accumulations in the normal metals\nand is nonzero even in equilibrium. It is conveniently\nwritten as a product of a frequency-dependent quantity\nπ(ω) and a site and scattering states dependent quantity\nσij[35]:\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (34)\nwith\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)\n−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] (35a)\nσijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓. (35b)Conversely, the self-energy matrices ˜Σ↑↓are due to trans-\nverse spin accumulation in the normal metals and, as a\nresult, vanish in equilibrium. Analogous to the decom-\nposition in Eq. (34), we write\n˜ΣK\n↑↓ij=−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (36)\nwhere\n˜π↑↓(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ℏω) (37a)\n˜σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (37b)\nThe noise matrices π(ω) and ˜ π(ω) are similar to what\nwas found in Ref. 35, and are calculated in the equi-\nlibrium, elastic, and inelastic transport regime in Sec.\nV B. Crucially, the shot noise differs on various sites, due\nto the site-dependence of σand ˜σ. At equilibrium, the\nfluctuation-dissipation theorem holds, e.g.\n2siγ2µ2\n0⟨Hf\nµiHf\nνi⟩=δµναii4kBTξ\u0012ℏω\n2kBT\u0013\n, (38)\nwhere ξ(x) =xcothx.\nIn the next section, we discuss the Keldysh action of\nthe model presented in this section and derive an effec-\ntive action by integrating out the fermionic degrees of\nfreedom.\nIV. KELDYSH THEORY AND EFFECTIVE\nACTION\nIn this section, we derive the semiclassical spin dynam-\nics by using an out-of-equilibrium path integral formal-\nism [30]. We introduce the closed contour action Sand\nthe partition function Z,\nZ=Z\nD[¯aa¯bb¯c↑c↑¯c↓c↓]eiS/ℏ. (39)\nThe action Sconsists of contributions from the localized\nmagnetic excitations aandb, and the spin-up c↑and\nspin-down electrons c↓from the scattering states. We\nwill integrate out the fermion operators and get an effec-\ntive action for the magnetic excitations aandb, which\nincludes effective transverse and longitudinal fields that\narise from the charge and spin accumulations in the nor-\nmal metals.\nWe follow Ref. 30 and replace the fields in Eq. (39)\nwith ” ±” fields residing on the forward and backward\npart of the Schwinger-Keldysh contour. The action in\nthe±basis is given in Appendix C. These fields are not\nindependent of each other and can be Keldysh rotated\ninto a new basis that takes into account the coupling\nbetween them. The rotated fields have the advantage\nof suggesting a transparent physical interpretation, cor-\nresponding to the semiclassical equations and quantum\ncorrections.6\nA. Keldysh action\nFor magnons, the classical ( cl) and quantum ( q) fields\nare defined linear combinations of the ±-fields, as de-\nscribed in detail in Appendix C. In Keldysh space, it is\nconvenient to also introduce the matrices\nγq=\u0012\n0 1\n1 0\u0013\nγcl=\u0012\n1 0\n0 1\u0013\n. (40)\nThe Keldysh rotated magnon action becomes\nSm=X\nat¯aq\na(iℏ∂t−EA\na)acl\na+X\nbt¯bq\nb(iℏ∂t−EB\nb)bcl\nb\n+X\nataq\na(iℏ∂t−EA\na)¯acl\na+X\nbtbq\nb(iℏ∂t−EB\nb)¯bcl\nb\n−2X\naa′t√sasa′Jaa′\u0002\n¯aq\naacl\na′+h.c.\u0003\n−2X\nbb′t√sbsb′Jbb′\u0002¯bq\nbbcl\nb′+h.c\u0003\n−2X\nabt√sasbJab\u0002\n¯aq\na¯bcl\nb+ ¯acl\na¯bq\nb+h.c.\u0003\n−γµ0ℏX\nat√sa\u0002\nHA\na−aq\na+HA\na+¯aq\na\u0003\n−γµ0ℏX\nbt√sb\u0002\nHB\nb−¯bq\nb+HB\nb+bq\nb\u0003\n, (41)where we wrote the time integral as a sum for compact\nnotation. In Eq. (41), h.c.denotes the hermitian conju-\ngate of the previous term. The fermion action becomes\nSe+S0=X\nst¯Csγcl(iℏ∂t−ϵ)Cs, (42)\nwhere we introduced vector notation for the 1 /2-fields,\n¯Cs= (¯c1\nsα,¯c2\nsα), and ϵis a diagonal matrix containing\nthe single-particle energies of the electrons. The Keldysh\nrotated first-order electron-magnon interaction is\nS1=−X\nat\nαβ1√sah\nacl\naWαβ\na↓↑¯C↓αγclC↑β+aq\naWαβ\na↓↑¯C↓αγqC↑β+h.c.i\n−X\nbt\nαβ1√sbh\n¯bcl\naWαβ\nb↓↑¯C↓αγclC↑β+¯bq\nbWαβ\nb↓↑¯C↓αγqC↑β+h.c.i\n, (43)\nand, finally, the second-order term reads\nS2=X\nat\nαβ1\n2sah\nWαβ\na↑↑\u0000¯AaγclAa¯C↑αγclC↑β+¯AaγqAa¯C↑αγqC↑β\u0001\n−Wαβ\na↓↓\u0000¯AaγclAa¯C↓αγclC↓β+¯AaγqAa¯C↓αγqC↓β\u0001i\n−X\nbt\nαβ1\n2sbh\nWαβ\nb↑↑\u0000¯BbγclBb¯C↑αγclC↑β+¯BbγqBb¯C↑αγqC↑β\u0001\n−Wαβ\nb↓↓\u0000¯BbγclBb¯C↓αγclC↓β+¯BbγqBb¯C↓αγqC↓β\u0001i\n,\n(44)\nwhere the magnon q/cloperators are consolidated in vec-\ntors ¯Aaand ¯Bb. The Keldysh rotated action proves to\nbe well-suited for the computation of an effective magnon\naction, a topic we delve into in the following section.B. Integrating out the fermionic degrees of\nfreedom\nFor the itinerant electrons in the normal metal and\nantiferromagnet, the total effective electron action is7\nSe,tot=Se+Sem, and can be expressed as\nSe,tot=X\nss′tt′¯Cs,tG−1\nss′,tt′Cs′,t′, (45)\nwhere the interacting Green function Gis given in terms\nof the noninteracting Green function G0and interaction\nterms as\nG−1=G−1\n0+˜W1+˜W2. (46)\nHere, ˜W1contains the first-order magnon operators on\nboth sublattices,\n˜W1=δ(t−t′)\u0002\nWA\n1+WB\n1\u0003\n, (47)\nwhere WA\n1andWB\n1are spin flip operators:\nWA\n1=−X\nxa1√saγx\u0002\nWa↑↓¯ax\naσ++Wa↓↑ax\naσ−\u0003\n(48a)\nWB\n1=−X\nxb1√sbγx\u0002\nWb↑↓bx\nbσ++Wb↓↑¯bx\nbσ−\u0003\n.(48b)\nIn Eq. (48), the variable x={cl, q}represents a Keldysh\nspace index, and σ±are the usual raising and lowering\nPauli matrices. Similarly, ˜W2contains the magnon oper-\nators to quadratic order for both sublattices,\n˜W2=δ(t−t′)\u0002\nWA\n2−WB\n2\u0003\n, (49)\nwith WA\n2andWB\n2given by\nWA\n2=X\naxy1\n2sa¯ax\naγxay\naγy\u0012\nWa↑↑ 0\n0−Wa↓↓\u0013\n(50a)\nWB\n2=X\nbxy1\n2sb¯bx\nbγxby\nbγy\u0012\nWb↑↑ 0\n0−Wb↓↓\u0013\n, (50b)\nwhere the spin structure is explicitly written out as a ma-\ntrix. The matrices WA(B)\n1 andWA(B)\n2 have a structure\nin the scattering states space from Wa(b), spin space from\nthe Pauli matrices, and Keldysh space from γx. The in-\nverse free electron Green function G−1\n0from Eq. (46) has\nthe conventional causality structure in Keldysh space,\nwith a retarded ( R), advanced ( A), and Keldysh ( K)\ncomponent:\nG−1\n0=\u0012\n[GR\n0]−1[GK\n0]−1\n0 [ GA\n0]−1\u0013\n, (51)\nand has equilibrium components that are diagonal in\nboth spin space and in the scattering states space,\n[G−1\n0]R(A)\nαβ,ss′=δαβδss′δ(t−t′)(iℏ∂t−ϵα±iδ),(52)\nwhere the upper sign corresponds to the retarded compo-\nnent, while the lower sign is applicable to the advanced\ncomponent. The Keldysh component includes informa-\ntion about the distribution function, and will be dis-\ncussed below, when we Fourier transform the Green func-\ntions.From the effective electron action in Eq. (45), it is\nevident that the partition function of Se,tottakes on a\nGaussian form with respect to the fermionic operators.\nHence, the fermionic integral in the partition function\ncan be evaluated exactly, with an inconsequential pro-\nportionality constant being disregarded:\nZ\nD[C]eiSe,tot/ℏ= eTr[ln[1+G0˜W1+G0˜W2]]. (53)\nIn Eq. (53), we have used the short-hand notation for\nthe functional integral measure of all fermionic states,\nD[C] =D[¯C↑C↑¯C↓C↓]. We have absorbed a normaliza-\ntion constant into the functional integral measure for sim-\nplicity. We note that as a consequence of the continuity\nof the time coordinate and scattering states energy that\nwe are employing, the unit matrix is a delta function in\ntime and energy, 1 ≡δ(t−t′)δ(ϵα−ϵβ), and thus quanti-\nties inside the logarithm carries dimension J−1s−1. The\ntrace, on the other hand, is an integral operator with unit\nJ s. As long as one interprets the logarithm in terms of\nits Taylor expansion, this does not lead to any problems,\nas the exponent of Eq. (53) becomes dimensionless for all\nterms in the expansion. The exponent is interpreted as\nan additional contribution to the magnon action,\ni\nℏSeff= Trh\nln[1 + G0˜W1+G0˜W2]i\n. (54)\nThe way forward is to treat this interaction as a pertur-\nbation, expanding the logarithm in first and second-order\ncontributions and disregarding higher-order terms,\nSeff≈ −iℏTrh\nG0˜W1i\n−iℏTrh\nG0˜W2i\n+iℏ\n2Trh\nG0˜W1G0˜W1i\n. (55)\nTo evaluate the trace in these terms, it is convenient to\nFourier transform all quantities from the time domain\nto the energy domain. This diagonalizes the noninter-\nacting Green functions, making calculations much more\nstraightforward.\nC. Fourier representation\nThe paper employs the Fourier transform convention\ndefined in Appendix D. In Fourier space, the fermion\nequilibrium Green function components are particularly\nsimple:\n[G0]R(A)\nαβ,ss′(ω) =δαβδss′(ℏω−ϵ±iδ)−1, (56)\nwhere δ > 0 is an infinitesimal quantity ensuring con-\nvergence. The Keldysh component accounts for non-\nequilibrium phenomena though the spin-dependent dis-\ntribution nss′αdefined in Eq. (17),\n[G0]K\nαβ,ss′(ω) =−2πiδαβδ(ℏω−ϵα) [δss′−2nss′α].(57)8\nThe Keldysh component has off-diagonal terms in spin\nspace if the distribution function nss′αhas off-diagonal\nelements, i.e. if there is a transverse spin accumulation\nin the normal metals.\nV. NON-EQUILIBRIUM SPIN DYNAMICS\nHaving derived the effective action as expressed in Eq.\n(55), we proceed by evaluating the traces and delving\ninto the resultant terms. The discussion unveils effective\nlongitudinal and transverse fields, which we ascribe to\nspin transfer torque and spin pumping originating from\nthe normal metal reservoirs.\nA. First-order contribution\nEvaluating the trace in the first order term in Eq.\n(55) corresponds to summing over the diagonal elements\nin spin space and Keldysh space, integrating over both\ntime variables, and summing over the space of scattering\nstates, we find\n−iℏTrh\nG0˜W1i\n=−X\naα2√saWαα\na↑↓n↓↑αZ\ndt¯aq\na(t)\n−X\naα2√saWαα\na↓↑n↑↓αZ\ndtaq\na(t)\n−X\nbα2√saWαα\nb↑↓n↓↑αZ\ndtbq\nb(t)\n−X\nbα2√saWαα\nb↓↑n↑↓αZ\ndt¯bq\nb(t).(58)\nHere, we have used the general Green function identity\nGR(t, t)+GA(t, t) = 0 [30], and written the time integra-\ntion explicitly. Comparing the first-order contribution in\nEq. (58) with the magnon action in Eq. (41), we observe\nthat the first-order effect of the spin accumulation in the\nnormal metal is equivalent to an effective deterministic\ntransverse magnetic field Hstt\ni, which act on a localized\nspin at site i={a, b}in the antiferromagnet. The ”stt”\nsuperscript indicates that this field will take the form of\na spin transfer torque, which will be elaborated on be-\nlow. The magnitudes of these effective transverse fields\nare given by\nγµ0Hstt\ni−=2\nsiℏX\nαWαα\ni↓↑n↑↓α (59a)\nγµ0Hstt\ni+=2\nsiℏX\nαWαα\ni↑↓n↓↑α, (59b)\nwhich implies that the Cartesian components read\nγµ0Hstt\nix=2\nsiℏX\nαRe\u0002\nWαα\ni↑↓n↓↑α\u0003\n(60a)\nγµ0Hstt\niy=2\nsiℏX\nαIm\u0002\nWαα\ni↑↓n↓↑α\u0003\n. (60b)Recalling that the spin accumulation is given by Eq. (24)\nand Eq. (21), we write the effective fields from Eq. (60)\nin the conventional spin transfer torque form:\nγµ0Hstt\ni=1\nℏX\nκ\u0002\nβR\niκz×µS\nκ+βI\niκz×(z×µS\nκ)\u0003\n,(61)\nwhere the appearance of zis a consequence of our the-\nory being restricted to small deviations for the equilib-\nrium magnetization ±z. This results in the spin transfer\ntorque given in Eq. (27). In Eq. (61), the superscripts\n”R” and ” I” denote the real and imaginary parts and the\nlead- and site-dependent constants βiκhave been intro-\nduced as sums over the transverse modes of the scattering\nmatrix elements,\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (62)\nand where we have assumed that the transverse spin dis-\ntribution functions n↑↓andn↓↑are only significant close\nto the Fermi surface, such that the scattering states ma-\ntrix elements are well approximated by their value at\nthe Fermi surface. The expression for the spin transfer\nfield in Eq. (61) is valid in both the elastic and inelastic\nregime, and vanishes in equilibrium. We note that the\ncoefficient βiκ, for i={a, b}, depends not only on the\npotential at lattice site ibut also indirectly of all lattice\nsites on both sublattices through the scattering states.\nTo the lowest order, the sublattice magnetizations are\nparallel and antiparallel to the z-axis, mA≈zand\nmB≈ −z. Thus, to the lowest order in the magnon\noperators, the expressions for the transverse fields are\nambiguous, and we can write the transverse field in Eq.\n(61) in terms of mAormB. To the lowest order in the\nmagnon operators, the Keldysh technique cannot be used\nto identify which sublattice the transverse fields in Eq.\n(58) originate from.\nB. Second order contribution\nThe second order contribution in Eq. (55) has contri-\nbutions from ˜W2,\nS21=−iℏTrh\nG0˜W2i\n, (63)\nas well as a contribution from ˜W1,\nS22=iℏ\n2Trh\nG0˜W1G0˜W1i\n. (64)\nProceeding in a manner analogous to the treatment of\nthe first-order term, the trace in S21is evaluated:9\nS21=−X\naαπ\nsa\u0002\nWαα\na↑↑(1−2n↑↑α)−Wαα\na↓↓(1−2n↓↓α)\u0003Z\ndt¯Aa(t)γqAa(t)\n+X\nbαπ\nsb\u0002\nWαα\nb↑↑(1−2n↑↑α)−Wαα\nb↓↓(1−2n↓↓α)\u0003Z\ndt¯Bb(t)γqBb(t). (65)\nFrom Eq. (41), it is apparent that the second-order terms in S21are equivalent with a longitudinal magnetic field,\nwith magnitude\nγµ0HA21\niz=−π\nℏsiX\nα\u0002\nWαα\ni↑↑(1−2n↑↑α)−Wαα\ni↓↓(1−2n↓↓α)\u0003\n, (66)\nwhich, in this reference frame, renormalizes the energies of localized magnon excitations. However, such longitudinal\nmagnetic fields should not affect the spin dynamics since they, in the instantaneous reference field, correspond to\ncontributions to the total free energy proportional to S2\ni.\nThe final contribution S22to the effective action contains inter-lattice and intra-lattice terms and can be written\ncompactly by introducing a field di={aa,¯bb}and summing over the two field components, i.e.P\nidi=P\naaa+P\nb¯bb:\nS22=Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↓(t′, t)Wi↓↑γxG0,↑↓(t, t′)γx′Wj↓↑i\ndx\ni(t)dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↑(t′, t)Wi↑↓γxG0,↓↑(t, t′)γx′Wj↑↓i\n¯dx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↓(t′, t)Wi↓↑γxG0,↑↑(t, t′)γx′Wj↑↓i\ndx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↑(t′, t)Wi↑↓γxG0,↓↓(t, t′)γx′Wj↓↑i\n¯dx\ni(t)dx′\nj(t′), (67)\nwhere the trace is taken only over the 2 ×2 Keldysh space and the space of scattering states α. The interlattice terms,\ni.e.d=d′, are discussed in Ref. 35 for a macrospin ferromagnet. Here, we summarize this discussion and highlight\nthe addition of the inter-lattice terms not present in the macrospin ferromagnet.\nEvaluating the trace in the first and second line of Eq. (67), we note that only the Keldysh component has off-\ndiagonal elements in spin space, and find a contribution only from x=x′=q,\n˜Sqq\n22=ℏZ\ndtdt′X\nijh\ndq\ni(t)˜ΣK\n↑↓ij(t, t′)dq\nj(t′)i\n(68a)\n˜S¯q¯q\n22=ℏZ\ndtdt′X\nijh\n¯dq\ni(t)˜ΣK\n↓↑ij(t, t′)¯dq\nj(t′)i\n, (68b)\nwhere the self-energies are\n˜ΣK\n↑↓ij(t−t′) =−2i\nℏ2√sisjX\nαβn↑↓αn↑↓βWαβ\ni↓↑Wβα\nj↓↑ei(ϵα−ϵβ)(t−t′)/ℏ(69a)\n˜ΣK\n↓↑ij(t−t′) =−2i\nℏ2√sisjX\nαβn↓↑αn↓↑βWαβ\ni↑↓Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (69b)\nThe reasoning behind identifying this self-energy as a Keldysh component is that it couples the quantum components\nof the fields, see Eq. (68). The terms in Eq. (68) do not have a direct analog in the magnon action in Eq. (41), and\ninterpreting these will be the subject of Sec. V C. The self-energies in Eq. (69) are invariant under a joint time and\nlattice site reversal, i.e. ˜Σij(t−t′) =˜Σji(t′−t). Moreover, due to the properties n↑↓=n∗\n↓↑andWαβ\ni↑↓= [Wβα\ni↓↑]∗, we\nsee that the self-energies are related by ˜ΣK\n↑↓ij(t−t′) =−[˜ΣK\n↓↑ij(t−t′)]∗, which will be important later.\nDisregarding terms of the order kBT/ϵFandµs/ϵF[35], we find that the Fourier-transformed self-energy becomes\n˜ΣK\n↑↓ij(ω) =−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (70)10\nwhere\n˜πκλ(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ω) ˜ σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (71)\nand where the matrix elements Ware evaluated at the Fermi surface. This is a straightforward generalization of\nthe macrospin ferromagnet case, with the addition of shot-noise contributions from inter-lattice and intra-lattice\ninteractions between different lattice sites. We can evaluate the quantity ˜ πκλ↑↓(ω) by using Eq. (17):\n˜π↑↓κλ(ω) =−uκ−uλ−Z\ndϵ[f↑κ(ϵ)−f↓κ(ϵ)] [f↑λ(ϵ+ℏω)−f↓λ(ϵ+ℏω)] (72)\nwhere we introduced the conventional ”lowering” vector u−=ux−iuy. This can be computed in equilibrium, elastic,\nand inelastic scattering cases, and results exactly similar to those in Ref. 35.\nWe now turn our attention to the third and fourth lines of the second-order action in Eq. (67). The contributions\nfrom the two lines are equal, which is evident from interchanging summation indices and rearranging terms. Their\ntotal contribution to the action S22can be split into contributions S¯qq\n22,S¯qcl\n22, and S¯clq\n22. The contribution Sclclvanishes,\ndue to the quantity GR(t′−t)GR(t−t′) being nonzero only for t=t′, which has measure zero, and similarly for\nGA. This ensures that the action satisfies the general requirement S[ϕcl, ϕq= 0] = 0 [30]. Introducing, for notational\nconvenience, the vector ¯Di=\u0000¯dcl¯dq\u0001\n, we find\nS¯qq\n22+S¯qcl\n22+S¯clq\n22=ℏZ\ndtdt′X\nij¯Di(t)ˆΣij(t−t′)Dj(t′), (73)\nwhere the self-energy matrix has structure in Keldysh space and in the sublattice space,\nˆΣij(t−t′) =\u0012\n0 ΣA(t−t′)\nΣR(t−t′) ΣK(t−t′)\u0013\nij, (74)\nand its components are given by\nΣK\nij(t−t′) =2i√sisjℏ2X\nαβ(n↑↑α+n↓↓β−2n↑↑αn↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(75a)\nΣR\nij(t−t′) =2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(75b)\nΣA\nij(t′−t) =−2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (75c)\nThe Keldysh component of this self-energy has the symmetry [ΣK\nij(t−t′)]∗=−ΣK\nji(t′−t). Imperatively, as a\nconsequence of this symmetry, the quantities\nΣK\nij(t−t′)−ΣK\nji(t′−t) = 2Re\u0002\nΣK\nij(t−t′)\u0003\n(76)\niΣK\nij(t−t′) + iΣK\nji(t′−t) =−2Im\u0002\nΣK\nij(t−t′)\u0003\n, (77)\nare real numbers, which will be important in the next section. We proceed by a similar analysis to what was done\nwith ˜Σ, writing it in terms of a shot-noise matrix. We assume that the matrices Wcan be approximated by their\nvalue on the Fermi surface, and write\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (78)\nwhere we introduced the matrices\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] σijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓.(79)\nThe matrix π(ω) can also be evaluated in equilibrium, and for elastic and inelastic scattering, and the results are\nagain exactly similar to those in Ref. 35.\nwhere ξ(x) =xcothxis an asymptotically linear function for high xandpss′κ= (1−uzκ)/2+uzκδssis a projection11\nfactor introduced for notational convenience.\nComparing with the magnetic action in Eq. (41), we\nnotice that the terms with the retarded and advanced\nself-energies are equivalent with longitudinal fields, which\nwe in the following will show consists of dissipative\nGilbert-like terms and non-dissipative field-like terms.\nFourier transforming and applying the identity (D5), the\nretarded and advanced self-energies from Eq. (75b) and\nEq. (75c) become\nΣR,A\nij=−2√sisjℏX\nαβn↑↑α−n↓↓β\nℏω+ϵα−ϵβ±iδWαβ\ni↓↑Wβα\nj↑↓.(80)\nThis self-energy has equilibrium contributions as well\nas non-equilibrium contributions, however, the non-\nequilibrium contributions scale as µ↑↑/ϵFandµ↓↓/ϵFand\nare disregarded in the following. The equilibrium part of\nEq. (80) becomes particularly transparent when expand-\ning to first order in the frequency ω:\nΣR/A\n↑↓ij(ω)≈ΣR/A\n↑↓ij(ω= 0)±iωαij, (81)\nwhere we introduced the frequency-independent matrix\nelement\nαij=2π√sisjX\nαβ[−f′(ϵα)]δ(ϵα−ϵβ)Wαβ\ni↓↑Wβα\nj↑↓,(82)\nwhich can be approximated to\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (83)\nwhere the scattering states matrix elements are evalu-\nated at the Fermi surface. We note from the identity\n[Wαβ\ni↑↓]∗=Wβα\ni↓↑thatαis a Hermitian matrix in the space\nof lattice sites, i.e. [ αij]∗=αji. The zeroth order term\nin frequency is\n[S¯qcl\n22+S¯clq\n22]0=ℏX\nijZ\ndω¯dq\ni(ω)ΣR\n↑↓ij(0)dcl\nj(ω)\n+ℏX\nijZ\ndω¯dcl\ni(ω)ΣA\n↑↓ij(0)dq\nj(ω),(84)which is a constant longitudinal field that plays no role\nin the instantaneous reference frame, as discussed above.\nThe first-order term in frequency is finite even in equi-\nlibrium,\n[S¯qcl\n22+S¯clq\n22]1=ℏX\nijαijZ\ndt¯Diγq∂tDj, (85)\nand takes the form of a Gilbert damping term, includ-\ning both inter-lattice and intra-lattice contributions. The\nspin transfer torque coefficient αand the spin pumping\ncoefficient βare related to each other as a consequence of\nthe Onsager reciprocal relations [36]. In Appendix B we\nderive this relation, which is given in Eq. (B11), and de-\nrive an optical theorem relating the scattering matrices,\ngiven in Eq. (B12).\nSummarizing this section, we have found that the cor-\nrections to the magnon action Smin the presence of spin\nand charge accumulations in surrounding normal metals\nisS1+S21+S¯qcl\n22+S¯clq\n22+˜Sqq\n22+S¯qq\n22, and found that the\nfirst three of these contributions appear like magnetic\nfields and (in the low-frequency limit) like Gilbert-like\ndamping terms in the effective magnon action. Impor-\ntantly, we find both longitudinal and transverse fields in\nthe general case. The last two contributions to the ac-\ntion consist of coupled quantum fields and are the result\nof purely quantum effects. These terms are the subject\nof the next section.\nC. Fluctuating fields\nFrom the effective action in the last section, we were\nable to associate the ( q, cl) and ( cl, q) terms with longi-\ntudinal fields by comparing them with the magnon ac-\ntion in Eq. (41). Now, we must address the issue of\nhow to interpret the ( q, q) terms, which lack an ana-\nlog in the action described in Eq. (41). In this section,\nwe derive fluctuating forces from these terms by employ-\ning a Hubbard-Stratonovich (HS) transformation on the\nquadratic fields in the effective action, introducing aux-\niliary fields in the process. Commencing with the con-\ntribution from the term S¯qq\n22, we introduce the complex\nauxiliary field h¯qq\ni(in units of inverse second) via a con-\nventional Hubbard–Stratonovich transformation:\neiS¯qq\n22/ℏ= exp\u0014Z\ndtdt′X\nij¯dq\ni(t)iΣK\nij(t−t′)dq\nj(t′)\u0015\n=1\ndet [−iΣK]ZY\niD[h¯qq\ni] exp\u0014\niZ\ndtX\nih¯qq\ni(t)¯dq\ni(t) +h.c.−Z\ndtdt′X\nij¯h¯qq\ni(t)[−iΣK\nij(t−t′)]−1hqq\nj(t′)\u0015\n,\n(86)\nwhere a shorthand notation for the measure was in-\ntroduced as D[h¯qq\ni] = Π k\b\nd[Imh¯qq\ni(tk)]d[Reh¯qq\ni(tk)]/π\t\n,where kis the index used to order the discretization of12\nthe time coordinate. From the Gaussian form of Eq. (86),\nthe correlators of the auxiliary field can be identified as\n⟨h¯qq\ni(t)⟩= 0 (87a)\n⟨h¯qq\ni(t)h¯qq\nj(t′)⟩= 0 (87b)\n⟨¯h¯qq\ni(t)h¯qq\nj(t′)⟩=−iΣK\nij(t−t′). (87c)\nThe second term in the exponent is quadratic in the newfields, and gives no contribution to the magnon action,\nwhile the first term is linear in the magnon field diand is\ninterpreted as an effective transverse field in the magnon\naction.\nThe contribution from the terms ˜Sqq\n22+˜S¯q¯q\n22is HS trans-\nformed by performing an unconventional transformation\nin the two complex fields ˜hqq\niand˜hqq\niseparately:\nei˜S22/ℏ=1q\ndet [−2i˜ΣK\n↓↑]q\ndet [−2i˜ΣK\n↑↓]ZY\niD[˜hqq\ni] exp\"\niZ\ndtX\ni\u0010\n˜hqq(t)˜hqq(t)\u0011\ni\u0012\nd(t)\n¯d(t)\u0013\ni+h.c.\n−Z\ndtdt′X\nij\u0010\n˜hqq(t)˜hqq(t)\u0011\ni \n0 −i˜ΣK\n↓↑(t−t′)\n−i˜ΣK\n↑↓(t−t′) 0!−1\nij ˜hqq(t′)\n˜hqq(t′)!\nj#\n,(88)\nagain interpreting the exponent as an effective action in-\ncluding the field ˜hqq\ni, which has the correlators\n⟨˜hqq\ni(t)⟩= 0 (89a)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↑↓ij(t−t′) (89b)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↓↑ij(t−t′) (89c)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩= 0. (89d)\nWe remark that the unconventional form of the Hubbard-\nStratonovich decoupling leads to non-zero correlators for\nequal fields, as opposed to the conventional approach\nwhere the non-zero correlators involve one field being\nthe complex conjugate of the other. The fields h¯qqand\n˜hqqare interpreted as fluctuating transverse fields with,\nin general, different amplitudes depending on the lattice\nsite, but with correlators between lattice sites. Compar-\ning the effective action in Eq. (86) and Eq. (88) with the\nmagnon action in Eq. (41), the components of the total\nfluctuating field Hfcan be identified as\nγµ0Hf\n+,i=−1√sih\n2˜hqq\ni+h¯qq\nii\n(90a)\nγµ0Hf\n−,i=−1√sih\n2˜hqq\ni+¯h¯qq\nii\n. (90b)In this expression, the factor of 2 arises from the uncon-\nventional nature of the Hubbard-Stratonovich transfor-\nmation in Eq. (88). The correlators between the Carte-\nsian components of the fluctuating field can be calculated\nusing Eq. (89) and Eq. (90),\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (91a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (91b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (91c)\nfrom which we conclude that the correlators in the fluc-\ntuating field Hfare real numbers. In Eq. (91), we omit-\nted the time arguments for notational simplicity. Fur-\nthermore, it is evident that for i=jandt=t′, the\ncorrelators in Eq. (91a) and (91c) are positive, aligning\nwith the conditions expected for representing the vari-\nance of a real field.13\nD. Equations of motion\nAfter HS decoupling the qqcomponents, the effective action reads\nSeff=−γℏµ0Z\ndt\"X\na√sa\u0000\nHstt\na++Hf\na+\u0001\n¯aq\na(t) +X\nb√sb\u0000\nHstt\nb−+Hf\nb−\u0001¯bq\nb(t) +h.c.#\n+ℏZ\ndt\"X\naa′βaa′¯aq\na∂tacl\na+X\nabβabaq\na∂tbcl\nb+X\nbaβba¯bq\nb∂t¯acl\na+X\nbb′βbb′¯bq\nb∂tbcl\nb′+h.c.#\n. (92)\nHaving cast the total action Sm+Seffin a form that is linear in the quantum fields ¯ aqand¯bqand their complex\nconjugates, we can integrate over these fields in the partition function, producing the functional delta function imposing\nthe semiclassical equations of motion for the fields aclandbcl[30]. Using acl\na=S+a/(ℏ√sa) and ¯bcl\nb=S+b/(ℏ√sb) in\nthe semiclassical limit, we find the coupled equations of motion:\ni∂tSi+=ℏ−1EiSi++siℏµ0γ\u0000\nHi++Hf\ni++Hstt\ni+\u0001\n−X\njβij∂tSj+, (93)\nas well as its complex conjugated counterpart. Both in the definition of this field and in Eq. (93), the upper sign holds\nfor sublattice A, and the lower sign holds for sublattice B. We find the Cartesian components by taking the real and\nimaginary parts and divide with ℏsito find an equation for the vector mi=Si/(ℏsi),\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni, (94)\nwhere\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n(95a)\nτf\ni=−γµ0z×Hf\ni (95b)\nτstt\ni=−γµ0z×Hstt\ni (95c)\nτsp\ni=X\njReβijz×∂tmj+X\njImβijz×(z×∂tmj) (95d)\nis microscopic expressions for the bulk torque τb, the fluctuating torque τf, the spin pumping torque τsp, and the\nspin transfer torque τstt.\nVI. CONCLUSION\nIn this paper, we have presented a general quantum\ntheory of spin dynamics in magnet-normal metal systems,\ngeneralizing earlier results to a general antiferromagnetic\nor ferrimagnetic bipartite lattice. Spin and charge ac-\ncumulations in the normal metals influence the magne-\ntization dynamics in the magnet through spin transfer\ntorque, and the damping is enhanced due to spin pump-\ning, including both inter- and intra-lattice contributions.\nWe derived expressions for transverse fluctuating fields\narising due to the electron magnon interactions. These\nfields have contributions from equilibrium terms as well\nas charge and spin accumulation in the normal metals.\nWe found site-dependent shot noise contributions that\nare non-negligible at low temperatures.ACKNOWLEDGMENTS\nThis work was supported by the Research Council\nof Norway through its Centers of Excellence funding\nscheme, Project No. 262633, ”QuSpin”.\nAppendix A: Holstein-Primakoff transformation\nIn this Appendix, we discuss the transformations used\nto diagonalize the magnon Hamiltonian of Eq. (9). To\ngo from the SU(2) spin operators to bosonic annihila-\ntion and creation operators, we employ the Holstein-\nPrimakoff transformation [37, 38] at sublattices AandB\nand expand to the lowest order in the bosonic operators,\nassuming the antiferromagnet is close to the N´ eel state,\ni.e. that all spins on sublattice A(B) is close to being\nparallel (antiparallel) to the z-direction. At sublattice A,14\nwe expand\nˆSa+=ℏ√\n2sa\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\nˆaa≈ℏ√\n2saˆaa (A1)\nˆSa−=ℏ√\n2saˆa†\na\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\n≈ℏ√\n2saˆa†(A2)\nˆSaz=ℏ(sa−ˆa†\naˆaa), (A3)\nwhere aaannihilates a localized magnon and sais the\ntotal spin at lattice site a. In the expansion of the square\nroots in Eq. (A1) and Eq. (A2), we assumed sa≫1 and\nexpanded the square root to lowest order in 1 /sA. We\nhave employed the standard raising and lowering spin\noperators, defined as S±=Sx±iSy.\nSimilarly, at sublattice B, we expand\nˆSb+=ℏ√\n2sbˆb†\nb \n1−ˆb†\nbˆbb\n2sb!\n≈√\n2sbˆb†\nb, (A4)\nˆSb−=ℏ√\n2sb \n1−ˆb†\nbˆbb\n2sb!\nˆbb≈√\n2sbˆbb, (A5)\nˆSbz=ℏ\u0010\n−sb+ˆb†\nbˆbb\u0011\n, (A6)\nwhere ˆbannihilates a localized spin-up magnon.\nAppendix B: Relating spin transfer torque and spin\npumping coefficients\nWe relate the spin transfer pumping coefficients de-\nfined in Eq. (82) to the spin transfer coefficients found\nin Eq. (62) in the case of one normal metal reservoir us-\ning the Onsager reciprocal relations [36]. We start by\ndefining the pumped spin current (in units of electrical\ncurrent, i.e. Ampere) into normal metal as the change in\ntotal spin inside the antiferromagnetic due to spin pump-\ning, i.e.\nIS=−e\nℏX\njSjτsp\nj. (B1)\nThe appearance of Sj=ℏp\nsj(sj+ 1) is due to the way\nwe have defined the torques in the main text, causing\nthem to have the dimension of inverse time. The dynam-\nics of the localized magnetic moment µj=−γSjmjand\nthe spin current are driven by the external effective field\nHeffand the spin accumulation µS, which are the ther-\nmodynamic forces in our system. In linear response, we\ncan then write the equations for the spin dynamics and\nthe spin current in matrix form:\n\u0012−γSi∂tmi\nIS\u0013\n=\u0012Lmm\nijLms\ni\nLsm\njLss\u0013\u0012µ0Heff\nj\nµS/e\u0013\n, (B2)\nwhere the matrix elements 3 ×3 tensors that effectively\napply the relevant cross products to make Eq. (B2) con-\nsistent with the Landau-Lifshitz equation, and where\nwe use the Einstein summation convention for repeated\nLatin indices.1. Identifying Lsm\nInserting the spin pumping torque from Eq. (29), the\nspin current becomes\nIS=−Xj∂tmj, (B3)\nwhere we defined the 3 ×3 matrix Xjas\nXj=e\nℏSjX\nih\nαR\nij˜O+αI\nij˜O2i\n, (B4)\nand the 3 ×3 matrix ˜Oimplements the cross product\nz×v=˜Ovand can be defined in terms of the Levi-\nCivita tensor. The LLG equation in the absence of spin\naccumulation (causing the spin transfer torque to vanish)\nreads\n(1−αb˜O)∂tmi=˜O(−γµ0Heff\ni), (B5)\nwhere αbis the (bulk) Gilbert damping constant. Hence,\nwe identify\nLsm\nj=γXj˜O(1−αb˜O)−1. (B6)\n2. Identifying Lms\nInserting the spin transfer torque from Eq. (27) into\nthe LLGS equation in the absence of an effective field,\nwe find\n∂tmi=ℏ−1(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\nµS,(B7)\nmeaning that we can identify the linear response coeffi-\ncient Lmsas (no Einstein summation)\nLms\ni=−Siγe\nℏ(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\n. (B8)\n3. Deriving relations from the Onsager reciprocal\nrelations\nWe are now looking to employ Onsager’s reciprocal\nrelation:\n[Lsm\ni({−mj})]T=Lms\ni({mj}), (B9)\nwhere the superscript Tindicates a matrix transpose in\nthe 3 ×3 Cartesian space. Using the matrix identity\n˜O3=−˜O, we find that Eq. (B9) implies that\nβI\nj˜O−βR\nj˜O2=X\nih\nαI\nij˜O−αR\nij˜O2i\n(B10)\nThis equality is satisfied if\nβj=X\niαij, (B11)15\nwhich generalizes the result from Ref. 35. Inserting the\ndefinitions of these coefficients in the low-temperature\nlimit, we find that\nX\nnWnn\nj↑↓= iπX\ninmWnm\ni↓↑Wmn\nj↑↓, (B12)\nwhich we classify as a generalized optical theorem, since\nin the diagonal case i=j, we can rewrite the imaginary\npart of this to\nIm\"X\nnWnn\ni↑↓#\n=πX\ninm|Wnm\ni↓↑|2, (B13)\nwhich is reminiscent of the optical theorem in wave scat-\ntering theory.\nAppendix C: Contour fields and Keldysh rotations\nIn this Appendix, we show how the action can be writ-\nten in the ±basis, and introduce the Keldysh rotated\nfields, which differ in the case of fermionic and bosonic\nfields. In the ±field basis, the action of the scattering\n(electron) states, corresponding to the Hamiltonian in\nEq. (7), reads\nSe+S0=X\nsZ∞\n−∞dt¯c+\ns(iℏ∂t−ϵ)c+\ns\n−X\nsZ∞\n−∞dt¯c−\ns(iℏ∂t−ϵ)cs−\n=X\nsξt¯cξ\ns(iℏ∂t−ϵ)cξ\ns, (C1)\nwhere now csis a vector containing the scattering fields,\n¯csdenotes its complex conjugate, and ϵis a diagonal\nmatrix containing all energy eigenvalues of the scatter-\ning states. In the final line, we have written the time\nintegration as a sum for concise notation. Additionally,\nwe introduced the sum over ” ±” fields as a sum over\nξ={+,−}, with an implicit negative sign before the ”-”\nfield, i.e.P\nξ. . .ξ=. . .+−. . .−. A similar notation will\nalso be used for the magnon fields below. The negative\nsign ( ξ=−) in the integral in Eq. (C1) and in the other\nactions below originates from reversing the integrationlimits on the backward contour. The magnon action is\nSm=X\nξabt[¯aξ\na(iℏ∂t−EA\nab)aξ\na+¯bξ\nb(iℏ∂t−EB\nab)bξ\nb]\n−2X\naa′Jaa′√sasa′¯aξ\naaξ\na′\n−2X\nbb′Jbb′√sbsb′¯bξ\nbbξ\nb′\n−2X\nξabtJab√sasb[aξ\nabξ\nb+ ¯aξ\na¯bξ\nb]\n−γµ0ℏX\nξatrsa\n2[HA\na−aξ\na+HA\na+¯aξ\na]\n−γµ0ℏX\nξbtrsb\n2[HB\nb−¯bξ\nb+HB\nb+bξ\nb]. (C2)\nThe first-order electron-magnon interaction is\nS1=−X\nξat\nαβr\n2\nsah\naξ\na¯cξ\n↓αWαβ\na↓↑cξ\n↑β+ ¯aξ\na¯cξ\n↑αWαβ\na↑↓cξ\n↓βi\n−X\nξbt\nαβr\n2\nsbh\n¯bξ\nb¯cξ\n↓αWαβ\nb↓↑cξ\n↑β+bξ\nb¯cξ\n↑αWαβ\nb↑↓cξ\n↓βi\n,\n(C3)\nand the second-order term is\nS2=X\nξat\nαβ1\nsa¯aξ\naaξ\nah\n¯cξ\n↑αWαβ\na↑↑cξ\n↑β−¯cξ\n↓αWαβ\na↓↓cξ\n↓βi\n−X\nξbt\nαβ1\nsb¯bξ\nbbξ\nbh\n¯cξ\n↑αWαβ\nb↑↑cξ\n↑β−¯cξ\n↓αWαβ\nb↓↓cξ\n↓βi\n.(C4)\nFor a general bosonic field ϕ, the classical ( cl) and\nquantum ( q) fields are defined as [30]:\nϕcl/q=1√\n2(ϕ+±ϕ−)¯ϕcl/q=1√\n2(¯ϕ+±¯ϕ−).(C5)\nIn our case, we have ϕ={a, b}. The upper (lower) sign\nholds for the classical (quantum) fields. For a fermionic\nfield c, the rotated fields are denoted by 1 and 2, and\ndefined as\nc1/2=1√\n2(c+±c−) ¯c1/2=1√\n2(¯c+∓¯c−).(C6)\nFor fermions, ¯ candcare independent variables, not re-\nlated by complex conjugation.\nAppendix D: Fourier transform\nFor a general function of relative time t−t′, we define\nthe Fourier transform between the relative time domain16\nand the energy domain as\nf(ω) =Z∞\n−∞d(t−t′)eiω(t−t′)f(t−t′), (D1)\nf(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′)f(ω). (D2)\nThe delta function can be represented as\nδ(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′), (D3)\nδ(ω) =1\n2πZ∞\n−∞d(t−t′)eiω(t−t′). (D4)Finally, we note a frequently employed identity,\n−iZ∞\n−∞d(t−t′)eiω(t−t′)θ(t−t′) = (ω+ iδ)−1,(D5)\niZ∞\n−∞d(t−t′)eiω(t−t′)θ(t′−t) = (ω−iδ)−1,(D6)\nwhere δis an infinitesimal positive quantity.\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu,\nB. Di´ eny, P. Pirro, and B. Hillebrands, Journal of Mag-\nnetism and Magnetic Materials 509, 166711 (2020).\n[2] A. Brataas, B. van Wees, O. 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White, Physical Review B 14,\n2939 (1976)."    },    {        "title": "2207.10775v1.Unusual_ferrimagnetism_in_CaFe2O4.pdf",        "content": "1  Unusual ferrimagnetism in CaFe2O4  Hiroki Ueda1,†,‡,*, Elizabeth Skoropata1,†,*, Cinthia Piamonteze1, Nazaret Ortiz Hernández1, Max Burian1, Yoshikazu Tanaka2, Christine Klauser3, Silvia Damerio4,#, Beatriz Noheda4,5, and Urs Staub1,*  1 Swiss Light Source, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland.  2 RIKEN SPring-8 Center, Sayo, Hyogo 679-5148, Japan.  3 Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland.  4 Zernike Institute for Advanced Materials, University of Groningen, 9747AG- Groningen, Netherlands.  5 CogniGron Center, University of Groningen, 9747AG- Groningen, Netherlands.   Abstract: Incomplete cancellation of collinear antiparallel spins gives rise to ferrimagnetism. Even if the oppositely polarized spins are owing to the equal number of a single magnetic element having the same valence state, in principle, a ferrimagnetic state can still arise from the crystallographic inequivalence of the host ions. However, experimental identification of such a state as “ferrimagnetic” is not straightforward because of the tiny magnitude expected for M and the requirement for a sophisticated technique to differentiate similar magnetic sites. We report a synchrotron-based resonant x-ray investigation at the Fe L2,3 edges on an epitaxial film of CaFe2O4, which exhibits two magnetic phases with similar energies. We find that while one phase of CaFe2O4 is antiferromagnetic, the other one is “ferrimagnetic” with an antiparallel arrangement of an equal number of spins between two distinct crystallographic sites with very similar local coordination environments. Our results further indicate two distinct origins of an overall minute M; one is intrinsic, from distinct Fe3+ sites, and the other one is extrinsic, arising from defective Fe2+ likely forming weakly-coupled ferrimagnetic clusters. These two origins are uncorrelated and have very different coercive fields. Hence, this work provides a direct experimental demonstration of “ferrimagnetism” solely due to crystallographic inequivalence of the Fe3+ as the origin of the weak M of CaFe2O4.  † These authors contributed equally to this work.  ‡ Present address: SwissFEL, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland.  # Present address: Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, 08193 Bellaterra, Spain.  * Correspondence authors: hiroki.ueda@psi.ch, elizabeth.skoropata@psi.ch, and urs.staub@psi.ch    2  Ferrimagnetism is a type of magnetic order in which the populations or amplitudes of oppositely-polarized spins are different, resulting in net magnetization (M) [1]. Antiparallel spins from different constituent magnetic elements [2], different valences of a magnetic element [1], and/or different crystallographic sites unbalancing the numbers of antiparallel spins [3] cause a ferrimagnetic order with a relatively large M. However, what if the number of parallel spins and antiparallel spins, both from a single magnetic element with the same valence, is the same but the magnetic sites that host them are crystallographically inequivalent? From the symmetry point of view, such a state breaks the time-reversal symmetry as ferrimagnets. Similar but distinct local coordination environments can result in different sizes of magnetic moments, even if the host ions have the same valence. However, experimental demonstration of such a state as being ferrimagnetic is not straightforward, as it requires independently quantifying two almost equivalent sites and grasping evidence that the sublattices host M, which is comparably as small as a potential impurity contribution. Tiny M displays a weak response to applied magnetic fields (H) and thus, naturally results in a large coercive field. Hence, such a “ferrimagnetic” state possesses two reversible states but can be robust towards H.  CaFe2O4 is a unique material exhibiting two distinct magnetic phases with a collinear spin arrangement, both recognized as antiferromagnetic (AFM) [4,5]. The overall magnetism comes from two Fe sites, Fe1 and Fe2, having octahedral coordination with six surrounding O2-. Both sites possess the formal oxidation state of 3+ (S = 5/2) and are located on Wyckoff positions type 4c of the orthorhombic space group Pnma (#62), which does not have a space-inversion center. Namely, the two Fe sites have nominally the same valence, and there are exactly the same number of atoms of each type in a unit cell. Fe1 has a more distorted octahedral coordination than Fe2 [6]. The sites form a zigzag chain along [001] in a sequence of Fe1-Fe1-Fe2-Fe2 as seen in Fig. 1(a) or 1(b), and thus, two intra-chain magnetic interactions exist: between two different Fe sites and between two Fe of the same type. The super-exchange interactions among Fe3+ depend on the bond angles of Fe-O-Fe. The magnetic interactions between the same sites of Fe3+ along the [001] chain are weak because the bond angles are close to 90°. The weak intra-chain interactions result in two energetically almost degenerate spin configurations that have either ferromagnetic (FM) or AFM patterns along [001] for Fe3+ located at the same site [see Figs. 1(a) and 1(b), respectively]. The former phase is called A phase [magnetic space group: Pn’ma’ (#62.448)] while the latter is called B phase [magnetic space group: Pnma’ (#62.445)] with spins pointing along [010] but a different stacking sequence along [001]: up-up-down-down and up-down-up-down, respectively. The B phase is the magnetic ground state since the exchange interactions are AFM, while the A phase is metastable due to single-ion anisotropy emerging by mixing higher-energy multiplet states [7]. The local FM spin configuration in the A phase (up-up) can be found as the antiphase domain boundary of the AFM B phase [up-down-up (domain 1) and up-down-up (domain2)].  CaFe2O4 shows a phase transition to the B phase at TNB ≈ 200 K upon cooling and coexistence of the A and B phases on further cooling below TNA ≈ 175 K [8]. The temperature range of the phase coexistence persists down to the lowest temperatures for single-crystal samples [8] while remains down to T* ≈ 130 K for powder samples, below which only the A 3  phase is present [9], as visualized in Fig. 1(c). It is proposed that the A phase is more stable in powder samples than in single-crystals samples because of the strain introduced by grinding [7]. Interestingly, despite the believed AFM nature, a small remanent M of ~0.02 μB, which points along [010], summed over eight Fe3+ in the unit cell [10] is consistently reported in CaFe2O4, which onsets at around 185 K. Despite more than a half-century of investigation [4,5], the origin of the small M remains unclear. Previous studies based on magnetometry and neutron scattering proposed contributions to M (i) from Fe2+-induced ferrimagnetic clusters due to oxygen vacancies [11] and (ii) from domain boundaries of an AFM magnetic structure that can be locally FM [12]. Magnetic diffuse scattering experiments have attributed M to the presence of FM domain boundaries which also display an enhancement in H resulting from an increase in domain density [8]. Recently, uncoupled spins from the AFM domains have been discussed as a possible origin, based on spin Hall magnetoresistance measurements [13]. Note that the small M is not due to a spin canting since M and the Néel vector are both along [010] [12].  As found in Fig. 1(a), all Fe1 (Fe2) spins in the A phase point in the same direction but are antiparallel between the two sites. Hence, whereas the B phase is AFM, the A phase with the up-up-down-down configuration matching the periodicity of the lattice displays a “ferrimagnetic” state due to the crystallographic inequivalence of the two Fe3+ sites despite the same numbers of antiparallel spins. Different local coordination environments with distinct orbital hybridization with surrounding O2- could provide the finite orbital angular momentum of Fe3+ or a different expectation value of the spin moment leading to different sizes of antiparallel magnetic moments even though the resulting slight difference between the two Fe3+ magnetic moments may not easily be detectable by neutron scattering. Note that such orbital hybridization can correlate to the reported mixing of the higher-energy multiplet states [7] as observed in cobaltites [14,15]. This orbital mixing is predicted to stabilize the A phase as discussed above [7] and can create a magnetic anisotropy even though the high-spin state of Fe3+ generally exhibits a negligible magnetic anisotropy due to the absence of orbital angular momentum.  X-ray magnetic circular dichroism (XMCD) is a powerful and beneficial spectroscopic technique for the study of magnetism. It is possible to measure element- and site-specific magnetic hysteresis curves [16] and to extract orbital angular momentum from the overall magnetic moment of an atom by applying the sum rule [17]. Even if the finite orbital angular momentum is too small to be detected, distinct local coordination environments between the two sites can be reflected in the spectrum, enabling us to independently quantify the two almost equivalent magnetic sites.  Here, we report our investigation of the origin of M in epitaxial CaFe2O4 films by means of synchrotron-based x-ray techniques, resonant elastic x-ray scattering (REXS) and XMCD. These two techniques are complementary since REXS on the (001) reflection, which is forbidden due to the 21' symmetry along [001], probes the AFM behavior from the B phase without any contamination from the A phase because the A phase does not break the 21' symmetry along [001], while XMCD probes M that could originate from the potentially “ferrimagnetic” A phase. Our results indicate that the B phase remains down to the lowest temperature with a developing population of the A phase upon cooling, which resembles the 4  behavior of single crystals. XMCD spectra show three contributions in M; two of them representing antiparallel Fe3+ moments from the Fe1 and Fe2 sites, as expected for the A phase, with the third one originating from Fe2+, likely arising from defects, e.g., oxygen vacancies. Even though the orbital angular momentum that differentiates the size of the magnetic moments between the two sites is too small to be detected within the experimental precision, the measured magnetic hysteresis curves evidently attest to the ferrimagnetic nature of the A phase.   \n Fig. 1 Magnetic structures of CaFe2O4 in (a) the A phase and (b) the B phase, where orange and blue spheres represent Fe3+ having a magnetic moment pointing to +b and –b, respectively, and (c) its phase sequence for powder samples (up) [9] and for single-crystal samples (down) [8]. A number written on the spheres denotes the Fe sites, either Fe1 or Fe2. (a) and (b) were drawn by VESTA [28].   Epitaxial films of CaFe2O4 with a thickness of ~100 nm were grown on a TiO2 (110) substrate by the pulsed laser deposition method. The films contain complex needle-like crystallines that organize forming domains because of the strain release processes and the epitaxial relation with the substrate. Details are described in Ref. [10]. Essentially for our study, there is a significant fraction of [001]-oriented domains along the surface normal, as displayed in Fig. 2(a). REXS experiments were performed at the RESOXS end-station [18] of the X11MA beamline [19] in the Swiss Light Source (Switzerland) and at the BL17SU [20] in the SPring-8 (Japan), and XMCD experiments were performed at the X07MA beamline [21] of the Swiss Light Source. The incident photon energy was set around the Fe L2,3 edges for both types of experiments, whose setups are shown in Figs. 2(b) and 2(c). XMCD spectra were measured at several temperatures below TNB in H up to ±6 T. The grazing incidence of x rays (~30°) allows us to detect M along [010], which is along the surface plane. We employed the total electron yield (TEY) and x-ray excited optical luminescence (XEOL) [22] detection modes, which provide surface- (~5-10 nm) and complete film depth sensitivity, respectively. To suppress charging affecting the TEY signals, 1.9 nm of Pt were deposited on the film surface. Representative spectra comparing TEY and XEOL data show excellent agreement (Supplemental Fig. S1). Due to the relatively large sample thickness, apparent self-absorption \nabc11112222Ca2+O2-11112222Fe3+(S// +b)Fe3+(S// -b)(a) (b) \nT BA+BATNB(≈ 200 K)TNA(≈ 175 K)T*(≈ 130 K )M. Songvilayet al. BA+BC. Stock et al. (c) Aphase Bphase 5  effects [22] hinder quantitative analysis of the XEOL spectra and hence, we show here only data taken with the TEY method.  \n Fig. 2 (a) Schematic representation of the CaFe2O4 film orientation grown on a TiO2 substrate, adapted from Ref. [10]. There are different crystallographic domains present, and the surface normal is either [001] or [302] oriented. Experimental setup for (b) REXS and (c) XMCD measurements. The (001) reflection appears around ~55° of θ at the Fe L3 edge. For the XMCD measurements, we employed grazing incidence of x rays around 30°, enabling us to examine magnetization along [010].   Figure 3(a) shows (00L) REXS profiles around L = 1 at various temperatures. A small intensity observed even above TNB is due to resonantly-allowed scattering from aspheric electron distribution ascribed to Fe 3d orbitals (see Supplemental Material for details). The presence of aspheric electron distribution is also confirmed from the observed quadrupole splitting in Mössbauer spectra [10]. The increase in the (001) reflection intensities for decreasing temperatures below TNB [see Fig. 3(b)] reflects the growth of the B phase. Note that the A phase does not contribute to the (001) reflection because of the 21' symmetry along [001]. The distinct origins of the scattering below or above TNB are also reflected in the differences in the energy spectra, displayed in Fig. 3(c). The presence of the B phase down to the lowest achieved temperatures resembles the behavior of single-crystal samples, which show the coexistence of the A and B phases below TNA [see Fig. 1(c)] [8].     \n6   Fig. 3 (a) REXS profiles of the (001) reflection taken at various temperatures. (b) Temperature dependence of (001) integrated intensities, and (c) photon-energy dependence while maintaining the diffraction condition for the (001) reflection at various temperatures. (a) and (b) were taken at the photon energy of 710.8 eV.   To visualize how the B phase develops upon cooling, we created two-dimensional maps of the (001) intensities at several temperatures across TNB with an x-ray beam size of ~15 μm ´ 30 μm scanning with a step size of ~7 μm and ~15 μm for horizontal and vertical directions, respectively, as displayed in Figs. 4(a)-4(d). Because of the various crystallographic domains in the CaFe2O4 film [10], we can only sample the crystallographic domain fraction that has the [001] axis out-of-plane. Therefore, the intensities are spatially inhomogeneous even at 200 K in the absence of magnetic order, as seen in Fig. 4(d). The irregular shape of the intensity distribution is consistent with the previously reported structural domain morphology [10]. To better examine the developing magnetic contributions, we normalized the intensities taken at each temperature to those taken at 200 K. Figures 4(e)-4(g), obtained from the maps in Figs. 4(a)-4(c), establish that the magnetic contribution develops everywhere over the scanned region but is not uniform. The nonuniform magnetic profiles can be due to (i) the different amplitudes of the magnetic contributions from in-plane crystallographic domains [10] and/or (ii) the spatially inhomogeneous evolution of the B phase. The latter implies the evolution of the A phase, which does not contribute to the (001) intensities, and is consistent with the XMCD data shown later. Note that (ii) is not caused by inhomogeneously distributed crystallographic domains with a different out-of-plane orientation because of the normalization by the 200 K data.  \n(a) \n(b) \n705710715720725024 30 K 100 K 150 K 200 K 300 KIntensity [arb. units]\nEnergy [eV](c) 7   Fig. 4 Two-dimensional maps of (001) intensities taken at (a) 40 K, (b) 100 K, (c) 150 K, and (d) 200 K, with the photon energy of 710.25 eV. Intensities are normalized by those of the 200 K data at (e) 40 K, (f) 100 K, and (g) 150 K. These maps were taken in the region surrounded by a white box in (h). Black areas in (a)-(g) lie outside of the sample.   The appearance of the “ferrimagnetic” A phase can be investigated by XMCD. Figure 5 displays XAS and XMCD spectra taken at T = 30 K and 150 K in H = 6 T after zero-field cooling (ZFC) from room temperature. The overall XAS with two well-resolved peaks (eg and t2g) at the Fe L3 edge is typical of a predominant Fe3+ character. As the two different Fe3+ sites are AFM coupled in the A phase and are expected to have slightly different spectral shapes due to the distinct local coordination environments, the two peaks with opposite signs (β and γ) in the XMCD can be assigned to these two sites. While the amplitude of γ is similar between 30 K and 150 K, that of β is different at the two temperatures and is comparable with the γ peak at 30 K, further supporting their origins from distinct Fe-sites. Note that we found a history dependence on XMCD spectra for the two peaks, consistent with the fact that the A phase is in competition with the B phase and thus the A-phase population can differ by the T and H history conditions. Shown in Fig. 6 are the XMCD spectra taken at the same measurement condition, at 150 K in 6 T, but after approaching with different T and H evolution; after cycling H (±6 T) at 2 K for the black curve, compared with no prior magnetic cycle for the green curve. Both the peaks are weaker in the green curve, indicating a smaller M and less population of the A phase with respect to the B phase. The history dependence is consistent with the inhomogeneous development of the B phase implied by the REXS data.  \n8   Fig. 5 (a) XAS and (b) XMCD taken at 30 K (red) and 150 K (black). XAS is obtained as the sum of two data with opposite helicity of circular polarization (C+/C-) and is normalized to its highest peak, whereas XMCD is obtained as the difference between C+ and C-.   \n Fig. 6 History dependence of XMCD spectra taken at 150 K after ZFC directly from room temperature (green) compared with the XMCD at 150 K after ZFC to 2 K, magnetic cycling (±6 T), and warming in +6 T (black). The black data is the same as Fig. 5(b) and Fig. 7(b).   For the assignment of the XMCD spectral features, multiplet simulations of the Fe sites were performed using the MultiX code [23]. Crystal field multiplet parameters of a typical Fe Oh site were used [24], but the local structure of the Fe1 and Fe2 sites was obtained from the bulk structure [25] in order to account for the differing octahedral distortions of the two crystallographic sites. Figure 7 shows the results of the multiplet simulations for the XMCD data shown in Fig. 5. The calculated site-selective XMCD spectra are shown in Figs. 7(c) and 7(d). Based on these results, we find that the Fe2 and Fe1 sites display a relative energy shift of ~0.45 eV accounting for the well-resolved β and γ peaks, respectively. On the other hand, the feature labeled as α is not reproduced well by the two Fe3+ sites. Such a \n9  lower-energy peak compared to that of Fe3+ is often assigned to Fe2+. However, multiple possible contributions from anisotropic Fe2+ in differently oriented domains hindered us to reproduce the feature uniquely with multiplet simulations. Nevertheless, the independent behavior of the peak α from the Fe3+ features is clearly visible in the experiment, as shown in Fig. 6. Thus, it is reasonable to attribute the peak α to Fe2+. Our observation of an XMCD signal from Fe2+ suggests a high-spin state, unlike the proposed low-spin state (S = 0) for the Fe2+ from magnetic susceptibility measurements of bulk samples [11]. Fe2+ can originate from oxygen vacancies as discussed previously [11] and can form a ferrimagnetic cluster with surrounding Fe3+. Note that the population of Fe2+ must be very small as the XAS is dominated by the typical Fe3+ spectra, which is also consistent with Mössbauer spectroscopy reports of a single order parameter of the Fe3+ sites below TNB [10,26] and no observable Fe2+ (detectable typically within few percent).  The peak β reflecting the Fe3+ moments of the Fe2 site is larger at 30 K than at 150 K while the peak γ reflecting the Fe3+ moments of the Fe1 site is almost the same between the two temperatures, indicating different temperature evolution of the Fe1 and Fe2 sites in the A phase. At +6 T and 150 K, we find the antiparallel orientation of the Fe1 and Fe2 sites with a clear imbalance of the site-specific magnetization with MFe1 > MFe2. At +6 T and 30 K, MFe2 becomes larger and both Fe3+ sites contribute approximately equally to the XMCD. This can be understood by the growth of the A phase when lowering the temperature, as reported in single-crystal samples; for higher T the Fe2-site moments (β) are significantly smaller than the Fe1-site moments (γ) in thin domains of the A phase because of the imbalanced population, whereas for lower T the population gets more balanced [see the sketch in Fig. 7(e)]. In other words, it suggests that the intrachain exchange interaction between two Fe1 is likely more FM than that between two Fe2. The favorable FM interaction between two Fe1 might correlate with more distortion in the octahedral coordination than Fe2, which can mix the higher-energy multiplet states into the ground state and stabilize the A phase with intrachain FM coupling between the same sites of Fe. The view in the sketch explains larger M at 150 K than 30 K [10] as two net M from the sublattices represented by the peaks β and γ are antiparallel to each other and β develops more and gets comparable with γ at lower temperatures. Since the local FM spin configuration in the A phase is the same as that in the antiphase domain boundary of the AFM B phase, the slightly different onset temperature of M (~185 K) compared to TNA might imply the short-range order of the A phase, which is likely present as the antiphase domain boundary of the B phase.  10    Fig. 7 XMCD spectra taken at (a) 30 K and 6 T, and (b) 150 K and 6 T, which are decomposed into three contributions, as found in (c) and (d). Broken curves correspond to calculated respective contributions to the spectra while a red curve corresponds to the sum of the contributions. (e) A sketch representing the growth of the A phase and resultant decrease in total magnetization.   We aimed to obtain direct evidence that the A phase is ferrimagnetic. The reversal of β and γ XMCD peaks should take place by sweeping H. The same fitting procedure as done for the data in Fig. 7 enables us to extract site-specific magnetic hysteresis curves. Figures 8(a) and 8(b) display XMCD spectra taken at 150 K and 30 K, respectively, in various H. Extracted magnetic hysteresis curves of the two Fe sites are shown in Figs. 8(c) and 8(d). It is observed that the XMCD peaks reverse their signs by sweeping H at 150 K, directly \nT= 150 KH= + 6 TT= 30 KH= + 6 T(a)(b)\n(c)(d)\n(e)High TMtot111111112222222222BAB\nLow TMtot111111112222222222BAB11  evidencing that these two AFM-coupled Fe3+ sites are responsible for M, namely the A phase of CaFe2O4 is ferrimagnetic.  Another feature is that M from the two Fe3+ sites does not reverse at 30 K even when going from +6 T to –6 T, in contrast to the data taken at 150 K. This observation is counterintuitive as the high-spin configuration of Fe3+ commonly exhibits a small anisotropy leading to small coercive fields due to the equal filling of the orbitals. However, the tiny overall magnitude of M from the two imperfectly compensated sublattice magnetizations leads to a strongly reduced force on M from H, naturally explaining an enlarged coercive field. In addition, the significant distortion of the FeO6 octahedron that could mix higher-energy multiplet states into the ground state through charge transfer (or orbital hybridization) with the surrounding O2- [7] may generate magnetic anisotropy that results in an enlarged coercive field.  The minor steps observed at H = 0 T in the 30 K Fe3+ data of Figs. 8(c) and 8(d) can be due to (i) the flipping of Fe2+ XMCD signals that spectrally overlap on the peaks β and γ and/or (ii) the flipping of Fe3+ moments coupling to Fe2+. Both possibilities imply a small coercive field of Fe2+ moments. Therefore, the uncoupled spins from the AFM phase, i.e., from the main Fe3+ spins, reported previously [13] should be ascribed to defective Fe2+ that can easily respond to H.  \n Fig. 8 Magnetic-field dependence of the XMCD signals at (a) 150 K and (b) 30 K. Hysteresis curve for Fe3+ (1) and Fe3+ (2) at (c) 30 K and (d) 150 K obtained from fits with calculated spectra to (a) and (b).   In conclusion, we have investigated the origin of the tiny remanent magnetization reported in nominally antiferromagnetic CaFe2O4 by synchrotron-based spectroscopic techniques. Although the numbers of antiparallel spins, up and down both hosted by Fe3+ with octahedral coordination, are exactly the same in the A phase, we clarified that the A \nT= 150 K+ 6 T0 T-1 T-6 T0 T+ 1 TT= 30 K30 K\n150 K(b)(c)\n(d)+ 6 T0 T-1 T-6 T0 T+ 1 T(a)12  phase is ferrimagnetic due to the difference in the local coordination environments of the two sites by measuring site-specific magnetic hysteresis loops. The magnetization magnitude is comparably small with an independent defect contribution (Fe2+) having a different coercive field. Such a ferrimagnetic state may be useful for future spintronics as (i) the coercive field is much larger than normal ferromagnets or ferrimagnets due to small overall magnetization, as evident in our study on Fe3+, which usually exhibits small coercive fields, demonstrating the robustness of its magnetic state under an external magnetic field, (ii) there are two switchable ferroic states (±M), and (iii) dominating antiferromagnetic interactions, allowing magnetic resonance frequencies in the sub-THz range as antiferromagnets. Our discovery may open up a new direction of material design for future devices.   Data availability  Experimental data are accessible from the PSI Public Data Repository [27].   References  1. M. L. Néel, Propriétés magnétiques des ferrites; ferrimagnétisme et antiferromagnétisme, Ann. Phys. 12, 137-198 (1948).  2. I. S. 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A. Green, F. Demmel, R. A. Ewings, P. Fouquet, M. Laver, Ch. Niedermayer, Y. Su, K. Nemkovski, J. A. Rodriguez-Rivera, and S.-W. Cheong, Solitary magnons in the S = 5/2 antiferromagnet CaFe2O4. Phys. Rev. Lett. 117, 017201 (2016).  9. M. Songvilay, S. Petit, M. Koza, S. Rols, E. Suard, V. Skumryev, C. Martin, and F. Damay, Disorder and magnetic excitations in CaCrxFe2–xO4 (x = 0, 0.5). Phys. Rev. B 101, 014407 (2020).  13  10. S. Damerio, P. Nukala, J. Juraszek, P. Reith, H. Hilgenkamp, and B. Noheda, Structure and magnetic properties of epitaxial CaFe2O4 thin films. npj Quantum Materials 5, 33 (2020).  11. R. Das, S. Debnath, G. Narsinga Rao, S. Narasimhan, and F. C. Chou, Ferrimagnetic cluster formation due to oxygen vacancies in CaFe2O4-δ. Phys. Rev. B 98, 144404 (2018).  12. C. Stock, E. E. Rodriguez, N. Lee, F. Demmel, P. Fouquet, M. Laver, Ch. Niedermayer, Y. Su, K. Nemkovski, M. A. Green, J. A. Rodriguez-Rivera, J. W. Kim, L. Zhang, and S.-W. Cheong, Orphan spins in the S = 5/2 antiferromagnet CaFe2O4. Phys. Rev. Lett. 119, 257204 (2017).  13. S. Damerio, A. A. Kaverzin, V. Ocelík, G. R. Hoogeboom, B. J. van Wees, and B. Noheda, Antiferromagnetic ordering and uncoupled spins in CaFe2O4 thin films probed by spin Hall magnetoresistance. Adv. Electron Mater. 8, 2100963 (2021).  14. R. H. Potze, G. A. Sawatzky, and M. Abbate, Possibility for an intermediate-spin ground state in the charge-transfer material SrCoO3. Phys. Rev. B 51, 11501-11506 (1995).  15. P. Ravindran, H. Fjellvag, A. Kjekshus, P. Blaha, K. Schwarz, and J. Luitz, Itinerant metamagnetism and possible spin transition in LaCoO3 by temperature/hole doping. J. Appl. Phys. 91, 291-303 (2002).  16. G. van der Laan and A. I. Figueroa, X-ray magnetic circular dichroism –A versatile tool to study magnetic. Coord. Chem. Rev. 277-278, 95-129 (2014).  17. P. Carra, B. T. Thole, M. Altarelli, and X. Wang, X-ray circular dichroism and local magnetic fields. Phys. Rev. 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Nolting, X-Treme beamline at SLS: X-ray magnetic circular and linear dichroism at high field and low temperature J. Synchrotron Rad. 19, 661-674 (2012).  14  22. C. Piamonteze, Y. W. Windsor, S. R. V. Avula, E. Kirk, and U. Staub, Soft X-ray absorption of thin films detected using substrate luminescence: a performance analysis. J. Synchrotron Rad. 27, 1289-1296 (2020).  23. A. Uldry, F. Vernay, and B. Delley, Systematic computation of crystal-field multiplets for x-ray core spectroscopies. Phys. Rev. B 85, 125133 (2012).  24. B. Liu, C. Piamonteze, M. U. Delgado-Jaime, R.-P. Wang, J. Heidler, J. Dreiser, R. Chopdekar, F. Nolting, and F. M. F. de Groot, Sum rule distortions in fluorescence-yield x-ray magnetic circular dichroism. Phys. Rev. B 96, 054446 (2017).  25. I. O. Galuskina, Y. Vapnik, B. Lazic, T. Armbruster, M. Murashko, and E. V. Galuskin, Harmunite CaFe2O4: A new mineral from the Jabel Harmun, West Bank, Palestinian Autonomy, Israel, Am. Mineral. 99, 965-975 (2014).  26. H. Yamamoto, T. Okada, H. Watanabe, and M. Fukase, Mössbauer effect study of spin relaxation in CaFe2O4. J. Phys. Soc. Jpn. 24, 275-279 (1968).  27. https://doi.org/10.16907/4a134d5f-29be-45c8-8c3c-b86065c34703  28. K. Momma and F. Izumi, VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272-1276 (2011).     Acknowledgements  The resonant x-ray diffraction experiments were performed at the X11MA beamline in the Swiss Light Source under proposal No. 20191307 and at the BL17SU in the SPring-8 under proposal No. 20200012. The x-ray magnetic circular dichroism measurements were performed at the XTreme beamline in the Swiss Light Source during in-house access. H.U. acknowledges the National Centers of Competence in Research in Molecular Ultrafast Science and Technology (NCCR MUST-No. 51NF40-183615) from the Swiss National Science Foundation and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 801459 – FP-RESOMUS. E.S. is supported by the NCCR Materials’ Revolution: Computational Design and Discovery of Novel Materials (NCCR MARVEL No. 182892) from Swiss National Foundation and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 884104 (PSI-FELLOW-III-3i). N. O. H. acknowledges financial support of the Swiss National Science Foundation, No. 200021_169017. M.B. is supported by the Swiss National Science Foundation through project Nos. 200021-196964 and 200021_169698, respectively. Financial support by the Groningen Cognitive Systems and Materials Center (CogniGron) and the Ubbo Emmius Funds of the University of Groningen is also gratefully acknowledged.   Additional information  Competing interests: The authors declare no competing interests.  "    },    {        "title": "2112.10036v1.Pressure_induced_charge_orders_and_their_coupling_to_magnetism_in_hexagonal_multiferroic_LuFe2O4.pdf",        "content": " \n \n1 \n Pressure -induced charge orders and their coupling to magnetism in \nhexagonal multiferroic LuFe 2O4 \nFengliang  Liu1,2,3#, Yiqing Hao1#, Jinyang Ni1#, Yongsheng Zhao2, Dongzhou Zhang4, Gilberto Fabbris5, \nDaniel Haskel5, Shaobo Cheng6, Xiaoshan Xu7, Lifeng  Yin1,8,9 ,10,11, Hongjun Xiang1,9, Jun Zhao1,9,10, Xujie \nLü2, Wenbin Wang8,9,10,11,*, Jian Shen1,8,9 ,10,11,* and Wenge  Yang2,* \n1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, \nChina  \n2Center for High Pressure Science and Technology Advanced Research (HPSTAR), Shanghai 201203, China  \n3Department of Physics, Nanchang University, Na nchang, 330031, China  \n4Hawaii Institute of Geophysics & Planetology, University of Hawaii Manoa, Honolulu, HI, USA.  \n5Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA  \n6Department of Condensed Matter Physics and Materials Sci ence, Brookhaven National Laboratory, Upton, \nNY 11973, USA  \n7Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA  \n8Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China  \n9Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China  \n10Shanghai Qi Zhi Institute, Shanghai 200232, China  \n11Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  \n \n#These authors  contributed equally.  \n* Corresponding authors. E-mail addresses: wangwb@fudan.edu.cn (W.B.W.) , \nshenjian5494@fudan.edu.cn (J.S.), yangwg@hpstar.ac.cn (W.G.Y.) . \n \nHexagonal LuFe 2O4 is a promising charge -order (CO) driven multiferroic material with high charge and spin \nordering temperatures. The coexisting charge and spin orders on Fe3+/Fe2+ sites result in novel \nmagnetoelectric behaviors, but the coupling mechanism between the charge and spin orders remains elusive. \nHere, by tuning external pressure, we reveal three correlated spin -charge ordered phases in LuFe 2O4: i) a \ncentrosymmetric incom mensurate three -dimensional CO with ferrimagnetism, ii) a non -centrosymmetric \nincommensurate quasi -two-dimensional CO with ferrimagnetism , and iii) a centrosymmetric commensurate \nCO with antiferromagnetism. Experimental in -situ single -crystal X-ray diffrac tion and X-ray magnetic \ncircular dichroism measurements combined with density functional theory calculations suggest that the  \n \n2 \n charge density redistribution caused by pressure -induced compression in the frustrated double -layer [Fe 2O4] \ncluster is responsible  for the correlated spin-charge phase transitions. The pressure -enhanced effective \nCoulomb interactions among Fe -Fe bonds drive the frustrated (1/3, 1/3) CO to a less frustrated (1/4, 1/4) CO, \nwhich induces the ferrimagnetic to antiferromagnetic transition . Our results not only elucidate the coupling \nmechanism among charge, spin and lattice degrees of freedom in LuFe 2O4 but also provide a new way to \ntune the spin -charge orders in a highly controlled manner.  \nINTRODUCTION  \nMultiferroric  materials have attracted lots of research interests during the last decades because of their great \npotential applications in electronic devices and spintronics1,2. Hexagonal LuFe 2O4 is a promising candidate \nmaterial for charge -order (CO) driven multiferro icity, which has been intensively investigated both from \nfundamental and applied perspectives3-10. Previous X-ray diffraction (XRD)4 and transmission electron \nmicroscopy (TEM) measurements11,12 have suggested that LuFe 2O4 exhibits three -dimensional (3D) CO at \nambient pressure, manifesting a periodic arrangement of low valence (Fe2+) and high valence (Fe3+) ions. \nIndication of the CO -driven ferroelectricity of LuFe 2O4 was also revealed in the observ ation of the \nspontaneous electronic polarization above room temperature3,13,14. In addition to the 3D order of Fe2+-Fe3+ \nions observed below the CO transition temperature TCO (~ 320 K), a quasi -2D (Q2D) ordering of Fe2+-Fe3+ \nwas also observed above TCO persisting up to ~ 525 K12. Furthermore, neutron diffraction measurements have \nrevealed ferrimagnetic order of Fe moments below the Neel transition temperature TN (~ 240 K)15. It is \ntherefore suggested that the correlations between charge and magnetic order associated with Fe2+ and Fe3+ \nions may play a crucial role in the multiferroicity in LuFe 2O44,16-19. \nApplication of external pressure is a powerful and clean tool to gain deep insights into the interplay \nbetween the charge, magnetic and structural degrees of freedom, because the strong frustration involved in \nthe spin and charge orders in the triangular lattice of LuFe 2O4 may result in highly tunable ground states16,19. \nHowever, high pressure diffraction measurements are often difficult  owing to the reduced beam flux and \nincreased  background  caused by  pressure cell s. Previous n eutron diffraction  measurement on powder \nsamples of LuFe 2O4 revealed ~ 30% reduction of the ferrimagnetic ordered moments  up to ~3 GPa20, and X-\nray powder diffraction showed  indications  for pressure -induced structur al phase  transitions21. However,  the \npositions and intensities of superlattice reflections observed in powder diffraction  show ed non-systematic \nevolution  with increasing pressure probably due poor powder averaging , thus preventing accurate description \nof the pressure -induced phases21. Moreove r, neutron and X-ray powder diffraction measurements  have \nrevealed a high -pressure polymorph phase LuFe 2O4-hp22,23 above 12 GPa. The LuFe 2O4-hp phase retained its \nstructure after pressure release, thus allowing  ex-situ measurement at ambient pressure22,23. This LuFe 2O4-hp \nphase adopts a rectangular Fe lattice23 and is not directly relevant to  the frustrated triangular lattice of \nLuFe 2O4 at ambient condition s. Therefore, despite the intensive  efforts, the pressure dependent evolution of \nthe frustrated charge and magnetic interactions in the multiferroic  LuFe 2O4 remains unclear .  \nHere, we investigated the evolution of charge and spin orders under pressure in LuFe 2O4 using the in -situ \nhigh-pressure single -crystal X-ray diffraction (HP -SXD) and high -pressure X-ray magnetic circular \ndichroism (HP -XMCD) spectroscopy. A series of pressure -induced charge -order phases, including a Q2D \nCO phase below 5.5 GPa and a 3D CO phase at higher pressur e (6.0 ~ 12.6 GPa), were identified by HP -\nSXD measurements. In addition, the HP -XMCD measurements suggest that the Q2D and 3D CO phases are \nassociated with ferrimagnetic and antiferromagnetic order, respectively. The pressure induced charge - and \nmagnetic -order phase transitions were further confirmed by density functional theory (DFT) calculations. \nThese results suggest that the CO phases are intimately coupled with magnetism, both of which can be \nmanipulated by external pressure in a highly controlled mann er in hexagonal layered multiferroic LuFe 2O4. \n \n \nRESULTS  \nLuFe 2O4 adopts a bi -layered triangular lattice structure with space group R-3m (No.166) at ambient \npressure. Considering the CO and the atom ic displacement it induces, the trigonal lattice splits into three 120o  \n \n3 \n twinned monoclinic lattices, each of which adopts the space g roup C2/m (No.12). The transformation \nbetween the trigonal and monoclinic lattices is shown in Fig. 1 a. The optimal stoichiometry of our sample \nwas confirmed by magnetic susceptibility and transmission electron microscopes measurements at ambient \npressure (Fig. S1) . Our high -pressure synchrotron X-ray diffraction measurement of LuFe 2O4 (Fig. S2) was \ncarried out on a high quality single -crystalline sample. Within  the representation of space group C2/m (Fig. \nS3), the lattice parameters are a = 5.957(2) Å, b = 3.434(4) Å, c = 8.642(3) Å, and = 103.28(2) ° at 300 K \nand 0.8 GPa. The lattice parameters a, b, and c shrink with increasing pressure. At 300 K and 12.6 GPa, the \nlattice parameters become a = 5.809(5) Å, b = 3.339(16) Å, c = 8.374(11) Å, and =103.39(9) ° with a volume \nreduction of 8.2(6)%.  \n \n \nFig. 1 Crystal structure and charge order evolution of LuFe 2O4 under high pressure. (a) Crystal \nstructure of LuFe 2O4 at ambient pressure. Dotted lines represent the trigonal lattices. Red, blue and green \nsolid lines represent three equivalent monoclinic lattices with different twinned directions. ( b-e) Single \ncrystal  X-ray diffraction (SXD) intensities in the ( HHL ) scattering plane measured near the charge ordering \n(CO) superlattice peaks at room temperature at 0.8, 3.1, 5.0 and 6.0 GPa, respectively. ( f-h) SXD intensity \nalong the Q = (1/3, 1/3, L), (1/3, 1/3, L) and (1/4, 1/4, L) direction measured at 0.8 GPa, 5.0 GPa and 6.0 \nGPa, respectively. (I) Summary of the charge order peak positions in HK-plane measured from 0.8 to 12.6 \nGPa. The black hexagon in center indicates ( H, H) = (1/3, 1/3).  \n \n \n \n4 \n Apart from the lattice shrinkage, a series of superlattice peaks were also observed with a wave vector kAP \n= (1/3, 1/3, 3/2) T in trigonal lattice or (0, 2/3, 1/2) M in monoclinic lattice at 0.8 GPa, as illustrated in Fig. 1 b \nand Fig. 1f. This wave vector is  close to that observed at ambient pressure where the charge order of low -\nvalence (Fe2+) and high -valence (Fe3+) Fe ions form s a √3×√3×2 super -lattice4 (Fig. 2 b). As pressure \nfurther increases, the superlattice peaks exhibit drastic broadening along the L-direction (Fig. 1( b-d)), \nindicating that the charge order becomes quasi -two-dimensional. Meanwhile, the maximum of peak intensity \nin L-direction moves from half  integer ( L = n+1/2) at ambient pressure to integer ( L = n) at 5.0 GPa (Fig. 1( d, \ng) and Fig. S4), indicating that the inter -plane polarization emerges under pressure (Fig. 2 d). Interestingly, \nsimilar quasi -two-dimensional charge order was also observed above the 3D charge ordering temperature of \n320 K in LuFe 2O4 at ambient pressure4. \n \n \nFig. 2 Charge order models of LuFe 2O4 corresponding to observed  SXD pattern from 0.8 GPa to 12.6 \nGPa. Red and blue spheres represent Fe2+ and Fe3+ ions, respectively. Sphere size in ( a, c, e) indicate the Fe \nions in different planes. Thick and thin solid lines represent nearest neighbored (NN) and next nearest \nneighbored (NNN) Fe -Fe pairs, respectively. Dashed lines represent lattice boundaries.  (a, b) \nCentrosymmetric CO -AP model with k ~ (1/3, 1/3, 3/2). (c, d) Non-centrosymmetric CO -2D model with k ~ \n(1/3, 1/3, 0) at P = 5.0 GPa. Noted the inter -layer arrangements in ( b) and ( d) are different due to change in \nL-component of the wave vector. (e, f) CO-HP model for k = (1/4, 1/4, 0), corresponding SXD patterns \nobserved between 6.0 and 12.6 GPa.  \n \nA closer inspection on the diffraction pattern actually reveals a tiny incommensurability of the charge order \nwave vector [ kAP = (1/3+ δ, 1/3+ δ, 3/2) T and k2D = (1/3 -δ, 1/3-δ, 0) T] at 0.8 GPa. The incommensurability \nincreases dramatically with increasing pressure, and eventually reaches a commensurate position kHP = (1/4, \n1/4, 0) T at pressures above 6.0 GPa. In contrast to the broadened diffraction patterns alo ng the L-direction for \nthe superstructure below 5.0 GPa (Fig. 1 d), the (1/4, 1/4, 0) T phase shows sharp peak features both along the \nin-plane ( HK) and out -of-plane ( L) directions (Fig. 1 (e, h)), indicating restoration of  robust 3D order . This \ncommensurate CO -HP phase persists up to 12.6 GPa without a major change in the primary structure.  \nThe evolution of the incommensurability of the superlattice peaks in the HK -plane under various pressures \nis summarized in Fig. 1 i. There are three 120 -degree twinned charge orders in LuFe 2O4, each with a unique \nk-vector, therefore the superlattice peak position differs for each charge order twin, forming a spiral -like \ndiffraction pattern in reciprocal space. As pressure increases, the centers of charge order peaks move away \nfrom (1/3, 1/3) along the 120 -degree directions in the HK-plane and eventually reach (1/4, 1/4), (1/2, 1/4), \n(1/4, 1/2) at 6.0 GPa. Therefore, the quasi -2D charge order phase observed in the 3.1 GPa to 5.0 GPa range \ncan be regarded as  an intermediate phase between the commensurate (1/3, 1/3) and (1/4, 1/4) charge -order \nphases.  \n \n \n5 \n We now discuss the possible charge order models for the high -pressure phases. We consider two valence \nstates of Fe ions (Fe2+ and Fe3+). Based on diffraction data, we found that the charge order of (1/4, 1/4, 0) T \nphase can be best described with the Pb2/c (BNS 13.71) black -white space group  (Fig. 2( e,f) and Fig. S5). \nThe CO phase models of CO -AP [kAP = (1/3, 1/3, 3/2) T], CO -2D [k2D = (1/3, 1/3, 0) T] and CO -HP [kHP = \n(1/4, 1/4, 0) T] are illustrated in Fig. 2( a,b), (c,d) and ( e,f), respectively. These CO models are further \nsupported by DFT calculation (Fig. 4d). The evolution from (1/3, 1/3) T to (1/4, 1/4) T can be understood \nthrough the Coulomb interactions on different types of Fe -Fe bonds. In CO -AP and CO -2D phases, 5/9 of \nthe nearest neighbor (NN), 2/3 of the 2nd NN and 5/9 of the 3rd NN Fe bonds are Fe2+-Fe3+ bonds. On the \ncontrary, in CO -HP phase, 2/3 o f the NN, 2/3 of the 2nd NN and 1/3 of the 3rd NN Fe bonds are Fe2+-Fe3+ \nbonds. Above observation provides a natural understanding, that the effective Coulomb interactions of NN \nand 3rd NN Fe -Fe pairs  are tuned by the compression in frustrated triangular d ouble -layer Fe 2O4 structure \ndue to increased pressure. We also calculated the interlayer next -nearest -neighbor ( VcNNN) and intralayer \nnearest -neighbor ( VabNN) Coulomb interactions at each pressure point. Our result shows that VcNNN/VabNN \nremains almost unchanged in the region of 1 to 5 GPa, and drops sharply above 6 GPa where the (1/4, 1/4) T \nCO phase is favored24 (Fig. S6).  \n \n \nFig. 3 Magnetic properties of LuFe 2O4 under pressure.  (a-c) Fe K -edge X -ray absorption near edge \nstructure (XAS) and XMCD spectroscopy data measured at T = 100 K, H = 5 T and P = 1.9 GPa ( a), 4.3 GPa \n(b), 9.5 GPa ( c), respectively. The pressures correspond to the three observed charge order phases (CO -AP, \nCO-2D, CO-HP) in LuFe 2O4. Insets of ( a-c) show the XAS and XMCD data of pre -edge region at 1.9 GPa, \n4.3 GPa and 9.5 GPa, respectively. ( d) Ferrimagnetic model for the centrosymmetric CO -AP phase ( e) \nFerrimagnetic model for the non -centrosymmetric CO -2D phase ( f) Antiferromagnetic model for the \ncentrosymmetric CO -HP phase for DFT calculations in pressurized LuFe 2O4. (Ferri: ferrimagnetic. AFM, \nanti-ferromagnetic).  \n \nIn order to inform on the magnetic ground states associated with these charge -ordered phases, we util ized \nHP-XMCD spectroscopy to monitor the evolution of net magnetization of LuFe 2O4 under pressure. Fig. 3a \nshows Fe K -edge isotropic absorption (XAS) and dichroic (XMCD) signals in LuFe 2O4 at 100 K and 5 T. \nThe measurements were performed at the Fe K-edge instead of the more commonly used Fe L-edge because \nsoft X-ray MCD is incompatible with the highly absorbing diamond anvil cell environment25-27. The dichroi c \nsignal at 1.9 GPa consists of a positive peak in the pre -edge region (7114.0 eV) and anothe r positive peak \nnear the main rising edge peak (7129.0 eV) with larger intensity  (Fig. 3a). With increasing pressure, these \ntwo dichroic peaks become much weaker and vanish at 9.5 GPa  (Fig. 3c). This result indicates non -zero net \nmagnetization at 1.9 GPa  and 4.3 GPa, and zero net magnetization at 9.5 GPa. The peaks of Fe K-edge XAS \nand XMCD signals observed at 1.9 GPa are in agreement with those observed at ambient pressure ( Fig. S7)28. \nTherefore, the magnetic order at 1.9 GPa  is best interpreted as a ferrimagnetic structure which was unveiled \n \n \n6 \n by the previous L -edge XMCD18 and neutron diffraction15 measurements at ambient pressure. Meanwhile, \nthe peak positions of dichroism peaks are unchanged at 1.9 GPa and 4.3 GPa (Fig. 3 (a-c)). This result suggests \nthat the ferrimagnetic arrangement of Fe2+ and Fe3+ moments is preserved at 4.3 GPa, but the net \nmagnetization is gradually reduced by pressure. Insets of Fig. 3(a-c) illustrate the pre -edge region of XAS \nand XMCD signals. The posi tions of XAS pre -edge peak and leading edge do not shift by pressure, which \nindicates that the valence ratio of Fe2+ and Fe3+ remains unchanged ( Fig. S8), in agreement with the CO \nmodels illustrated in Fig. 2.  \n \n \nFig. 4 Phase diagram of LuFe 2O4 under pressure.  (a) Relative reduction of lattice parameters a and c by \nincreased pressure measured between 0.8 and 12.6 GPa. ( b) Evolution of in -plane incommensurate CO wave \nvector δ by pressure. δ is defined by (1/3+ δ, 1/3+ δ) in CO -AP and (1/3 -δ, 1/3 -δ) in CO -2D. The \ncommensurate CO wave vector (1/4, 1/4) of CO -HP is represented by δ=1/12. ( c) The pressure dependence \nof XMCD spectral weight. ( d) The pressure dependence of relative enthalpies between the charge order \nphases mentioned in Fig. 2 and Fig. 3d-f. (ferri: ferrimagnetic. AFM:anti -ferromagnetic).  \n \nMore insight into the nature of the magnetic ground states associated with the CO phases in pressurized \nLuFe 2O4 can be obtained by DFT calculations. For CO -AP phase and CO -2D phase, DFT calculations show \nthat LuFe 2O4 adopts the “2:1 ferrimagnetic state” as ground state. In this state, as shown in Fig 3( d, e), \nmajority spin orientation consists of all Fe2+ ions plus  one-third of total Fe3+ ions while the minority spin \norientation consists of remaining Fe3+ ions. For CO -HP phase, DFT calculation shows that the \nantiferromagnetic structure (Fig. 3f) is the magnetic ground state ( Fig. S9). In this state, both Fe2+ and Fe3+ \nare evenly separated in opposite spin orientations, resulting in the centrosymmetric electromagnetic order. \nThis result is consistent with the XMCD measurement, where the net magnetization  is completely suppressed \nby pressure.  \n \nDISCUSSION S \nCombining the  HP-SXD, HP -XMCD and DFT calculations, we have found a series of 3D -2D-3D charge \norder transitions in the hexagonal LuFe 2O4. The emergen ce of the CO-HP phase at high pressure is \naccompanied by a ferri -magnetic to antiferromagnetic transition (Fig. 3( d-f)). We calculate the pressure \ndependence of the enthalpy for each spin -charge ordered phase using GGA+ U method29,30, as illustrated in \nFig. 4d. According to the calculation, at ambient pressure, the 3D centrosymmetric CO phase (CO -AP) \ncombined with 2:1 ferrimagnetic order (Fig. 3d) is most favored. Then, as the lattice shrinks with applying \nhydrostatic pressure (Fig. 4a), a 2D non -centrosymmetri c CO phase (CO -2D) with 2:1 ferrimagnetic order \n(Fig. 3e) becomes the ground state.  At higher pressure, a 3D centro -symmetric CO phase (CO -HP) with \nantiferro -magnetic order (Fig. 3f) becomes the ground state. During the CO -AP to CO -2D phase transition, \nthe Fe-Fe bonds within [Fe 2O4] bilayer structure are compressed by pressure. Such compression allow s \ncharge transfer among low valence and high valence  Fe sites  in the frustrated lattice , which could result in \n \n \n7 \n damping of out -of-plane correlations and incommensurability in ab -plane (Fig. 1( b-d)). As the pressure \nfurther increases, the effective Coulomb interaction among Fe -Fe bonds reaches a critical point where the Q \n= (1/3, 1/3) charge orders cannot be sustained. Thus the system evolves into a less fr ustrated Q = (1/4, 1/4) \nCO-HP charge order where all Fe layers are charge neutral. Due to the strong coupling between charge and \nspin correlations, such drastic redistribution of charge density in turns causes substantial change in the \nmagnitude of spin -spin interactions. As a result, the magnetic moments on Fe2+ and Fe3+ are re -arranged and \nthe magnetic structure evolves from ferrimagnetism to anti -ferromagnetic order. The DFT calculation s and \nexperimental evidence indicate  that the observed charge orderin g and magnetic transitions are coupled and  \nstrongly correlated with the compression of  the crystalline lattice. Our calculation reveals that the enthalpies \nof the three phases (CO -AP, CO -2D, CO -HP) are close to each other (Fig. 4d). This is not surprising as the \ndouble -layer triangular structure of LuFe 2O4 induces strong frustration in both charge and spin exchange \ninteractions.  \nIn summary, we have conducted a systematic study on the evolution of the crystal structure, charge orders \nand spin orders in hexagonal LuFe 2O4 under pressure using the in -situ HP -SXD and HP -XMCD \nmeasurements. With increasing pressure, the system exhibits three correlated charge ordered ground states: \n1) the centrosymmetric 3D CO -AP phase with ferrimagnetic order, 2) the non -centrosymmetric CO -2D phase \nwith ferrimagnetic order, and 3) the centrosymmetric 3D CO -HP phase with antiferromagnetic order. These \nresults suggest strong coupling between charge and magnetic orders in hexagonal LuFe 2O4. The evolution of \nthe charge and magneti c order under pressure is further supported by the DFT calculations, which show that \nthe phase transitions are the result of tuned frustrated charge and spin interactions induced by the compression \nof triangular Fe 2O4 double layer structures. Our study pro ves that hydrostatic pressure is a powerful tool to \nunveil novel charge and magnetic states in hexagonal LuFe 2O4 and other candidate multiferroics , and also \nprovides new insights on realiz ing the potential charge -order induced multiferroicity in LuFe 2O4. \n \nMETHODS  \nSample Preparation  \nSingle crystals of LuFe 2O4 were grown by the floating -zone method using a CO/CO 2 (~ 2.7:1) mixed \natmosphere to control oxygen stoichiometry. Our electron probe micro analyzer (EPMA) measurement on \nthe single crystalline sample shows almost optimal stoichiometry of Lu 1.01(1) Fe2O3.97(4) . \nHigh -pressure measurement  \nSingle -crystal X -ray diffraction (SXD) experiment was conducted at synchrotron radiation beamline 13 -BM-\nC, Advanced Photon Source (APS), Argonne National Lab (ANL), using  monochromatic X-rays with 0.434 \nÅ wavelength, with the pressure increasing from 0.8 GPa up to 14.5 GPa. The 1 -deg step scan, wide -step \nscan and whole -range scan were performed with a scanning angle range of ± 35 degrees at each pressure \npoint31. Symmetry a nalysis was conducted using Sarah software32 and Bilbao Crystallographic Server33, and \nstructure refinements were conducted using FULLPROF program suite34. High -pressure synchrotron X -ray \nMagnetic Circular Dichroism (XMCD) spectroscopy measurements with th e energy scanning across the Fe \nK-edge were conducted at beamline 4 -ID-D, APS, ANL. Circularly polarized X -rays were generated with a \ndiamond phase retarder. To obtain the XMCD signal, the X -ray helicity was modulated at 13.1 Hz, and \nXMCD signals were dete cted with a phase lock -in amplifier. In addition, XMCD scans were repeated with \nopposite applied field direction to ensure artifact -free XMCD signals35. Corresponding to the three observed \nCO phases in pressurized LuFe 2O4, the XMCD data were collected at 1 .9 GPa, 4.3 GPa, and 9.5 GPa. In \nthese measurements, magnetic field was set to +5T/ -5T and temperature to 100 K. X -ray Absorption \nSpectroscopy (XAS) data is collected simultaneously during the XMCD measurement and obtained by \naveraging X -ray absorption for  opposite X -ray helicities. The pressure was determined by the in -situ ruby \nfluorescence measurement system36 with the pressure uncertainty less than 5%. An offline ruby system was \nused in the SXD measurement, before and after each manual pressure change a t ambient temperature. For  \n \n8 \n the low temperature XMCD measurements, online membrane and ruby fluorescence systems were used to \napply and measure pressure, respectively (See details  in supplementary Section S 1). \n \nDensity Functional Theory (DFT) calculations  \nThe first -principle density functional theory calculations are based on the projector augmented wave (PAW) \nmethod37 encoded in the Vienna ab initio simulation package (VASP)38. The exchange -correlation functional \nof the Perdew -Becke -Erzenh (PBE)39 form is adopted and the plane -wave cutoff energy is set to 500 eV. To \nproperly describe the strong electron correlation in the Fe 3 d states, the GGA plus on -site repulsion U method \n(GGA+ U)29 was employed with the effective U value ( Ueff = U - J) of 4 eV. Calculati ons with various Ueff \nshow that the main results remain valid when Ueff is varied between ~3.1 and 5.5 eV. The structural \noptimizations are carried out until the forces acting on atoms are smaller than 0.01 eVÅ-1. To obtain the \nenergy of each charge order structure, GGA+ U calculations were carried out in two steps30. For each charge \norder structure, we first optimize the chosen charge order structure using a large U (e.g., Ueff = 7.5 eV). Then \nwe re -optimize the charge order structure with a smaller U (i.e., Ueff = 4 eV) using the converged charge \ndensities obtained with the large U. It is noted that a larger Ueff was employed only to generate an initial \ncharge density with the desired charge order structure. The enthalpies of each charge order phase under \nvarious pressure are calculated by follow expression: H = U – PV, where H is enthalpy, U is total energy and \nPV is product of pressure and volume. 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We \nacknowledge Dr. Lili Zhang, Dr. Sheng Jiang and Dr. Aiguo Li of 15 U1 at SSRF, and Dr. Changyong Park \nat HPCAT, APS, ANL for the technical supports on the high pressure XRD experiment. Gas loading by \nSergey N. Tkachev is also acknowledged. HPCAT operations are supported by DOE -NNSA under Award \nNo. DE -NA0001974 and DOE -BES under Award No. DE -FG02 -99ER45775, with partial instrumentation \nfunding by NSF. The SXD measurement was conducted at 13BM -C, APS, ANL. 13BM -C operation is \nsupported by COMPRES through the Partnership for Extreme Crystallography (PX2) project, under NSF \nCooperative Agreement EAR -1661511 and by GSECARS under NSF EAR -1634415. The XMCD \nexperiment was conducted at beamline 4ID -D, APS, ANL. This research used resources of the Advanced \nPhoton Source, a U.S. Department of Energy (DOE) Office of Science User Facili ty operated for the DOE \nOffice of Science by Argonne National Laboratory under Contract No. DE -AC02 -06CH11357.  \n \nAUTHOR CONTRIBUTIONS  \nWenbin Wang, Jian Shen and Wenge Yang proposed and conceived this project. Fengliang Liu conducted \nthe high-pressure XRD an d XMCD experiments. Yiqing Hao, Fengliang Liu, Wenbin Wang, Jun Zhao, \nLifeng  Yin, and Wenge Yang analyzed the data. Jinyang Ni and Hongjun Xiang performed the DFT \ncalculations. Dr. Dongzhou Zhang supported the SXD measurement, Drs. Gilberto Fabbris and Dan iel Haskel \nsupported the XMCD measurement, Dr. Yongsheng Zhao offered help in the XMCD measurements. Drs. \nShaobo Cheng, Xiaoshan Xu and Xujie Lü  offers valuable discussion during this project. Fengliang Liu, \nYiqing Hao, Wenbin Wang, Wenge Yang and Jian She n wrote the original version of the manuscript, and \nreceived feedbacks from all authors.  Fengliang  Liu, Yiqing Hao  and Jinyang Ni  contributed equally  to this \nwork.  \n \nCONFLICT OF INTEREST  \nThe authors declare no conflict of interests.  \n \nADDITIONAL INFORMATION  \nSupplementary Information  \nSupplementary data associated with this article can be found, in the  supplementary files.  \n \n \n \n \n "    },    {        "title": "0909.0209v1.Magnetism_in_Re_based_ferrimagnetic_double_perovskites.pdf",        "content": "arXiv:0909.0209v1  [cond-mat.mtrl-sci]  1 Sep 2009Magnetism in Re-based ferrimagnetic double perovskites\nA. Winkler,1N. Narayanan,1D. Mikhailova,1K. G. Bramnik,1H. Ehrenberg,1H. Fuess,1\nG. Vaitheeswaran,2,3V. Kanchana,2F. Wilhelm,4A. Rogalev,4A. Kolchinskaya,1and L. Alff1,∗\n1Technische Universit¨ at Darmstadt, Petersenstr. 23, 6428 7 Darmstadt, Germany\n2Royal Institute of Technology(KTH), Brinellv¨ agen 23, 100 44 Stockholm, Sweden\n3ACRHEM, University of Hyderabad, Hyderabad 500 046, India\n4European Synchrotron Radiation Facility (ESRF),\n6 Rue Jules Horowitz, BP 220, 38043 Grenoble Cedex 9, France\n(Dated: received 29 April 2009)\nWe haveinvestigated spin andorbital magnetic momentsof th eRe5dionin thedoubleperovskites\nA2FeReO 6(A= Ba, Sr, Ca) by X-ray magnetic circular dichroism (XMCD) at t he ReL2,3edges.\nIn these ferrimagnetic compounds an unusually large negati ve spin and positive orbital magnetic\nmoment at the Re atoms was detected. The presence of a finite sp in magnetic moment in a ’non-\nmagnetic’ double perovskite as observed in the double perov skite Sr 2ScReO 6proves that Re has\nalso a small, but finite intrinsic magnetic moment. We further show for the examples of Ba and Ca\nthat the usually neglected alkaline earth ions undoubtedly also contribute to the magnetism in the\nferrimagnetic double perovskites.\nPACS numbers: 75.25.+z, 75.30.-m, 75.50.-y\nI. INTRODUCTION\nOrdered double perovskites of the composition\nA2MNO6(withAan alkaline earth, Ma magnetic tran-\nsition metal ion, and Na non-magnetic ion) have come\nagain into the focus of research because of their inter-\nesting magnetic properties. First, in Sr 2FeMoO 6a large\nroom-temperature magnetoresistance was observed [1].\nSecond, within the group of ferrimagnetic double per-\novskites materials with higher Curie-temperatures, TC,\nthan in the simple perovskites (e.g. doped manganites)\ncan be obtained. At the moment the highest TCvalues\nhave been reported for Sr 2CrReO 6(TC≈635K) [2, 3, 4]\nand Sr 2CrOsO 6(TC≈725K) [5, 6, 7]. Third, the mech-\nanism leading to magnetic coupling is believed to be as-\nsociated with a strong tendency to a half-metallic nature\nofthe chargecarriersatthe Fermilevel[8, 9, 10]. Clearly,\nthese materials are interesting candidates for spintronic\napplications [11], in particular when having in mind fully\nepitaxial structures based on perovskite materials.\nRecently, Majewski et al.and Sikora et al.have\nproposed a simple scaling law between the Curie-\ntemperature and the induced magnetic moment at the\nnon-magnetic site in the double perovskite structure\n[4, 12, 13]. Philipp et al.have discussed that a high\nCurie-temperature is associated with a tolerance factor\nclose to one for the corresponding crystal [14]. The only\nexception for this rule is found in the series A2FeReO 6\n(A= Ba, Sr, Ca). In this particular FeRe-system it\nis the strongly monoclinically distorted Ca-based com-\npound having an anomally high TC, namely about 540K\n[15, 16, 17] (comparing to about 400K for Sr 2FeReO 6\n[16] and 325K for Ba 2FeReO 6[18, 19]). The dimen-\n∗Electronic address: alff@oxide.tu-darmstadt.desionless tolerance factor, f, inA2FeReO 6whose devia-\ntion from unity implies structural distortion varies from\naboutf= 1.057 forA= Ba over f= 0.997 forA=\nSr tof= 0.943 for A= Ca [14]. In general, the\nBa-based ferrimagnetic double perovskites are close to\na structural transition into a hexagonal lattice where\nferro(i)magnetism is not allowed for symmetry reasons;\nthe Sr-based compounds are always close to a perfect cu-\nbic structure with maximal TC, and the Ca-based double\nperovskites are orthorhombically or monoclinically dis-\ntorted, with still a large but - due to the reduced ex-\nchange - clearly reduced ferrimagnetic transition temper-\nature. The exceptional large TCof Ca2FeReO 6is accom-\npanied by an insulating state at low temperatures, in\ncontrast to Sr 2FeReO 6or even the similarly monoclini-\ncally distorted Ca 2FeMoO 6which both are metallic [20].\nThe metal-insulator transition in Ca 2FeReO 6has been\nreported to occur between 100 and 150K [15, 16, 20].\nThis behavior has been attributed to strongly enhanced\nelectron-electron correlations on the Re site due to a re-\nduced transfer integral between Fe and Re corresponding\nto an extremely large effective Coulomb repulsion, Ueff,\nofabout 4eV on both ions [20]. This, however, is in some\ncontradiction to the observed high Curie-temperature,\nwhich is believed to be a consequence of a kinetic energy\ngain due to the hybridization of the Fe 3 dand Re 5 d t2g-\norbitals. The prediction that a decreased band-filling is\nfavorable for TC[21], which could be used to conciliate a\nhighTCwith a reduced Re-Re overlap, has turned out to\nbe not valid: an increased band-filling actually leads to\na strong TCenhancement for both the FeMo-system [22]\nand the CrW-system [23]. Within the kinetically driven\nexchangemodel[8, 9, 10], theincreaseof TCismorenatu-\nrallyexplained asaconsequenceofincreasedband-filling.\nNote, that the casesof Ca 2FeReO 6and Sr 2CrOsO 6, both\nbeing insulating and having a high TCat the same time,\narecompletelydifferent: InthecaseofSr 2CrOsO 6having2\nTable I: Summary of sample properties from x-ray diffrac-\ntion at 300K (calculated by Rietfeld-refinement) and SQUID\nmagnetometry.\nmaterial TCsymm. lattice antisites\n(K) (˚A) (%)\nBa2FeReO 6317Fm3ma= 8.0571(2) 0.9\nSr2FeReO 6418Fm3ma= 7.8752(4) 2.6\na= 5.3992(2)\nCa2FeReO 6556P21/nb= 5.5269(2) 3.6\nc= 7.6826(3)\na= 5.6760(2)\nSr2ScReO 6-P21/nb= 5.6534(2) 7\nc= 7.9862(3)\nonlyatinyrhombohedraldistortion,theOs5 dt2gbandis\ncompletelyfilled, whileforCa 2FeReO 6itisthestructural\ndistortion that drives the metal-insulator transition. Re-\ncently, it was suggested that in double perovskites with\nheavy ions as Re a large orbital contribution to the mag-\nnetic moment leads to an enhanced total magnetization\nabovetheintegervaluethatisexpectedforahalf-metallic\nmaterial [25]. This elsewhere predicted and calculated\n[26] strong influence of spin-orbit coupling leads to a\nquasi-half metallicity which still is from the view point of\napplications in spintronics very high (above 90%). An-\nother point of interest is the possibility of an intrinsic\nenhancement of the Re spin magnetic moment due to\nthe peculiar Re5+state in the ferrimagnetic double per-\novskites. In this study, we present the XMCD analysis\nof the system A2FeReO 6(A= Ba, Sr, Ca), compare the\nexperimental data to theoretical predictions calculated\nwithin the full-potential linear muffin-tin orbital method\n(FP-LMTO) [27] with included spin-orbit coupling, and\ncomplete the so far suggested scaling law [4] by using the\nidentical method to extract separately spin and orbital\nmagnetic moments. Furthermore, we search for a con-\ntribution of the alkaline earth element to the magnetic\nbehavior, and also look for an intrinsic Re moment in\na suited double perovskite compound with Mbeing a\nnon-magnetic ion: Sr 2ScReO 6.\nII. EXPERIMENTAL\nA summary of the sample properties is given in Ta-\nble I. All values where a comparison can be made to\nliterature values are in good agreement with these data\n[2, 15]. Note that the small amount of antisite disorder\ndoes not affect our results. The XMCD measurementson\nthe ReL2,3edges were performed at the European Syn-\nchrotron Radiation Facility (ESRF) at beam line ID12\n[29]. The spectra were recorded within the total fluores-\ncenceyield detection mode. The XMCDspectrawereob-\ntained as direct difference between consecutive XANES/s49/s48/s53/s53/s48 /s49/s48/s54/s48/s48 /s49/s49/s57/s53/s48 /s49/s50/s48/s48/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48\n/s82/s101/s32 /s76\n/s50/s32/s101/s100/s103/s101/s82/s101/s32 /s76\n/s51/s32/s101/s100/s103/s101\n/s32/s67/s97\n/s50/s70/s101/s82/s101/s79\n/s54/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s66/s97\n/s50/s70/s101/s82/s101/s79\n/s54\n/s32/s83/s114\n/s50/s70/s101/s82/s101/s79\n/s54\nFigure 1: XMCD spectra for A2FeReO 6(A=Ba, Sr, Ca).\nscans (X-ray Absorption Near Edge Spectrum) recorded\nwith opposite helicities of the incoming x-ray beam. To\nensure that the XMCD spectra are free from any exper-\nimental artefacts the data were collected for both direc-\ntions of the applied magnetic field of 6T (parallel and\nantiparallel to the x-ray beam). The degree of circular\npolarization of the monochromatic x-ray beam was 98%.\nThe measurements were performed at about 10K for all\nsamples ( T≪TC), if not indicated otherwise. Since the\nsamples measured in backscattering geometry were very\nthick, the spectra were first normalized to the edge jump\nof unity and then corrected from self-absorption effects.\nThe edge jump intensity ratio L3/L2was then normal-\nized to 2.19/1 [30]. This is different from the statistical\n2:1branching ratiodue to the difference in the radial ma-\ntrix elements of the 2 p1/2to 5d(L2) and 2p3/2to 5d(L3)\ntransitions. The XMCD measurement as a function of\napplied field suggests that our samples are closer to sat-\nuration at 6T as is concluded by de Teresa at al.[25]\nfrom high-field SQUID measurements. This issue has to\nbe clarified in future by high-field XMCD measurements.\nIII. RESULTS AND DISCUSSION\nIn this paper, the XANES spectra themselves are\nnot further discussed. As shown in Fig. 1, for FeRe-\ncompounds at both absorption edges we find a rather\nintense XMCD signal. This is a clear evidence for the\nexistence of a magnetic moment at the Re 5 dshell. For\nall three compounds, the XMCD spectra at the L2edge\nare largest (as expected for m= 1 orbitals) and similar\nin shape. In Ca 2FeReO 6the size of the XMCD signal\nis by a factor of 2 smaller compared to the two other\nFeRe compounds. At the L3edge, the Ca-based double\nperovskite again stands out by a pronounced peak with\nnegative XMCD signal which is absent for Sr 2FeReO 6\nand Ba 2FeReO 6. The data at the L3edge look slightly\ndifferent in amplitude as compared to previously pub-\nlished data [13]. However, the data are consistent in that\nthe integrated XMCD intensity at the L3edge is nega-\ntive only in the case of Ca 2FeReO 6. In this sense, all\ndata support the unusual behavior of Ca 2FeReO 6, which\ncannot only be attributed to the different ionic size of the3\nTable II: Measured (exp., normalized to 5K) and calculated\n(th., calculated within the generalized gradient approxim a-\ntion including spin-orbit coupling (GGA+SO)) magnetic mo-\nments at the Re site for different double perovskites at about\n10K. For a detailed discussion of the applied band-structur e\ncalculation see e.g. [26, 28]. Calculation in [31] is GGA with\nspin-orbit coupling. The number of d-holes was taken from\nthe band-structure calculation. In our case this number was\naround 5 .3. The error of the measured values is estimated as\n2.5%.\nmaterial mS(µB/f.u.)mL(µB/f.u.)|mL/mS|\nexp. Ba 2FeReO 6 −0.56 0 .15 0 .27\nSr2FeReO 6 −0.74 0 .21 0 .28\nCa2FeReO 6 −0.47 0 .16 0 .34\nSr2CrReO 6[4] −0.68 0 .25 0 .37\nSr2ScReO 6(80K) 0 .013 −0.002 0 .15\nth. Ba 2FeReO 6 -0.65 0.19 0.29\nSr2FeReO 6 -0.68 0.15 0.22\nSr2FeReO 6[31] -0.85 0.23 0.27\nSr2CrReO 6[26] -0.85 0.18 0.21\n/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52\n/s49/s48/s53/s50/s53 /s49/s48/s53/s53/s48 /s49/s48/s53/s55/s53 /s49/s49/s57/s53/s48 /s49/s50/s48/s48/s48/s48/s46/s48/s48/s46/s57/s49/s46/s56\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s56/s48/s32/s75/s101/s108/s118/s105/s110\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s114\n/s50/s83/s99/s82/s101/s79\n/s54/s82/s101 /s32/s76\n/s50\n/s32/s82/s101 /s32/s76\n/s51\nFigure 2: XANES and derived XMCD spectra at the Re L2\nandL3edges of Sr 2ScReO 6.\nAsite ions.\nIn Fig. 2 we show XANES and XMCD spectra for the\ncompound Sr 2ScReO 6. This compound is important be-\ncause the absence of any free electrons at Sc3+which\nhas a 3d0configuration will lead to a complete break-\ndown of the inducedmagnetic moment at the Re site.\nThis compound therefore allows the measurement of the\nintrinsic magnetic moment of Re5+(also in contrast to\nRe6+compounds as Sr 2MgReO 6). Previously, Kato et\nal.have calculated from a Curie-Weiss fit to the suscep-\ntibility aneffective magneticmomentofRe in Sr 2ScReO 6\nof about 1.1 µB/f.u., as expected within the ionic picture\n[32]. In contrast, our data show the existence of a much\nsmaller, but finite intrinsic moment at the Re site, in-\ndicating an increased tendency to magnetic ordering of\nRe5+. Since this moment is present above the antiferro-\nmagnetic transition temperature, it is not related to spin\nglass behavior. The spin magnetic moment is about 50\ntimes smaller than corresponding induced moments on/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50\n/s53/s50/s53/s48 /s53/s50/s55/s53 /s53/s54/s50/s53 /s53/s54/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s66/s97\n/s50/s70/s101/s82/s101/s79\n/s54/s97/s41\n/s66/s97/s32 /s76\n/s50\n/s66/s97/s32 /s76\n/s51\n/s52/s48/s51/s53 /s52/s48/s52/s48 /s52/s48/s52/s53 /s52/s48/s53/s48 /s52/s48/s53/s53 /s52/s48/s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s54/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110/s67/s97\n/s50/s70/s101/s82/s101/s79\n/s54\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s67/s97/s32 /s75/s98/s41\n/s32\nFigure 3: XANES and derived XMCD spectra at the a) Ba\nL2andL3edges of Ba 2FeReO 6and b) at the Ca Kedge of\nCa2FeReO 6.\nRe5+, and the orbital magnetic moments even by a fac-\ntor of 100. However, due to the high sensitivity of the\nset-up at ESRF, one can unambiguously prove the exis-\ntence of this moment. In contrast to the opposite sign of\ntheinducedmagneticmomentwithrespecttotheapplied\nfield, the spin magnetic moment at the Re in Sr 2ScReO 6\nis aligned with the field. This is expected because the\nkinetic exchange via fully polarized spin down is not at\nwork. This intrinsic moment of Re5+therefore has to be\nconsidered as an indicator of the tendency to unusually\nhigh magnetization of Re based double perovskites.\nAs a last point, we address magnetism in the earth al-\nkaline ions itself, which usually are completely neglected\nin the magnetic scenario. The XANES and XMCD spec-\ntra of the Ba L2andL3edges of Ba 2FeReO 6and of the\nCaK-edge of Ca 2FeReO 6are shown in Fig. 3. The 5 d\nspin magnetic moment (calculated with 9 as the number\nofd-holes corresponding to the band-structure calcula-\ntion) of Ba is µS=−0.0065 and the 5 dorbital magnetic\nmoment µL=−0.0013 (both in µB/f.u.),|µL/µS| ≈0.2.\nThe theoretical predictions calculated as described else-\nwhere [26, 28] are µS=−0.0084 and µL=−0.0014\nwhich is in fair agreement with our experimental data.\nFor Ca 2FeReO 6we can only qualitatively say that a fi-\nnite magnetic moment is observed, because the K-edge\nprobes only the 4 porbital magnetism. Since the Ledges\nare experimentally not accessible, a quantative analysis\ncannotbe done. Theobservationofa magneticallypolar-\nized density of states gives clear evidence for a magnetic\ninteraction of the earth alkaline ions with the other ions.\nThe magnetic contribution of Ba in this case is a fac-\ntor of 2 smaller than the contribution of the intrinsic Re\nmoment. Naturally, one expects that the magnetic con-4\ntribution increases with ionic size due to the increased\nexchange with the neighboring ions. The clear orbital\ncontribution in Ba 2FeReO 6is not unexpected due to the\nheavy ionic mass. Our data provide a test for a detailed\ntheoretical study of the magnetism in the double per-\novskites, and underlines the importance of taking spin-\norbit coupling into account. Note, that for example in\nCrO2, where the importance of oxygen in the magnetic\nmechanism is undoubted, comparable values of spin and\norbital moments of the oxygen ion have been measured\n[33] as compared to our results on Ba in Ba 2FeReO 6.\nIn Table II we summarize our results for the spin and\norbital magnetic moments at the Re site as derived from\nthe XMCD measurements by applying the standard sum\nrules [34, 35] and compare them to theoretical values.\nAlso, the ratio |mL/mS|is calculated, since this quan-\ntity is not affected by possible uncertainties in the calcu-\nlated number of holes. In general, the calculated dataare\nin surprisingly good agreement with the measured data.\nOne of the main reasons certainly is, that spin-orbit cou-\npling is taken into account from the beginning. Note,\nthat in the hard x-ray range the sum rules apply with\nhigh validity due to the large spin-orbit splitting of the\ncore level.\nLetusfinallydiscussagainourdataforthe threeFeRe-\nbased compounds. Our data are in good qualitative and\nquantitative agreement with literature data with one ex-\nception: Ca 2FeReO 6. While Sikora et al.[13] find, that\nthe spin magnetic moment of Re in Ca 2FeReO 6scales\nwith the high TC, in our case it has the lowest spin mag-\nnetic moment, letting Ca 2FeReO 6stand out from the\nscaling law [4, 12, 13] which so far holds in all other\ncases. This behavior is certainly more natural, since oneexpects that a reduced exchange will also lead to a re-\nduced spin magnetic moment on the Re site. Note that\nthe ratio of orbital and spin magnetic moments are con-\nsistent with the previous data. As suggested previously\nby Kato et al.[16], a Re t2gorbital ordered state or its\nglass-state analog associated with the monoclinic lattice\ndistortion occurs, pointing out the importance of corre-\nlation effects in this compound. Recently, Sikora et al.\n[36] proposed a scenario with a complex competition be-\ntween two phases with different electronic and crystallo-\ngraphic structure. Our data give further indication that\nCa2FeReO 6is exceptional among the double perovskites\ndue to the strong octahedral-site distortions.\nIV. SUMMARY\nIn summary, we have elucidated the Re magnetic mo-\nments in the FeRe-based series of double perovskites as a\nfunction of the earth alkaline ion, confirming the excep-\ntional position of Ca 2FeReO 6. We have measureda finite\nintrinsic magneticmomentattheRe5+siteinSr 2ScReO 6\nindicating the tendency to enhanced magnetic moments\nobserved in Re based double perovskites. Furthermore,\nfor the first time we were able to measure by XMCD the\nmagnetic moments directly at the alkaline earth site it-\nself. Our result shows that the usually neglected Ca and\nBa ions play a role in the magnetic scenario of the kinet-\nically driven exchange model, comparable in size to the\nrole of oxygen.\nThis work was supported by the ESRF (HE-\n2114/2115/2379).\n[1] Kobayashi K I, Kimura T, Sawada H, Terakura K and\nTokura Y 1998 Nature395677\n[2] Kato H, Okuda T, Okimoto Y, Tomioka Y, Takenoya\nY, Ohkubo A, Kawasaki M and Tokura Y 2002\nAppl. Phys. Lett. 81328\n[3] AsanoH, KozukaN, Tsuzuki A and Matsui M 2004 Appl.\nPhys. Lett. 85263\n[4] Majewski P, Gepr¨ ags S, Sanganas O, Opel M, Gross R,\nWilhelm F, Rogalev A and Alff L 2005 Appl. Phys. Lett.\n87202503\n[5] Krockenberger Y, Mogare K, Reehuis M, Tovar M,\nJansen M, Vaitheeswaran G, Kanchana V, Bultmark F,\nDelin A, Wilhelm F, Rogalev A, Winkler A and Alff L\n2007Phys. Rev. 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Lett. 89062509\n[14] Philipp J B, Majewski P, Alff L, Erb A, Gross R, Graf\nT, Brandt M S, Simon J, Walther T, Mader W, Topwal\nD and Sarma D D 2003 Phys. Rev. B 68144431\n[15] Westerburg W, Lang O, Ritter C, Felser C, Tremel W\nand Jakob G 2002 Solid State Comm. 122201\n[16] Kato H, Okuda T, Okimoto Y, Tomioka Y, Oikawa K,\nKamiyama T andTokura Y2002 Phys. Rev. B 65144404\n[17] Michalik J M, De Teresa J M, Blasco J, Algarabel P A,\nIbarra M R, Kapusta Cz and Zeitler U 2007 J. Phys.:\nCondens. Matter 19506206\n[18] Prellier W, Smolyaninova V, Biswas A, Galley C, Greene\nR L, Ramesha K and Gopalakrishnan J 2000 J. Phys.:\nCondens. Matter 129655\n[19] Azimonte C, Cezar J C, Granado E, Huang Q, Lynn J\nW, Campoy J C P, Gopalakrishnan J and Ramesha K\n2007Phys. Rev. Lett. 98017204\n[20] Iwasawa H, Saitoh T, Yamashita Y, Ishii D, Kato H,\nHamada N, Tokura Y and Sarma D D 2005 Phys. Rev. B\n71075106\n[21] Chattopadhyay A and Millis A J 2001 Phys. Rev. B 64\n024424\n[22] Navarro J, Frontera C, Balcells Ll, Mart´ ınez B and\nFontcuberta J 2001 Phys. Rev. B 64092411\n[23] Gepr¨ ags S, Majewski P, Gross R, Ritter C and Alff L\n2005J. Appl. Phys. 9908J102\n[24] Philipp J B, Reisinger D, Schonecke M, Marx A, Erb A,\nAlff L, Gross R and Klein J 2002 Appl. Phys. Lett. 79\n3654\n[25] De Teresa J M, Michalik J M, Blasco J, Algarabel\nP A, Ibarra M R, Kapusta C and Zeitler U 2007\nAppl. Phys. Lett. 90252514\n[26] Vaitheeswaran G, Kanchana V and Delin A 2005\nAppl. Phys. Lett. 86032513\n[27] Wills J M, Eriksson O, Alouani M and Price O L 2000\nElectronic Structure and Physical Properties of Solids\nEd. Dreyss´ e H, Springer, Berlin[28] Vaitheeswaran G, Kanchana V and Delin A 2006 J. of\nPhys.: Conf. Series 2950\n[29] Rogalev A, Goulon J, Goulon-Ginet Ch and Malgrange C\n2001Magnetism and Synchrotron Radiation Eds. Beau-\nrepaire E et al565, Springer, p 666\n[30] Wilhelm F, Poulopoulos P, Wende H, Scherz A, Baber-\nschke K, Angelakeris M, Flevaris N K and Rogalev A\n2001Phys. Rev. Lett. 87207202\n[31] Jeng Horng-Tay and Guo G Y 2003 Phys. Rev. B 67\n094438\n[32] Kato H, Okuda T, Okimoto Y, Tomioka Y, Oikawa K,\nKamiyama T andTokura Y2004 Phys. Rev. B 69184412\n[33] Huang D J, Jeng H-T, Chang C F, Guo G Y, Chen J,\nWu W P, Chung S C, Shyu S G, Wu C C, Lin H-J and\nChen C T 2002 Phys. Rev. B 66174440\n[34] Thole B T, Carra P, Sette F and van der Laan G 1992\nPhys. Rev. Lett. 681943\n[35] Carra P, Thole B T, Altarelli M and Wang X 1993\nPhys. Rev. Lett. 70694\n[36] Sikora M, Mathon O, van der Linden P, Michalik J\nM, de Teresa J M, Kapusta Cz and Pascarelli S 2009\nPhys. Rev. B 79220402(R)"    },    {        "title": "2005.00871v1.Optically_driven_ultrafast_magnetic_order_transitions_in_two_dimensional_ferrimagnets.pdf",        "content": " \n \n 1 Optically driven  ultrafast magnetic order  transitions  \nin two-dimensional  ferrimagnet s \nJunjie He,1,* Thomas Frauenheim1,2,* \n1Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, 2835 , \nBremen, Germany.  \n2Beijing Computational Science Research Center (CSRC) , Beijing  100193  and Shenzhen  Computational \nScience  and Applied Research (CSAR) Institute , Shenzhen  5181 10, China . \n \nE-mail: junjie.he.phy @gmail.com ; thomas.frauenheim@bccms.uni -bremen.de  \n \n                                           ABSTRACT  \nLaser -induced switching and manipulation of the spins  in magnetic materials are of great interest  to \nrevolut ionize future magnetic storage technology  and spintronics with the fastest speed  and least power  \ndissipa tive. Inspired by the recent discovery of  intrinsic two-dimensional  (2D) magnets, which  provide a \nunique plat form to explore the new phenomenon for light-control magnetism in the 2D limit , we \npropose to realize light can  efficiently tune magnetic states of 2D ferrimagnets  in early  time. Here, using \nthe 2D ferrimagnetic MXenes  (M2M’X2F2, M/M ’=Cr, V, Mo; X=C/N)  as prototype system s, our real-\ntime density functional  theory  (TDDFT) simulation show that laser pulses can directly induce ultrafast \nspin-selective charge transfer between two magnetic sublattices  (M and M ’) on a few  femtoseconds,  and \nfurther generate dramatic changes in the magne tic structure of these MXenes , including a magnetic \norder  transition  from ferrimagnetism  (FiM) to transient ferromagneti sm (FM).  The microscopic \nmechanism underpinning this ultrafast switching of magnetic order  in MXenes  is governed  by optically \ninduced inter-site spin transfer (OISTR)  effect, which theoretically enables  the ultrafast direct  optical  \n \n 2 manipulation of the magnetic state in MXenes -based 2D materials . Our results open new opportunities \nfor exploring the manipulat ion of  the spin  in 2D magnets  by optical approaches .  \nKEYWORDS  : TDDFT , 2D magnets,  MXenes , spin transfer, photo -induced  spin dynamics  \nTOC  \n \n \nIntroduction  \nIn 2017, two experimental groups have independently observed the intrinsic two -dimensional (2D) \nferromagnetism (FM) in exfoliated van der Waals (vdW) CrI 3 and CrGeTe 3 crystal by magneto -optical \ntechnique. 1,2 Subsequently, a large variety of 2D magnetic vdW crystals, including FePS 3,3 NiPS 3,4  MnPS 3, \n5 Fe3GeTe 2,6 VSe 2,7 and MnSe 28 etc. have been realized by using only adhesive tape, chemical vapour \ndeposition or molecular beam epitaxy. Interestingly, both VSe 2 and MnSe 2 were claimed to be a 2D \nferromagnet with Curie temperature (T C) above room temperature.7,8 Until now, the family of 2D magnetic \ncrystals is still rapidly growing, and many more new 2D magnets have been predicted and await discovery. \n \n \n 3 9,10,11,12,13,14 These 2D magnets with excellent properties in optical, magnetic, magneto -electric, and magneto -\noptic areas , offer new opportunities for both explor ing the manipulation of  spin or magnetism in the 2D \nlimit  and for developing the next  generation spintronic devices . Following these exciting  discoveries, \nconsiderable studies have bee n devoted to  manipulat ing spin interaction  in 2D magnets , e.g., Dzyaloshinskii -\nMoriya interaction,  interlayer magnetic order , and magnetic anisotropy  by mechanical strain,  gate voltage , \nand magnetic  field.15,16,17,18 For example, it demonstrated  that the interlayer magnetic order  of bilayer CrI 3 \ncan be  reversibly tuned from the antiferromagnetic (AFM) to the  FM phase by  applying  electrostatic doping \nor vertical electric field  gate voltage .16,17,18 \n       The light represents the fast est means to manipulate the spin structures  of matter at ultrashort time \nscales  (from femtosecond to attosecond ) that promises to revolutionize future  recording and information \nprocessing by achieving the fastest possible and least power dissipative.19 Laser induced s pin injection in \nFM/NM interface ,20 all-optical magnetization switching ,21 optically tunable magneti c anisotropy  in CrI 3 \nmonolayer, 22 photo -induced topological  states 2D materials,23 and light can manipulate the interfacial \nmagnetic proximity coupling  and valley polarization  WSe 2/CrI 3 heterostructures24 are p rominent examples of  \nlight-controlled properties in materials.  Using time-dependent density -functional theory  (TDDFT) \nsimulation, Dewhurst et al. have shown  ultra-short laser  can directly  induce a spin transfer  between sub -\nlattices  causing significant magnetic order  transition , i.e., the switching  from Ferrimagnetic (FiM)  to FM \norder  in a multi -component  magnetic system, which is governed by the  previously unknown  optically \ninduced inter -site spin transfer (OISTR)  effect in  sub-exchange and sub -spin-orbit timescales .25,26 Indeed, \nrecently, several  experiment al group s have demonstrated  the ultrafast  laser  can  direct ly and coherent ly \nmanipuate  spin of magnetic multilayer  and H eusler compounds  on sub-femtosecond time scales , and \ninduced spin transfer between two magnetic  subsystems , which  confirmed  the OISTR effect .19,27,28,29  These   \n \n 4 breakthroughs suggest  a search for ultrafast optically tunable magnetism in 2D  systems . However , to realize  \nthe OISTR effect, it is highly  desir able to develop  multi -component  (two or more ) magnetic  system s but it  is \nrare in 2D magnets .  \n        Recently, 2D transition metal carbides  or nitrides called  by MXenes have emerged as a new type \nof 2D materials with the general formula  Mn+1XnTz, where M represents an early transition metal  (such \nas Ti, V, Cr, or Mo), X is carbon  and/or nitrogen, and T stands for the surface terminations  (e.g., -OH, -\nO, or -F).30,31,32 The MXenes attract great attention due  to their potential applications in sensors, \ncatalysis, energy  storage and nanoelectronics.32 The chemical and physical properties of MXenes can be \ntuned through the choice of transition metals  and surface chemical group s.12,14,33,34,35 For example, the \nferromagnetic behavior  of bare Cr 2C and Cr 2TiC 2 has been shown to become semiconduct ing \nantiferromagnetic upon -F, -OH and -Cl functionalization .12,14 Particularly , the Mo 3N2F2 MXene  has \nbeen  found to be a ferrimagnetic half -metallicity with a  high Curie temperature .36 Therefore, MXene s \nrepresent unusual  a class of  multi -component  magnetic  material s in 2D system, which provid e an \nexcellent  platform to study the optically tunable  magnetism  and spin transfer  in 2D system.  \nIn this work, we propose to realize the optically manipulate magnetic order transition in MXenes . \nFirstly, t he ground states  DFT  calculations reveal that 2D MXenes , including Cr2VC 2F2, Mo 2VC 2F2, \nMo 2VN 2F2, Mo 3C2F2 and Mo 3N2F2, have unusual  ferrim agnetic order . Then, o ur real time time-\ndependent DFT (TDDFT) simulations on FiM MXenes have shown  that l aser pulses induce spin -\nselective charge transfer  and further generate dramatic changes in the magnetic  structure of these \nMXenes , including a magnetic order  transition  from FiM to transient FM at early time . The microscopic \nmechanism underpinning this ultrafast switching of magnetic order is governed  by optically induced  \n \n 5 inter-site spin transfer (OISTR)  effect . The results open new opportunities to manipulate  the 2D \nmagnetism by optical  approaches  \n \nFig 1: The top  (a) M2M’X 2F2 (M=Cr, Mo; M’=Cr, Mo, V; X=C/N) . (b) The ligand environment for M and M’  is \nshown .  Schematics of magnetic configurations for considered MXene s: (c) FM, (d) FiM1, (e) FiM2,  (f) AFM1, (g) \nAFM2.  The  2×1 supercells are employed for the total energy calculation.   \nResults and discussions  \nThe atomic structure MXenes -based 2D systems , including Cr 2VC 2F2, Mo 2VC 2F2, Mo 2VN 2F2, \nMo 3C2F2, and Mo 3N2F2 show in Figure 1 . The M and M’ localized in the distinctive  octahedral  \nenvironment, where M is bonded to six C/N atoms  and M’  atom , but M’  is bonded to three  C/N and \nthree F  atoms , respectively . The distinctive  chemical environment  render s the different local magnetic \nmoment for M and M ’ atoms.  Geometry optimization carried out at PBE +U level gives lattice constant  \nand geometry parameter s as summarized  in Table S1. For Mo 3N2F2 MXene , the calculated  constant \nlattice  is in good agreement  with previous calculations.36 The calculated local magnetic moments  for M \n \n \n 6 and M ’ atoms are show n in Table 1 . The M and M ’ atoms  of MXenes  predominantly  contribute to the \ntotal magnetic moments while the neighboring X  and F atoms have only a small contribution. To study  \nthe magnetic ground state structures of the  M2M’X2F2 MXenes , the collinear ferromagnetic  (FM)  and \nantiferromagnetic  (AFM ) and ferromagnetic  (FiM) states were considered , respectively . To elucidate \nthe spin configurations  of these phases, we schematically plotted the arrangement  of the local magnetic \nmoments of M and M ’ atoms  with colored  arrows as shown in Figure 1 c-g, which is considering  a 2×1 \nsuper cell of MXenes . The lowest energy has  been found for the F iM1 state and this has been set as \nreference  energy. For instance, for Cr2VC 2F2 MXenes , the EFM, EFiM2, EAFM1, and EAFM2 states have \nenergies 0.81, 0.25,  0.83, and 0.35 eV higher, respectively, than the F iM1 state. These results clearly \ndemonstrate that the  FiM1 state configuration for all considered MXenes  is the magnetic  ground state \nwith high magnetic stability , similar  to previously reported results for Mo 3N2F2 MXenes.36 The \nmagnetic anisotropy energy  (MAE)  of materials, which determines  the orientation of the magnetization \nat low -temperature,  is an important parameter  for their applications in high -density  storage or quantum \nspin processing.  The non -collinear magnetic calculations  were performed for magnetization along \nX[100], Y[010] , and Z[001] directions  for MXenes , which is summarized  in Table S1 . The resul ts \nindicate  that the easy axis of Cr2VC 2F2, Mo 3C2F2, and Mo 3N2F2 MXenes  is along the [001] orientation , \nwhile easy axis of Mo 2VN 2F2 and Mo 2VC 2F2 is along the [ 100] and [010] orientation.  The band \nstructure s with the orbital projection  for MXenes in FiM state are reported in the supporting  materials  \n(see Figure S1).  They all show the metallic  feature . Interestingly , both Mo 3N2F2 and Mo 2VN 2F2 are \npredicted  to be  half-metallic , which is a great potential for spintronic  application s. \nTable 1: Calculated structural and magnetic characteristics  of five different MXenes. a \n  \n \n 7 Structure  EFM  EFiM1  EFiM2  EAFM1  EAFM1        \nCr2VC 2F2 0.81 0 0.25 0.83 0.35 2.39 -0.43 \nMo 2VC 2F2 0.54 0 0.27 0.58 0.16 1.03 -0.96 \nMo 2VN 2F2 2.33 0 2.46 1.52 1.52 1.91 -1.90 \nMo 3C2F2 0.96 0 1.07 0.89 1.18 1.39 -0.25 \nMo 3N2F2 1.30 0 0.75 0.90 1.09 1.21 -0.73 \n \na Relative energies for FM ( EFM) , FiM1 ( EFiM1), FiM2 ( EFiM2), AFM1 ( EAFM1)  and AFM2 ( EAFM2) are \nshown in eV. Local m agnetic moment s of M (  ) and M ’ (   ) atoms  for M 2M’X2F2 MXenes are also \ngiven  in μB. \n \nFigure 2: Ultrafast laser-induced ultrafast change of magnetic order. ( a) Time evolution of local magnetic  moment  \nwith (full lines)  and without (dashed lines)  spin-orbital coupling  for Cr1, Cr2, V, C and F  atoms , which at t = 0 fs (i.e., \nin the ground state) is ferrimegn ets.  At ∼6 fs, the  V atom demagnetizes and then remagnetizes but with the reversed \nspins  direction . The magnetic order  thus switches from FiM to transient FM order . (b) The relative  magnetization \n \n \n 8 dynamics for Cr1, Cr2 and V atoms . The snapshots of magnetization density at (c) t = 0  fs, (d) t= 5.8 fs, and (e) t= 15.2 \nfs. The iso-surfa ce was set to be 0.012  e/Å3 \nNext, w e will explore  the real -time evolution of the element resolved magnetism of the  FiM \nMXenes  monolayer induced by the electron dynamics under the  influence of an external light with a \nselected frequency. We performed real -time TDDFT calculations, in which the linearly polarized (in -\nplane polarization) laser pulse with full width at half -maximum  (FWHM ) of 3.63 fs, photon energy of  \n1.63 eV  and fluence of 34.2 mJ/cm2 (the vector potential  A(t) of the laser pulse  are shown in Figure \nS2(c)).  We first examined t he real-time spin  dynamics  of local magnetic moment of Cr and V atoms for \nCr2VC 2F2 MXenes  are shown in Fig ure 2a  under the in fluence of a laser pulse.  We can see that  both the \nlocal moment Cr and V atoms change dramatically, while the global moment showed no change.  The top \nand bottom  Cr atoms (Cr1 and Cr2)  demagnetize strongly  from ~2.4 μB of initial magnetic  moment to \nfinal ~1.4 μB. Most interestingly, at ∼6 fs (slightly before peak intensity of the laser pulse) the V atom  \ndemagnetizes and then remagnetizes  (from ~ -0.4 μB to ~ 0.9 μB) but with the reversed spin  direction , \nthus leading to the FiM-FM magnetic order transitio n. After 10 fs, the Cr2VC 2F2 show s transient  FM \norder  for a long time (at least up to 100 fs, at which point  our simulation ended ).  Under the influence of \nan external laser field, the relative local  moments , i.e., the quantity ( M(t)/M(0)) changes dramatically; \nthe moment of Cr  atoms  show  about 50 % decrease , while the ultrafast loss of moment  for V atoms and \nthen increase  (~200%)  but with the reversed spin  direction  as shown in Figure 2b.  We also simulate the \nreal-time spin dynamics of Cr 2VC 2F2 MXenes  with spin-orbital coupling (SOC)  as shown in Figure 2a.  \nBecause of the total spin moment  will no longer  a good quantum number , the spin-flip of photo -excited  \nprocesses  in Cr 2VC 2F2 MXenes  will be allowed. It is clear that that the effect of SOC will be not notable   \n \n 9 at least below  36 fs. The dynamics of the magnetization density  as a function of time are shown in \nFigure 1 d. The s witching of the spin direction of V at t = 15.2 fs can also be seen in this plot.  \nThe magnetic states transition from FiM to FM in Cr 2VC 2F2 can be understood by optically  \ninduced inter -site spin transfer (OISTR) mechanism , which  is found  previously to dominate the early \ntime spin dynamics of magnetic  metal multilayers and  the Heusler compounds .25,26  The ground -state \ndensity of states  (DOS) for Cr and V atoms shown in  Figure 3a .   Because the Cr and V are \nferrimagnetic  coupling , the majority spin is oppositely oriented in each  atoms : spin -up is the majority in \nthe Cr1 and Cr2  atoms,  and spin -down is the majority in V atom. From DOS, t he occupied  Cr atoms are \ndominated by  majority states, whereas there is considerable  empty  minority  orbital  of V around [1, 3] \neV energy window .  Such ground states electronic structure will enable efficient photo -driven spin  \nselective  charge  transfer  from the majority Cr  to minority V atoms according to  the OISTR mechanism  \nas shown in  Figure 3b, result ing in the loss of magnetic moment for Cr atoms and the gain of mag netic  \nmom ent for V atoms, respectively. The OISTR mechanism  in magnetic order transition of  Cr2VC 2F2 \nMXenes  is in agreement with the previous ly reported magnetic metal  multilayers and  Heusler alloys .25,26 \nTo obtain  further  analysis for the chang ing magnetic moment  for Cr 2VC 2F2 MXenes , we show  time-\ndependent occupation changes  (Δn(t)) of  the V and Cr atoms , which is defined as the difference  of the \ntime-resolved occupation function n (t) at time t concerning  the unexcited Cr 2VC 2F2 MXenes , i.e.,  Δ\nn(t)= n (t)−n(0). As shown in Figure 3 , the most notable transient change below  the Fermi level occurs \nin the minority channel of Cr and minority channel s of V atoms . One can see that photo -excitation \nresults in  a significant loss in the number of  majority  spin carriers as a function of time, which \ncorresponds to  the demagnetization  for Cr atoms . These loss minority spin carriers are transferred  from \nthe Cr to the V  sub-lattice, thus, leading to the  enhanced  magnetic moment  for V atom , and further  \n \n 10 induced the direction reverse . It is also evidenced an obvious  gain in the number of  minority  spin \ncarriers of V  (See Figure 3d ).  For other possible excitation, the Δn(t) for the Cr minority and V \nmajority spin channel is relatively small as shown in Figure S3, which does not sign ificantly affect the \ntotal spin -transfer  process between Cr and V atoms. T he time -dependent change in the spin -down  and \nup electrons for V and Cr  are presented in Figure S4  relative to the ground -state for Cr2VC 2F2 MXenes . \nIt is clear from Fig ure S4 that majority  electrons of Cr are transferred  to the minority of V atoms, which \nis consistent with  the OISTR mechanism.  \n \nFigure 3: (a) Projected density of states (DOS) calculation for Cr1 and V in Cr2VC 2F2 MXenes . And, the favorable  \nspin transfer from Cr majority  to V minority is marked . (b) Schematic overview of the OISTR effect in Cr2VC 2F2. The \noptical excitation leads to an effective spin transfer from  the occupied Cr majority  into the V minority channel. TD -\nDFT calculations of the difference of the transient occupation compared with the unexcited case in  the (c) Cr1 minority \n \n \n 11 channel  and (d) V minority channel  at selected  time steps. For Δn(t), a negative signal arises corresponding to a loss \nof minority electrons, while a simultaneous positive signal correlating to spin gain.  \nWe now turn to explore the light response  on magnetism  for Cr 2VC 2F2 MXenes with respect to \nthe changing laser pulse parameter.  Figure 4c has plotted the magnetization dynamics  of Cr and V \natoms in Cr2VC 2F2 MXenes  under the influence of laser pulses of  various  frequencies  (fixed FWHM) \nand FWHM (fixed frequency) . Here, their time dependence of the ext ernal  vector field  is shown in \nFigure S2. It is clear that the demagnetization of Cr and V atoms is highly sensitive to  the frequency  of \nlaser , indicating that the frequency of the pulse can also be used to  tailor and optimize the process of \nphoto -induced magnetic transition for FiM MXenes. For the higher energy in 1.09, 1.63, 2.18 , and 2.72 \neV, the local magnetic moment of Cr and V atoms are obviously demagnetized, whereas only relatively \nsmall  loss of local  magnetic moment is observed for energy in 0.54 eV. On the other hand, t he \ndemagnetization time will not strongly dependent on the FWHM. We can see V and Cr atom s are shown \ndramatic demagnetization for various FWHM  value s. However, the demagnetization proc ess for var ious \nFWHM take s place after a time lag between the laser pulse and the maximum of the laser pulse  (See \nFigure S2). Such a time lag effect between  the laser pulse and the  demagnetization  in 2D MXene system \nalso is similar to the Fe, Co , and Ni metal.37  \n \n 12  \nFigure 4: The spin d ynamics of local  magnetic  moment of Cr1 and V atoms in Cr 2VC 2F2 MXenes under the \ninfluence of seven different laser pulses given by (a) frequency /fluence,  and (b) FWHM/ fluence . \n \n \n \n 13 Figure 5: Time evolution of local magnetic  moment of M and M’  atoms  for (a) Mo 3C2F2, (b) Mo 3N2F2, (c) Mo 2VC 2F2 \nand (d) Mo 2VN 2F2, respectively.  \nTo probe the validity of optically induced magnetic order transition  in other  2D MXenes materials, \nwe have performed real-time TDDFT calculations of the laser induced  change in magnetic structure for \nan extended set of 2D systems : Mo 3C2F2, Mo 3N2F2, Mo 2VC 2F2, and Mo 2VN 2F2, which have a FiM \nground state . For purpose of companion, the  same  linearly polarized (in -plane polarization) laser pulse \nwith FWHM  of 3.63 fs, photon energy of  1.63 eV , and fluence of 34.2 mJ/cm2 are employed as driven \nlaser field. Under the influen ce of an external laser field,  the local  moment changes  for these four \nmaterials  dramatically  change ; the middle layered metal (M ’) demagnetize and then remagnetize with \nreverse spin direction, while the upper and lower metal (M) shown  relatively small demagnetization as \nshown in Fig ure 5 and Figure S 5. The results demonstrate that OISTR also  dominates early -time spin \ndynamics in these FiM MXenes, which is  the same mechanism as found in the case of  Cr2VC 2F2 \nMXenes . It is also important to mention  that linearly polarized light in the x -direction  for present work  \nwas used . We find that changing polarization  direction (e.g., y-direction)  of this linearly polarized light  \ndoes not affect  the process of photo -induced magnetic order transition  of MXenes . Note  that our \nsimulations just  performed at early  spin dynamics (up to  100 fs ) following the laser pulse. This time -\nscale is completely dominated by direct  optical manipulation without the  rotation of atomic magnetic \nmoments .29 After  this time, the phonon -mediated spin -flip p rocesses  will efficiently affect the \nmagnetization  of materials , and which can lead to further a reduction in magnetization; however, such \nprocesses are not included in our TDDFT simulations.  \nTo explore how the optical manipulation of magnetism in 2D FM and AFM materials, we take \nMn 2CF2 and Cr 2CF2 as an example to simulate their spin dynamics by TDDFT with the same laser  \n \n 14 parameter as Cr 2VC 2F2 as shown in Figure S6.  Both my calculation and previous reports indicate that \nthe Mn 2CF2 and Cr 2CF2 have FM and s ymmetrical AFM order.  From Figure S6, the Mn 2CF2 show \nalmost identical demagnetization of Mn atoms, result in no change  in magnetic order.  The AFM Cr 2CF2 \nwill symmetrical demagnetization because of spin transfer  between  majority  and minority of Cr atoms \nwill be identical in both spin channels . Our simulation in 2D FM and AFM materials is in agreement \nwith the previous ly proposed demagnetization mechanism in bulk materials (e.g. NiO).20 Thus, \ndeveloping suitable 2D ferromagnetic  systems will be crucial  to explore ultrafast optical switch of \nmagnetic order in 2D limit, which will trigger the further materials search and  design for photo -induced \nmagnetic states transition in 2D magnets as well as their van der Waals heterostructures.  \nExperimentally , our work  can be verified  by ultrafast extreme ultraviolet (EUV)  high harmonic pulses  \nand time-resolved magnetic circular di chroism (MCD), which have been widely applied to  explore  the \nelement -specific spin dynamics in multi -component  magnetic systems .19,29 \n     In conclusion, we performed DFT and real-time TDDFT  calculations for  ground -states  properties  \nand optical  manipulation of magnetism  in MXenes -based 2D system . Our DFT calculations show  that  \nM2M’C2F2 MXenes, including Cr2VC 2F2, Mo 2VC 2F2, Mo 2VN 2F2, Mo 3C2F2, and Mo 3N2F2, have  \nunusual ferrimagnetic  order. Using TDDFT, we predicted that l aser pulses can directly induce ultrafast \nspin-selective charge transfer between two magnetic sublattices  (M and M ’) on a few  femtoseconds,  and \nfurther generate dramatic changes in the magnetic  structure of these MXenes , including a magnetic \norder  transition  from ferrimagnetic  (FiM) to transient ferromagnetic (FM).  The spin transfer of magnetic \nsublattice in MXenes can further  be tuned by laser parameter, which is achievable with current \nexperimental techniques . The microscopic mechanisms for ultrafast switching of magnetic order  are \ndiscussed based on  optically induced inter -site spin transfer (OISTR)  effect, which theoretically enable s  \n \n 15 the ultrafast direct  optical manipulation of the magnetic state in 2D magnets. Our results open  new \nopportunities to manipulate  the spin in 2D magnets as well as the potential applications  in spintronic s.  \nMethods and c omputational details  \nThe structure  optimizations , magnetic ground states , and band structures for MXenes  were \nperformed using the Vienna ab initio simulation package (VASP)38,39 within the generalized gradient \napproximation , using the  Perdew -Burke –Ernzerhof (PBE) exchange -correlation functional .40 \nInteractions  between electrons and nuclei  were  described by the projector -augmented wave (PAW) \nmethod.  The criteria of energy and atom force convergence were  set to 10−6 eV and 0.0 01 eV/Å, \nrespectively. A plane -wave  kinetic energy cutoff of 500 eV was employed.  The vacuum space of 15  Å \nalong  the MXenes normal was adopted for calculations on monolayers . The Brillouin zone (BZ) was \nsampled using 15 ×15×1 Gamma -centered Monkhorst -Pack grids for the calculations of relaxa tion and  \nelectronic structures .  To account  for the energy of localized 3 d orbitals of TM atoms properly, the  \nHubbard “U” correction is employed within the rotationally  invariant DFT + U approach.41 A correction \nof U = 3 eV  for V, Cr , and Mo  is employed based on the relevant  previous reports .11,12,14 \nTo identify the spin dynamics  in these MXenes materials under the influence of ultrafast laser pulses , \nwe have  performed  time-dependent density functional theory (TDDFT)  calculations . The time  evolving  \nstate functions (  ) are calculated by solving the time  dependent Kohn −Sham  (KS)  equation as follows:  \n         \n     \n       \n          \n          \n             \n                               (1) \nwhere         and σ represent  a vector potential  and Pauli matrices. The KS effective potential         \n=                            can be  decomposed into the external potential      , the classical \nHartree potential   , and the exchange -correlation (XC) potential     , respectively . The KS magnetic  \n \n 16 field can be written as         =                   , where      and     represent  the magnetic field \nof the applied laser pulse plus possibly an additional magnetic field and XC magnetic field , respectively .  \nThe last term in Eq. (1) stand for the SOC .  We only time propagate the electronic system while keeping \nthe nuclei fixed .  \n     With the   s in hand a time-resolved DOS, shown in Figure  2, can be calculated using the following  \nexpression : \n                                                                                              (2) \nwith \n                                                                                                    (3) \nhere, spin -resolved and time -dependent            is calculated from the projection of the time -propagated \norbitals              onto the ground -state Kohn -Sham orbitals at t = 0. The dynamical evolution of the \nelectronic struc ture for  MXenes  after photo -excitation s also can be analyzed by the time -dependent \nchanges of        .  \n     We employed  a fully non -collinear version of TDDFT  by full-potential  augmented plane -wave ELK \ncode .42 A regular mesh  in k-space of 7 × 7 × 1 is used, and a time step of Δt = 0.1 a .u. is employed for \nthe real time TDDFT simulation . A smearing width of 0.027 eV is used.  Laser pulses used in the present \nwork are linearly polarized (in -plane p olarization) with selected  frequency . All calculations were  \nperformed usi ng adiabatic local spin density  approximation (ALSDA)  with Hubbard U (ALSDA+U), \nwith U=3 eV for Cr, V , and Mo atoms.  \nReferences   \n \n 17                                                  \n1 Huang, B. ; Clark,  G.; Navarro -Moratalla,  E.;  Klein,  D.R. ;  Cheng,  R.; Seyler,  K.L.; Zhong,  D.;  \nSchmidgall, E. ; McGuire,  M. A.;  Cobden,  D. H. ;  Yao,  W.  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The ELK code, http://elk.sourceforge.net, 2017 .   1 Supp orting Information  “Optically driven  ultrafast \nmagnetic order transitions  in two-dimensional \nferrimagnet s” \nJunjie He,1,* Thomas Frauenheim1,2,*  \n1Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, 2835 , \nBremen, Germany.  \n2Beijing Computational Science Research Center (CSRC) , Beijing  100193  and Shenzhen  Computational \nScience  and Applied Research (CSAR) Institute , Shenzhen  5181 10, China . \n \nE-mail: junjie.he.phy @gmail.com ; thomas.frauenheim@bccms.uni -bremen.de  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  2  \nTable S1 : Calculated constant lattice and magnetic anisotropy of five different MXen es. The L and h  is \nthe lattice constant and height of MXenes in Å. The magnetic anisotropy between Y and Z ( EYZ) and X \nand Z ( EXZ) directions are in meV. The ES stand for the electronic structure. \nStructure  L  h         easy axis ES \nCr2VC 2F2 3.04 7.04 -1.04 -2.40 [001]  Metal  \nMo 2VC 2F2 3.26 7.07 -5.70 2.38 [010]  Metal  \nMo 2VN 2F2 3.24 7.27 2.58 2.90 [100]  Half-metal  \nMo 3C2F2 3.29 7.10 -42.11  -43.66  [001]  Metal  \nMo 3N2F2 3.35 6.87 -0.21 -0.26 [001]  Half-metal  \n \n \nFigure S1 : Band structure with orbital projection  for spin up of (a) Cr 2VC 2F2, (b) Mo 3C2F2, (c) \nMo 3N2F2, (d) Mo 2VC 2F2, (e) Mo 2VN 2F2; Spin down of (f)  Cr2VC 2F2, (g) Mo 3C2F2, (h) Mo 3N2F2, (i) \nMo 2VC 2F2, (g) Mo 2VN 2F2. \n \n \n 3  \nFigure S2:  The applied pump laser pulse  in Cr 2VC 2F2. The energy and FWHM of laser was \nmarked, respectively.  \n \n \n \n \n 4  \nFigure S 3: TDDFT calculations of the difference of the transient occupation compared with the \nunexcited case in  the  (c) Cr1 minority channel and (d) V minority channel at characteristic time steps. \nFor Δn(t), a  negative signal arises corresponding to a loss of minority electrons, while a simultaneous \npositive signal correlating to spin gain. The same x -axis scale in Δn(t) is selected with Figure 3cd  \n \n \nFigure S4: The time -dependent change in the (a) spin-up and ( b) spin-down electrons on V, Cr1 and \nCr2 atoms  relativ e to the ground -state for Cr2VC 2F2. \n \n 5  \nFigure S 5:  The relative local  moment (i.e., the quantity ( M(t) /M(0)) for  (a) Mo 3C2F2, (b) \nMo 3N2F2, (c) Mo 2VC 2F2 and (d) Mo 2VN 2F2. \n \n \nFigure S 6:  Photo-induced spin dynamics for (a) ferromagnetic MXenes Mn 2CF2 and (b) \nantiferromagnetic  MXenes Cr 2CF2. \n \n \n"    },    {        "title": "0808.1399v1.Formation_of_guided_spin_wave_bullets_in_ferrimagnetic_film_stripes.pdf",        "content": "arXiv:0808.1399v1  [cond-mat.other]  10 Aug 2008Formation of guided spin-wave bullets in ferrimagnetic film stripes\nA.A. Serga,1M.P. Kostylev,2,∗and B. Hillebrands1\n1Fachbereich Physik and Forschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2School of Physics, The University of Western Australia, Cra wley WA 6009, Australia\nThe formation of quasi-2D nonlinear spin-wave eigenmodes i n longitudinally magnetized stripes of\na ferrimagnetic film, so-called guided spin-wave bullets, w as experimentally observed by using time-\nand space-resolved Brillouin light scattering spectrosco py and confirmed by numerical simulation.\nThey represent stable spin-wave packets propagating along a waveguide structure, for which both\ntransversal instability and interaction with the side edge s of the waveguide are important. The\nexperiments and the numerical simulation of the evolution o f the spin-wave excitations show that the\nshapeoftheformedpacketsandtheirbehaviorarestronglyi nfluencedbytheconfinementconditions.\nThe discovery of these modes demonstrates the existence of q uasi-stable nonlinear solutions in the\ntransition regime between one-dimensional and two-dimens ional wave packet propagation.\nPACS numbers: 75.30.Ds, 76.50.+g, 85.70.Ge\nStable two-dimensional localized nonlinear spin-wave\nexcitations, so-called spin-wave bullets, have been pre-\nviously observed in thin ferrimagnetic films of yttrium-\niron-garnet (YIG) magnetized along the propagation di-\nrection [1, 2, 3]. These films were practically unbounded\nin both in-plane directions compared to the transver-\nsal size of the spin-wave packets and the wavelength of\nthe carrier spin wave. In contrary, in a one-dimensional\nwaveguidestructure, where the width is comparabletoor\nsmaller than the spin-wave wavelength, only quasi one-\ndimensional nonlinear spin-wave objects were observed,\nwhich are spin-wave envelope solitons. [4, 5]. Both for\nsolitons and bullets linear pulse spreading in the direc-\ntion of propagation (so called longitudinal direction) due\nto wave dispersion is compensated by longitudinal non-\nlinear compression. As for the transverse in-plane direc-\ntion, solitons are meant to have a stable transverse dis-\ntribution of their dynamic magnetization coinciding with\nthe profile of the lowest linear spin-wave eigenmode of\nthe waveguide. On the contrary, bullets show transverse\nnonlinear instability of attractive type which overcom-\npensates transverse diffraction broadening of the wave\npacket. The nonlinear compression would lead to a wave\npacket collapse if the medium is lossless. Weak mag-\nnetic losses in a real magnetic film ensure a fine balance\nof nonlinear narrowing of the packet and of its diffrac-\ntion spreading for some distance of propagation [1]. This\nresultsinaquasi-2Dspatiallylocalizedbell-shapedwave-\nform which is stable during the lifetime of the bullet.\nHere we report on the experimental observation of a\nstable spin-wave packet propagating along a waveguide\nstructure, for which both transversal instability and in-\nteraction with the side edges of the film waveguide are\ncrucial. The structure can be considered as a transitional\ncase between the 2D case of a continuous film and the\nquasi-1D case of a narrow stripe. We show that in this\ncase the nonlinear wave dynamics is distinctively differ-\nent from both the soliton and the bullet cases, andcan beconsidered as efficient coherent nonlinear mixing of spin-\nwave eigenmodes of the waveguide. Our theory shows\nthat this mixing is due to a specific interaction – the\npseudo-linear generation of higher order modes by the\nfundamental one.\nThe experiment was carried out using a longitudinally\nmagnetized long YIG film stripe of 2.5mm width and\n7µm thickness. The magnetizing field was 1831Oe. The\nspin waves were excited by a rf magnetic field created\nwith a microstrip antenna of 25 µm width placed across\nthe stripe and driven by input microwave current pulses\n20ns in duration at a carrier frequency of 7.125GHz.\nThe spatio-temporal behavior of the traveling spin-wave\npackets was investigated by means of space- and time-\nresolved Brillouin light scattering spectroscopy [6].\nThe results of our measurements are demonstrated in\nFig. 1. The panels show the spatial distribution of the\nintensity of spin-wave packets at different times of their\npropagation from the left to the right along the stripe\nwaveguide. The left vertical set of diagrams corresponds\nto the linear case. The input microwave power for this\nset is 20mW. The right set is for nonlinear propaga-\ntion. Thesedatawerecollectedapplyinganinput driving\npower of 376mW.\nDifferences between these two cases are clearly seen.\nThe linear spin-wave packet is characterized by a trans-\nverse profile very similar to one half of the period of the\nsine function, while the cross section of the nonlinear\npacket has a pronounced bell-like shape. Furthermore,\nthe intensity of the linear packet decays monotonically\nwith time. That is obviously due to magnetic damping\nin the film. The intensity of the nonlinear packet ini-\ntially increases because of its strong transverse compres-\nsion (see the second diagramfrom the top in Fig. 1), then\ngraduallydecaysasthe dampingovercomesthe nonlinear\ncompression for larger propagation times.\nFormationofastablenonlinearspin-waveobjectisalso\nevidenced in Fig. 2 which shows the measured peak in-2\nFIG. 1: Observation of linear propagation and of formation\nof the guided spin-wave bullet in an YIG film stripe waveg-\nuide using time- and space-resolved Brillouin light scatte ring\nspectroscopy. For parameters see main text.\ntensity of the nonlinear wave packet and its width (i.e.\npacket size in the transverse direction) as a function of\npropagation path. At the beginning of the propagation\npath one clearly sees an increase of the peak intensity\non top of the exponential decay. The increase in peak\nintensity is due to transverse compression of the packet,\nevidenced in the same figure by a decrease of the packet\nwidth. Down the propagation path the width stabilizes,\nand in the range of distances z=1-4mm it does practi-\ncally not change, apart from small superimposed oscilla-\ntions which will be discussed below.\nBoth figures provide a clear proof of development of\ntransverseinstability and bullet formation. Interestingly,\nthe 2D bell-like shape survives even at the end of the\npropagation path ( z >4mm) when the packet intensity\nhas decreased more than ten times and the nonlinearity\ncontribution to the spin-wave dynamics has been consid-\nerably diminished.\nImportantly, in Fig.2 oneobservessuperimposedoscil-\nlatoryvariations in the intensity and in the packet width,\nboth in the quasi-linear regime of the packet propagation\n(z >4mm) andinthe highlynonlinearregime z <4mm.\nIn the entire range the oscillations in intensity and in\nwidth are in anti-phase to each other. This effect has not\nbeenobservedincaseofconventionalbulletformation[1].\nOn the contrary, a similar picture is usually observed for\nlinear guided spin waves in narrowfilm waveguides[7, 8],\nwhere it is explained as a beat of phase-correlated linear\nwaveguide width modes propagating at the same carrier\nfrequency. This effect evidences the importance of the/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s49\n/s87 /s105/s100/s116/s104/s80/s101/s97/s107/s32\n/s105/s110/s116/s101/s110/s115/s105/s116/s121/s80/s101/s97/s107/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s68/s105/s115/s116/s97/s110/s99/s101/s32/s40/s109/s109/s41/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s32/s87/s105/s100/s116/s104/s32/s40/s109/s109/s41\nFIG. 2: Measured peak intensity and width for the nonlinear\nwave packet shown in the right panels of Fig. 1 as a function\nof propagation path. The dashed vertical lines show positio ns\nof local minima of intensity and of the local maxima of the\npacket width.\ninfluence of confinement on the nonlinear evolution of\nthe packet transverse profile and suggests that interac-\ntion of the linear eigenwaves of the waveguide – the so-\ncalled width modes – underlies nonlinear wave dynamics.\nTherefore we term this nonlinear object a “guided spin-\nwave bullet”.\nThe theoretical description ofthe observedphenomena\nis based on concepts developed in Refs. [9] and [10]. The\ningredients of the model are (i) effective dipole pinning\nof magnetization, which results in a tri-linear interaction\nas the initial interaction for development of the trans-\nverseinstability of the wavepacketprofile, (ii) the width-\nmode group velocity matching, and (iii) the nonlinear\nextension of spectrum of width modes. In Ref. [9], we\ntheoretically studied linear propagating eigenwaves of a\nmagnetic stripe. The eigenwaves represent guided modes\nwith discrete transverse wavenumbers. For a stripe of\nrectangular cross-section the modes are characterized by\na standing-wave type dynamic magnetization distribu-\ntion across the stripe cross-section and by a monochro-\nmatic propagating wave with the longitudinal wavenum-\nberkzalong the stripe. Importantly, for stripes with a\nlarge ratio pof width to thickness the thickness distri-\nbution of the dynamic magnetization is practically ho-\nmogeneous, whereas in the direction of the stripe width\nthe standing spin waves possess considerably decreased\namplitudes at the edges due to dynamic demagnetiza-\ntion effects [11]. Previous calculations (see Fig. 3 in [9])\nshow that the assumption of a totally pinned magneti-\nzation at the stripe edges results in good approximation\nfor the transverseprofilesof the propagatingeigenmodes.\nAdopting the assumption of totally pinned edge spins,\nthe transverseprofileofthe dynamic magnetizationisde-\nscribed by an integer number of half-periods nof the sine\nfunction with the lowest transverse wavenumber ky≡kn\nbeingkn=1= 1·π/w, wherewis the stripe width. For3\nthe aspect ratio of our waveguide, p= 357, this works\nwith good accuracy.\nAs one sees from Fig. 1, this conclusion is in a good\nagreement with the packet profiles measured in the lin-\near regime. The nonlinear dynamics is now described\nby a theory which is analogous to the one developed in\nRef. [10]. We assume that the dynamic pinning of mag-\nnetization is conserved in the weakly nonlinear regime.\nThen thetransverseevolutionofthe nonlinearpacketcan\nbe considered as interaction of linear eigenmodes which\nare pinned at the stripe edges. Indeed, in Fig. 1 one\nclearly sees that in the nonlinear regime the dynamic\nmagnetization at the stripe edges practically vanishes.\nThen the description for the propagating modes results\nin an evolutional equation for the spin-wave precession\nangleφwhich reads:\ni∂/∂tF n,k[φ(y,z)]+(ωn,k+iη−ω)Fn,k[φ(y,z)] (1)\n+TFn,k[|φ(y,z)|2φ(y,z)] =fn,k,t.\nIn this expression k≡kz,ωn,kis the eigenfrequency of\nthen-th guided mode for the longitudinalwavenumber k,\nηistherelaxationfrequencyforthefilm, and ωisthe car-\nrier frequency of the microwave signal f(y,z,t)exp(iωt)\napplied at the vicinity of z=0 which excites the input\nspin-wave packet, and Tis the nonlinear coefficient anal-\nogousto the nonlinearcoefficient ofthe spin-waveversion\nof the Nonlinear Schr¨ odinger Equation [12]. The opera-\ntionFn,kdenotes a 2D Fourier transform. It is the dis-\ncrete sine transform with the basis functions sin( kny) in\nthe direction of stripe width y, and is the Fourier integral\nover continuous wavenumbers kwith the basis functions\nexp(ikz) along the stripe.\nThe expression Eq. (1) can be easily transformed into\na system of dynamic equations for amplitudes of guided\nmodesφn(z,t) =Fn[φ(y,z,t)] coupled by the four-wave\nnonlinear interaction. The analysis of the system shows\nthat the formation of the two-dimensional waveform can\nbe considered as an extension of the spectrum of the\nwidth modes. The partial waveforms carried by the in-\ndividual width modes have the same carrier frequencies\nequal to that of the external excitation signal ωand the\ncarrier wave numbers which satisfy the dispersion rela-\ntions for the modes ωn,k=ω. In the linear regime all the\nmodes are independent. In the nonlinear (high ampli-\ntude) regime the width modes become mutually coupled\nwhich ensures intermodal coherent energy transfer.\nThe efficiency of mode coupling in the pulse regime\ndepends on two major factors: the mode group velocity\nmatching and the type of nonlinear interaction. The ge-\nometry of a relatively wide stripe is very favorable for\nhaving maximum contributions from both.\nLet us first discuss the type of nonlinear mode inter-\naction. The spin-wave packet immediately after having\nbeen launched into the stripe is carried by the lowest\n(fundamental) width mode ( n= 1)φ1(z,t). Thereforeit is necessary to consider the nonlinear interaction of\nhigher-order width modes with this particular mode.\nThe interactions of the fundamental width mode with\nall even modes is not important for symmetry reasons.\nThe nonlinear interaction of modes of the same type of\nsymmetry is described by the parametric term as well\nas by an additional pseudo-linear (tri-linear) term. The\nparametric interaction is of convective instability type\nand is the same parametric instability which triggers for-\nmation of conventional bullets in continuous films. This\nconventional process is described by a pair of complex-\nconjugated equations with a parametric term propor-\ntional to the square of amplitude of the pumping wave\n(see e.g. Eqs. (4) and (5) in [13]). In this interaction\nthe packet carried by the fundamental mode plays the\nrole of the pumping wave. Its energy is transferred to\nthe partial waveforms carried by the higher-order width\nmodes. There is a threshold associated with this para-\nmetricprocessduetonaturaldampinginthe medium. In\nthewaveguidestructurethethresholdisofthesameorder\nof magnitude as the modulation instability threshold in\ncontinuous films (see e.g. Eq. (10) in Ref. [1]). Further-\nmore,aninitialperturbationintheformofnon-vanishing\namplitude of a higher-order mode is needed to start the\nparametric amplification. This perturbation usually is\nprovided by thermal excitation. Therefore an amplified\nhigher-ordermode, which is group velocity matched with\nthe pumping wave, needs a large distance of propagation\nin order to reach a noticeable level. The energy of the\npumping wave decreases down the propagation path due\nto losses in the medium. If the parametric amplification\ngain is small because of small supercriticality the higher-\norder mode cannot reach an amplitude comparable with\nthatofthefundamentalmodebeforetheamplitudeofthe\npumping wave packet carried by the fundamental mode\nfallsbelowthe thresholdofparametricinstability andthe\ngain ceases.\nWhat distinguishes the confined waveguide geometry\nis that the nonlinear mixing starts as a pseudo-linear\n(tri-linear) interaction of the fundamental with the next-\norder symmetric mode which is the third mode φ3(z,t):\n∂φ3/∂t+v3∂φ3/∂z+iωnl\n3φ3=S13(z,t). Herev3is the\ngroup velocity of the third mode, ωnl\n3is its nonlinear\nfrequency shift, and S13is the tri-linear inhomogeneous\nterm. This term has the form of a linear source of exci-\ntation with amplitude proportional to |φ1(z,t)|3moving\nwith the group velocity of the fundamental mode. The\npresence of this pseudo-linear interaction at the early\nstage of the bullet formation is entirely due to the ef-\nfective dipolar pinning of the magnetization at the stripe\nedges. If the edge spins were unpinned, the interaction\nof all the width modes would be purely parametric. The\npseudo-linear excitation introduced by this term into the\ndynamic equations is a threshold-freeprocess, so it works\neven below the threshold of parametric instability. Fur-\nthermore, in contrast to the parametric process this pro-4\ncess does not require non-zero initial amplitude of the\namplified waveform to start the amplitude growth. This\nmechanism ensures rapid growth of the symmetric n= 3\nmode driven by the n= 1 mode up to the level where the\nparametric mechanism starts to work efficiently. After\nthat the fundamental mode jointly with the n= 3 mode\nis capable to rapidly generate a large set of modes with\nyet higher odd numbers nthrough both pseudo-linear\nand parametric mechanisms.\nLet us now discuss the importance of the mode group\nvelocity matching. Our theory shows that the efficiency\nof both nonlinear interaction mechanisms (parametric\nand tri-linear) strongly depends on the group velocity\ndifference of interacting modes and the initial length of\nthe nonlinear packet. In wider stripes the group veloc-\nities of the width modes are closer to each other. As\nthe nonlinearity is of attractive type, nonlinear correc-\ntions to the group velocities partially compensate for the\ngroupvelocitymismatch. Asaresultthenonlinearlygen-\nerated higher-order partial waveforms remain for some\ntime within the pump packet. Again, due to the attrac-\ntive character of the nonlinearity the modes have initial\nphases such as their transverse profiles are summed up\nconstructively in the middle of the stripe width and a\nbullet-like total wave packet is formed as confirmed by\ndirect numerical solution of Eq.(1) shown in Fig. 3.\nIn the narrower stripes (1mm in width as in Fig. 3,\npanel 2) the mode group velocity difference is larger and\ncannot be compensated by the nonlinear corrections to\nthe group velocities for the same initial intensity of the\nwave packet. Thus the group velocity matching is not\nensured. As a result the nonlinearly generated higher-\norder modes leave the area of the interaction before they\nreach significant amplitudes. For the same length of the\ninitial packet φ1(t= 0,z) the extension of the spectrum\nof width modes does not occur. The waveform φ1(t,z)\nremains unaffected by the interactions with the higher-\norder modes. It undergoes only a nonlinear longitudi-\nnal compression and forms a quasi-1D wave packet – the\nspin-wave envelope soliton – which has a stable sine-like\nprofile in the y-direction.\nThe excellent agreement of the simulation results with\nthe experimental data shown in Fig. 3 provides evidence\nfor the validity of the developed theory.\nIn conclusion, the formation of quasi-2D nonlinear lo-\ncalized wave packets – guided spin-wave bullets – was\nstudied in spin-wave waveguides. Our experimental and\ntheoretical investigations show that formation of these\nstable nonlinear objects is strongly affected by the trans-\nverse confinement of the medium. A specific magneto-\nstatic effect – the effective dipole pinning of the mag-\nnetization at the edges of the stripe, the width-mode\ngroup velocity matching of different discrete waveguide\nmodes, and the extension of the width mode spectrum\ndue to nonlinear mode-mode energy transfer are essen-\ntial for the nonlinear evolution of the initial spin-wave\nFIG. 3: Lateral shapes of the nonlinear SW packets. 1 and 2\n– theoretical results calculated for the ferrite stripes of width\nof 2.5mm and 1mm, respectively. 3 and 4 – experimental\nprofiles observed in YIG waveguides of width of 2.5mm and\n1mm, respectively. 1 and 3: bullets. 2 and 4: solitons.\nexcitation. Both the experimentally detected properties\noftheevolutionandthetheoreticallyrevealedmechanism\nofformationshowthattheobservednonlinearwavepack-\nets can be treated as “guided spin-wave bullets” which\nare specific for laterally confined magnetic films.\nSupport by the DFG within the SFB/TRR 49, the\nAustralian Research Council, the University of Western\nAustralia, and the Russian Foundation for Basic Re-\nsearch is gratefully acknowledged.\n∗on leave from St.Petersburg Electrotechnical University,\n197376, St.Petersburg, Russia\n[1] O. B¨ uttner, M. Bauer, S.O. Demokritov, B. Hille-\nbrands, Yu.S. Kivshar, V. Grimalsky, Yu. Rapoport, and\nA.N. Slavin, Phys. Rev. B 61, 11576 (2000).\n[2] A.A. Serga, B. Hillebrands, S.O. Demokritov, and\nA.N. Slavin, Phys. Rev. Lett. 92, 117203 (2004).\n[3] A.A. Serga, B. Hillebrands, S.O. Demokritov,\nA.N. Slavin, P. Wierzbicki, V. Vasyuchka, O. Dzyapko,\nand A. Chumak, Phys. Rev. Lett. 94, 167202 (2005).\n[4] A.N. Slavin, O. B¨ uttner, M. Bauer, S.O. Demokritov,\nB. Hillebrands, M.P. Kostylev, B.A. Kalinikos, V. Gri-\nmalsky, and Yu. Rapoport, Chaos 13, 693 (2003).\n[5] M. Chen, M.A. Tsankov, J.M. Nash, and C.E. Patton,\nPhys. Rev. B 49, 12773 (1994).\n[6] S.O. Demokritov, B. Hillebrands, and A.N. Slavin, Phys.\nRep.348, 441 (2001).\n[7] O. B¨ uttner, M. Bauer, C. Mathieu, S.O. Demokritov,\nB. Hillebrands, P.A. Kolodin, M.P. Kostylev, S. Sure,\nH. D¨ otsch, V.Grimalsky, Yu.Rapoport, andA.N.Slavin,\nIEEE Trans. Magn. 34, 1381 (1998)\n[8] V.E. Demidov, U.-F. Hansen, O. Dzyapko, N. Koulev,\nS.O. Demokritov, and A.N. Slavin, Phys. Rev. B 74\n092407 (2006).\n[9] M.P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti,\nT. Ono, and R.L. Stamps, Phys. Rev. B 76, 054422\n(2007).\n[10] M. Kostylev, V.E. Demidov, U.H. Hansen, and\nS.O. Demokritov, Phys. Rev. B 76, 224414 (2007).\n[11] K.Y. Guslienko, S.O. Demokritov, B. Hillebrands, and\nA.N. Slavin, Phys. Rev. B 66, 132402 (2002).\n[12] A. Zvezdin and A. Popkov, JETP 57, 350 (1983).\n[13] B.A. Kalinikos, N.G. Kovshikov, M.P. Kostylev, and\nH. Benner, JETP Lett. 64, 171 (1996)."    },    {        "title": "1104.4032v1.Magnetic_properties_of_the_ferrimagnetic_cobaltite_CaBaCo4O7.pdf",        "content": "arXiv:1104.4032v1  [cond-mat.str-el]  20 Apr 2011Magnetic properties of the ferrimagnetic cobaltite\nCaBaCo 4O7\nZhe Qua,∗, Langsheng Linga, Lei Zhanga, Li Pib,a, Yuheng Zhanga,b\naHigh Magnetic Field Laboratory, Chinese Academy of Science s,\nHefei, Anhui, 230031, China\nbHefei National Laboratory for Physical Sciences at the Micr oscale,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\nAbstract\nThe magnetic properties of the ferrimagnetic cobaltite CaBaCo 4O7are sys-\ntematically investigated. We find that the susceptibility exhibits a dow nward\ndeviation below ∼360 K, suggesting the occurrence of short range magnetic\ncorrelations at temperature well above TC. The effective moment is de-\ntermined to be 4.5 µB/f.u, which is consistent with that expected for the\nCo2+/Co3+high spin species. Using a criterion given by Banerjee [Phys.\nLett.12, 16 (1964)], we demonstrate that the paramagnetic to ferrimagn etic\ntransition in CaBaCo 4O7has a first order character.\nKeywords: A. magnetically ordered materials, D. phase transitions\n1. Introduction\nTransition metal oxides with geometry frustration have attracte d consid-\nerable interest over decades. [1, 2, 3, 4] They commonly exhibit th e per-\nsistence of strong spin fluctuations at low temperatures. As a res ult, the\nlong-range magnetic order is at least partially suppressed and vario us short\nrange correlated phases such as spin liquid, spin glass or spin ice deve lop.\nIn some cases, frustration can be partially or entirely released, eit her by\nstructural distortions that lift the ground-state degeneracy, or by the ”order-\nby-disorder” mechanism, [5] resulting in the establishment of a long- range\nmagnetic order.\n∗Corresponding author. Tel: +86-551-559-5640; Fax: +86-551- 559-1149.\nEmail address: zhequ@hmfl.ac.cn (Zhe Qu)\nPreprint submitted to Solid State Communications October 1 1, 2018Two well-known structural topology causing the presence of geom etry\nfrustrationaretwo-dimensionaltriangularlatticeandtwo-dimens ionalkagome\nlattice. Compositions whose structural motif embraces triangular or kagome\nlayers are of great interest as model systems and have been the f ocus of\nnumerous studies. In this respect, the recently discovered ”114 ” cobaltite\nCaBaCo 4O7[6, 7] is particularly interesting because its crystal structure is\nbuilt up of an alternate stacking of triangular or kagome layers form ed by the\nCoO4tetrahedra (shown in theinset to Fig. 1). There is very largedistor tion\nin the crystal, characterized by a strong buckling of the kagome lay ers. [6, 7]\nIn addition, it exhibits charge ordering, with Co2+sitting on two sites and\n”mixed valent” cobalt Co3+/Co2+Lsitting on two other sites. [7] Due to the\nlarge structural distortion and the charge ordering, the geomet ry frustration\nis lifted, resulting in a ferrimagnetic ground state at low temperatur es. [6, 7]\nAlthough significant progress has been made in understanding the m ag-\nnetic properties in CaBaCo 4O7, a few questions remain to be answered. For\nexample, does the system shows short-range magnetic correlatio ns above TC\nlike their ”114” cousins such as YBaCo 4O7? [8, 9] Why the obtained effec-\ntive moment differs significantly from the expected value in CaBaCo 4O7? [6]\nWhat’s the nature of the paramagnetic to ferrimagnetic transition ?\nTo address these questions, we systematically measured the magn etic\nproperties of CaBaCo 4O7. It is found that the susceptibility exhibits an\ndownward deviation below ∼360 K, suggesting the occurrence of short range\nmagnetic correlations at temperature well above TC. By extending the mag-\nnetization measurement up to 800 K, the effective moment is determ ined to\nbe 4.5µB/f.u through a Curie-Weiss analysis, which is consistent with that\nexpected for the Co2+/Co3+high spin species. Using a criterion given by\nBanerjee, [10] we demonstrate that the paramagnetic to ferrima gnetic phase\ntransition in CaBaCo 4O7is a first order one.\n2. Experiment\nPolycrystalline sample of CaBaCo 4O7was prepared by using the con-\nventional solid-state reaction method described in Ref. [6]. Stoich iometric\nproportionsofhighpurity CaCO 3, BaCO 3andCo 3O4were mixedandheated\nat 900oCin air to decarbonation. They are then pelletized, and then sin-\ntered at 1100oCin air for 12 hours and quenched to room temperature. The\nstructure and the phase purity of the samples were checked by po wder X-ray\ndiffraction (XRD) at room temperature. Magnetization measureme nts were\n2performed with a commercial superconducting quantum interfere nce device\n(SQUID) magnetometer (Quantum Design MPMS 7T-XL) and a Physic al\nPropertyMeasurement System(QuantumDesignPPMS-16T)equip ped with\na vibrating sample magnetometer (VSM).\n3. Results and Discussion\nFigure 1 displays the powder XRD pattern of CaBaCo 4O7at room tem-\nperature. Rietveld refinement [11, 12] of the XRD pattern confirm s that the\nsample is single phase with an orthorhombic structure ( Pbn21space group).\nThe lattice parameters are determined to be a= 6.2871 ˚A,b= 11.0106\n˚Aandc= 10.1945 ˚A, which agree well with previous reports within the\nexperimental error. [6, 7]\nThe temperature dependence of the magnetization M(T) between 2 K\nand 400 K under 0.1 T is shown in the upper panel of Fig. 2. They are me a-\nsured during field cooling sequence (FCC), during warming after field cooling\nsequence (FCW) andduring warming afterzerofield coolingsequenc e (ZFC),\nrespectively. All curve shows a rapid increase below ∼70 K, suggesting the\noccurrence of the transition into a magnetically ordered state. At 5 K, the\nsaturated magnetic moment is still relatively small, only ∼1.1µB/f.u.under\n14 T (see the lower panel of Fig. 2), agreeing with a ferrimagnetic gr ound\nstate. The Curie temperature, defined as the temperature corr esponding to\nthe maximum in the dM/dTcurve, is determined to be 60 K (see the inset to\nFig. 2). Below TC, we observe significant irreversibility between the magne-\ntization curve measured after ZFC and FCC histories. This is attribu ted to\nthe large coercive field compared to the applied field. [6, 13] As show n in the\nlower panel of Fig. 2, the coercive field of CaBaCo 4O7is about 2 T at 5 K,\nwhich is much larger than the applied field of 0.1 T. Therefore, the mag netic\ndomains will be locked in random direction during ZFC sequence while be\naligned to the same direction during FCC or FCW sequences, resulting in the\nlarge irreversibility below TC. All these results are consistent with previous\nreport, [6] confirming that our sample is of high quality.\nA close look on the temperature dependence of the magnetization r eveals\nmoreinformation. Theinset toFig. 3displays theenlargedviewofthe M(T)\ncurves between TCand 400 K. One can see that while the magnetization\ndecreases with increasing temperature above TCthe slope of the M(T) curve\ndoes not decrease monotonously as that expected for a pure par amagnetic\nstate where the Curie-Weiss law predicts χ∝C/(T−TCW) (HereCis\n3the Curie constant and TCWis the Curie-Weiss temperature). In order to\nunderstand this behavior, we extend the measurement of the M(T) during\nFCC sequence up to 800 K and perform the Curie-Weiss analysis. As s hown\nin Fig. 3, 1/ χshows an upward deviation from the linearity below ∼360 K.\nSince the deviation temperature is much higher than the Curie tempe rature,\nthis behavior could not be attributed to the critical enhancement o f the\nspin fluctuations due to the approach to the paramagnetic to ferr imagnetic\ntransition but suggests the occurrence of short range magnetic correlations.\nThe Curie-Weiss temperature TCWis determined to be ∼-890 K. This gives\nf=TCW/TC∼14.8 which means CaBaCo 4O7is strongly frustrated. The\neffective moment is determined to be ∼4.5µB/f.u., which agrees well with\nthevalueofa1:1combinationofCo2+/Co3+highspinspecies expected based\non the chemical formula. The Curie-Weiss temperature and the effe ctive\nmoment obtained here are different from previous report. [6] This should be\nunderstood because short-range magnetic correlations might ap pears in their\nfitting temperature region.\nIn order to obtain further information on the paramagnetic to fer rimag-\nnetic transition in CaBaCo 4O7, we use the criteria proposed by Banerjee to\ndetermine the order of this transition. By considering the essentia l similarity\nbetween the Landau-Lifshitz [14] and Bean-Rodbell [15] criteria, B anerjee\nshows that the slope of the H/MversusM2curves near the critical tem-\nperature can distinguish the first-order magnetic transition from the second\norder ones: a negative slope means the former and a positive slope m eans the\nlatter. [10] We then measured the initial isothermal magnetization c urves at\ntemperatures in the vicinity of the Curie temperature. Before eac h run, the\nsample is warmed up to 200 K and then cooled to the measuring temper ature\nunder zero field to ensure a perfect demagnetization of the sample . The data\nare summarized in the inset to Fig. 4. It is noted that the M(H) curve ex-\nhibitsapeculiar behaviorthatitsslopeshowsadecrease beforeanin crease at\nintermediate fields. This behavior was also observed in MnAs, where a first\norder transitionoccursatits Curietemperature andisused totes t Banerjee’s\ncriteria. [10, 15] We replotted the M(H) curves as H/Mvs.M2in Fig. 4.\nNegative slope is clearly observed between 64 and 70 K, which confirm s that\nthe paramagnetic to ferrimagnetic transition occurred in CaBaCo 4O7has a\nfirst order character according to the criterion.\n44. Conclusion\nIn conclusion, we systematically investigate the magnetic propertie s of\nCaBaCo 4O7. TheCurie-Weiss temperatureisdetermined tobe ∼-890Kand\nthe effective moment be ∼4.5µB/f.u.. The susceptibility shows downward\ndeviation from the Curie-Weiss law below ∼360 K, hinting that short range\nmagnetic correlations might occur at temperature much higher tha nTC=\n60 K. The paramagnetic to ferrimagnetic transition in CaBaCo 4O7is found\nto have the first order character.\n5. Acknowledgments\nThis work is financially supported by the National Key Basic Research of\nChina under Grant No. 2007CB925001 and 2010CB923403, and by N ational\nNatural Science Foundation of China under contract No. 1100419 8.\nReferences\n[1] A. P. Ramirez, Annu. Rev. Mater. Sci. 24, (1994) 453.\n[2] P. Schiffer and A. P. Ramirez, Comments Condens. Matter Phys. 18,\n(1996) 21.\n[3] J. E. Greedan, J. Alloys Compd. 408-412, (2006) 444.\n[4] R. Moessner and A. P. Ramirez, Phys. Today 59 (2), (2006) 24.\n[5] J. Villain, R. Bidaux, J. P. Carton, and R. Conte, J. Phys. (Franc e)41,\n(1980) 1263.\n[6] V. Caignaert, V. Pralong, A. Maignan, andB.Raveau, SolidState Com-\nmun.149, (2009) 453.\n[7] V. Caignaert, V. Pralong, V. Hardy, C. Ritter, and B. Raveau, P hys.\nRev. B81, (2010) 094417.\n[8] M. Soda, Y. Yasui, T. Moyoshi, M. Sato, N. Igawa, and K. Kakura i, J.\nPhys. Soc. Jpn. 75, (2006) 054707.\n[9] P. Manuel, L. C. Chapon, P. G. Radaelli, H. Zheng, and J. F. Mitche ll,\nPhys. Rev. Lett. 103, (2009) 054707.\n5[10] Phys. Lett. 12, (1964) 16; see also, for example, S. V. Vonsovskii, Mag-\nnetism(Wiley, New York, 1974), Vol. 2, Chap. 25.\n[11] A. C. Larson and R. B. Von Dreele, General Structure Analysis System\n(GSAS)(Los Alamos National Laboratory Report LAUR 86-748, 2000).\n[12] B. H. Toby, J. Appl. Cryst. 34, 2001 (210).\n[13] Z. R. Yang, S. Tan, and Y. H. Zhang, Appl. Phys. Lett. 79, (2001) 3645.\n[14] L. D. Landau, Zh. Eksp. Teor. Fiz. 7, (1937) 19; ibid. 7, (1937) 627;\nibid.E. M. Lifshitz, 11, (1941) 269; ibid.V. L. Ginzburg, 17, (1947) 833;\nS. V. Vonsovskii, Izv. Akad. Nauk SSSR, Ser. Fiz. 11, (1947) 485.\n[15] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, (1962) 104.\n620 40 60 80 100 \n  Intensity (arb. units) \n2θ (degree) Ba \nCa K Co \nT Co a\nc\nFigure 1: (Color online) Powder XRD patterns of CaBaCo 4O7. The solid curve is the\nbest fit from the Rietveld refinement using GSAS, with Rp= 11.68% and Rwp= 10.08%.\nThe vertical marks indicate the position of Bragg peaks and the bot tom curves show the\ndifference between the observed and calculated intensities. Inset shows the structure of\nCaBaCo 4O7viewed along baxis. K Co and T Co represent kagome layer and triangular\nlayer of CoO 4tetrahedra, respectively.\n70 100 200 300 0.0 0.2 0.4 0.6 0.8 \n-10 -5 0 5 10 -1.0 -0.5 0.0 0.5 1.0 H = 0.1 T \n ZFC \n FCC \n FCW \n M ( µb/f.u.) \nT (K) 40 60 80 100 \n  dM/dT \nT (K) \n  M ( µb/f.u.) \nH (T) T = 5 K \nFigure 2: (Color online) Upper panel: The magnetization as function o f the temperature\nunder an applied field of 0.1 T. inset shows the dM/dTas function of the temperature.\nLower panel: the magnetization as function of the field measured at 5 K.\n80 200 400 600 800 10 -2 10 -1 10 0\n1/ χ (T f.u./ µB)\nχ ( µB/f.u. T) \nT (K) 020 40 60 H = 0.1T \n100 200 300 400 2.0 2.5 3.0 3.5 \n  M (10 -3  µb/f.u.) \nT (K) \nFigure 3: (Color online) The susceptibility and the reciprocal of the s usceptibility as\nfunction of temperature between 2 and 800 K under 0.1 T measured in FCC sequence.\nThe solid lines represent the Curie-Weiss fitting. Inset shows the en larged view of the\nM(T) curve to highlight the deviation from the Curie-Weiss fitting.\n0.0 0.2 0.4 0.6 010 20 \n50 K \n H/M (T f.u./ µB)\nM 2 ( µB2/f.u. 2)70 K \n0 1 2 3 4 50.0 0.2 0.4 0.6 \n  \n70 K 50 K M ( µB /f.u.) \nH (T) \nFigure 4: Inset shows the initial isothermal magnetization curves a t temperatures in the\nvicinity of the Curie temperature TC= 60 K at an interval of 1 K. The main panel shows\nthese curves replotted as H/Mvs.M2.\n9"    },    {        "title": "1210.4368v2.Mn__2_FeSbO__6___a_ferrimagnetic_ilmenite_and_an_antiferromagnetic_perovskite.pdf",        "content": "Mn 2FeSbO 6: a ferrimagnetic ilmenite and an antiferromagnetic perovskite\nR. Mathieu,1,\u0003S. A. Ivanov,1, 2I. V. Solovyev,3G. V. Bazuev,4P. Anil Kumar,1P. Lazor,5and P. Nordblad1\n1Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden\n2Department of Inorganic Materials, Karpov' Institute of Physical Chemistry,\nVorontsovo pole, 10 105064, Moscow K-64, Russia\n3National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan\n4Institute of Solid-State Chemistry, Ural Branch of the Russian Academy of Science, 620999 Ekaterinburg, GSP-145, Russia\n5Department of Earth Sciences, Uppsala University, Villav agen 16, SE-752 36 Uppsala, Sweden\n(Dated: August 18, 2021)\nDue of its polymorphism, Mn 2FeSbO 6can be synthesized at high pressures and temperatures as\na ferrimagnetic ilmenite or an antiferromagnetic perovskite. The structural phase transformation\nis discussed in detail, and magnetic structures are proposed for both phases. The high-pressure\nMn2FeSbO 6polymorph is a rare example of A2B0B00O6perovskite with solely Mn cations on the\nA-site. Fe and Sb cations are ordered on the B-sites. Theoretical calculations for the perovskite\nphase suggest a complex magnetic structure, holding an electronic polarization.\nPACS numbers: 75.47.Lx,75.50.Gg,75.85.+t\nThere is an ongoing quest for new materials show-\ning several, possibly interacting, ferroic properties. Such\nmultiferroic materials may e.g. exhibit ferromagnetism\nand ferroelectricity[1], as well as magnetoelectric e\u000bects\nrelated to the correlation of spin and dipole ordering[2{\n4]. Multiferroic and/or magnetoelectric materials have\nfundamental and applied importance in the \felds of\nstrongly correlated transition metal oxides and spintron-\nics. A strategy to obtain such new materials is to de-\nsign complex crystal structures including several mag-\nnetic cations, and small diamagnetic cations favoring po-\nlar distortions such as Te6+, Nb5+or Sb5+[5].\nMn 2FeSbO 6(MFSO) ilmenite-type mineral, called\nmelanostibite[6], contains two magnetic ions (Mn2+and\nFe3+) on di\u000berent crystallographic sites. MFSO cannot\nbe synthesized under ambient pressure. Yet it was shown\nthat MFSO ilmenite could be synthesized with relatively\nhigh purity using thermobaric treatments[6, 7]. MFSO\nilmenite was found to order ferrimagnetically below 270\nK. By increasing the pressure employed during the ther-\nmobaric synthesis, a double perovskite phase of MFSO\nmay be stabilized, with Mn cations on the A-site, and\nFe, Sb cations ordered on the B-site.\nWe show in the present article that the perovskite\nphase has magnetic properties quite di\u000berent from those\nof the ilmenite one, namely is antiferromagnetic below 60\nK. We discuss the phase transformation from ilmenite to\nperovskite in MFSO based on X-ray powder di\u000braction\nresults. Using theoretical calculations, we predict a\ncomplex magnetic structure, and electronic polarization\nof the perovskite phase.\nMn 2FeSbO 6ilmenite, i.e. with the same structure as\nthe mineral material, was synthesized as ceramic using\nthe following thermobaric treatment: pressure P= 3\nGPa, temperature T= 1000oC, duration d= 30 mins\n(see Ref. [6, 7] for details). By increasing the pressure\nFIG. 1. X-ray di\u000braction patterns at 295 K and Rietveld\nre\fnements of perovskite (main frame; Rp= 0.031,Rwp=\n0.049, andRb= 0.035) and ilmenite (inset; Rp= 0.046,Rwp\n= 0.067, and Rb= 0.041) phases.\nto 6 GPa (and duration to 40 mins), a perovskite phase\nof MFSO could be stabilized [7]. Crystal structure and\nstoichiometry of ilmenite and perovskite MFSO were\ninvestigated by X-ray powder di\u000braction (XRPD) on a\nD8 Bruker di\u000bractometer using CuK \u000b1radiation and\nmicroprobe energy-dispersive spectroscopy analysis.\nMagnetization and heat capacity measurements were\nperformed using a superconducting quantum inter-\nference device (SQUID) MPMS magnetometer and\nPPMS physical property measurement system from\nQuantum Design Inc. Complementary high-temperature\nmagnetization measurements were performed using the\nVSM/oven option of the PPMS. Electronic structure\ncalculations were performed to predict the magnetic\nordering of the perovskite phase. Crystal and some of\nthe magnetic structures were drawn using VESTA[8].arXiv:1210.4368v2  [cond-mat.str-el]  18 Oct 20122\nFIG. 2. (Color online) Polyhedral representations of di\u000berent polymorphs in comparable orientation[9]. The color of the\npolyhedra re\rects the central cation; (a,b) red: Mn, blue: Fe/Sb, (c) blue: Fe, yellow: Sb; Mn polyhedra are omitted for clarity.\nSmall red spheres indicate oxygen atoms.\nTABLE I. Structural parameters at room temperature. V=Z is the unit cell volume per formula unit, \u001ais the density. Minimum-\nmaximum values of M-O bond lengths, M-O-Mbond angles, and M-Mdistances (M= Fe or Mn) are indicated.\nilmenite perovskite\nSpace group R\u00163 P21=n\na, b, c ( \u0017A) ;\f(deg.) 5.2379(4), 14.3519(6) 5.2459(4), 5.3987(4), 7.6332(5) ; 89.67(6)\nCoordination number for Mn / Fe 6 / 6 8(+4) / 6\nV/z ( \u0017A3) ;\u001a(g/cm3) 56.8 ; 5.23 54.1 ; 5.89\nMn-Mn / Mn-Fe / Fe-Fe distances ( \u0017A) 3.129-3.981 / 2.985-3.989 / 3.048 3.847-3.683 / 3.099-3.519 / 5.246-5.398\nMn-O / Fe-O distances ( \u0017A) 2.295-2.564 / 1.738-1.962 2.124-2.764 / 1.949-2.094\nMn-O-Mn / Mn-O-Fe / Fe-O-Fe angles (deg.) 60.6-151.6 / 58.8-145.3 / 55.3-136.1 78.9-156.4 / 51.7-140.7 / 125.5-135.3\nPolyhedral volume ( \u0017A3) ; distortion Mn/Fe-O 6 17.80 ; 0.043 / 7.72 ; 0.076 22.67 ; 0.067 / 10.87 ; 0.005\nMn / Fe cation shift from centroid ( \u0017A) 0.191 / 0.053 0.161 / 0\nThe XRPD patterns shown in Fig. 1 were obtained\nfor ilmenite and perovskite compounds. The data sets\nwere successfully re\fned using the Rietveld method, per-\nmitting the determination of the structural parameters\nlisted in Table I. As shown in the table, the perovskite\nMFSO adopts a monoclinically distorted phase, with \f\nangle close to 90o(i.e. almost orthorhombic). The XRPD\ndata evidences the ordering of the Fe/Sb cations on the\nperovskiteB-site.\nMFSO is dimorphic; the ilmenite phase is the low-\npressure polymorph, while the perovskite one is the high-\npressure one. The transformation of ilmenite into per-\novskite phase at high hydrostatic pressures is accompa-\nnied by a decrease in the cell volume and distances be-\ntween the layers of the hexagonal anion packing, i.e. by\nthe formation of a denser structure (see Table I). From a\ncrystallochemical point of view, the reconstructive phase\ntransition from ilmenite to perovskite phase in MFSO\noccurs by incorporation of Mn cations from interlayerpositions into the layers of the closest packing. This re-\ndistribution may occur since Mn2+cations have no site\npreference, and can occupy sites of various coordination.\nThe hexagonal type of packing then changes to the cubic\none.\nThe ilmenite structure is based on a hexagonal\nclose-packed array of oxygen. Cations occupy 2/3 of\nthe octahedral sites leaving 1/3 of the octahedral sites\nvacant. Mn and Fe cations are distributed between\nalternating layers such that each Mn octahedra shares\na face with the Fe octahedra above or below (but\nnot both) and also shares edges with three other Mn\noctahedra in the same layer (see Fig. 2). The Fe\noctahedra have similar linkages. The octahedral holes\nof the oxygen array are linked by face-sharing into\nchains along the c-axis that are \flled in the sequence\nvacancy-Mn-Fe-vacancy-Fe-Mn-vacancy, etc. In the\n(0001) planes, the ilmenite structure has the same\ntype of cations. We can assume that in the \frst stage\nof the high-temperature/high-pressure-induced phase3\ntransformation, the ilmenite structure is turned into the\nLiNbO 3-type structure (Mn-Fe-vacancy-Mn-Fe-vacancy-\nMn-Fe-vacancy, etc) in which two di\u000berent cations (Mn\nand Fe) are arranged alternatively in (0001) planes[9]\n(see Fig. 2). This mechanism requires Mn and Fe cation\nrearrangement and is accompanied by a topological\nchange in the structure to accommodate cation re-\ndistribution. Several possibilities may be proposed to\ndescribe those cation displacements: exchange of two\nedge-shared cations (using the vacant site) and two\nface-shared cations, di\u000busion of cations and octahedral\ntilting. The cation di\u000busion involves breaking bonds,\nwhich requires high temperatures due to high activation\nenergy. The octahedral tilting may help lowering the\nactivation energy creating of the necessary topological\naccommodation for Mn and Fe cations during the phase\ntransition. The perovskite structure is derived from the\nideal cubic Pm\u00003mstructure by a combination of\nin-phase (resp. out-of-phase) octahedral tilting about\n<001>(resp.<110>) axis of the prototype perovskite\nstructure. The larger Mn cations occupy the cubocta-\nhedral sites, although as a consequence of the tilting of\nthe FeO 6octahedra (around 19 deg.). The Mn cations\nare e\u000bectively eight coordinated with 4 more oxygen\nanions located at greater distances (more than 3 \u0017A). The\nMn-O bond length increases in the transformation from\nilmenite to perovskite, increasing the Mn coordination.\nThe Fe coordination however does not change.\n0 100 200 3000100200C (J/mol K)\nT(K)020 40 60 800306090\nT(K)0 100 200 30000.10.2M/H (emu/g/Oe)\nT(K)0 100 200 300234x 10−4\nT(K)(a) ilmenite\n(c) ilmenite(b) perovskite\n(d) perovskite\nFIG. 3. (Color online) Upper panels: Zero-\feld-cooled (ZFC)\nand \feld-cooled (FC) magnetization curves for MFSO in (a,\nH= 20 Oe) ilmenite and (b, H= 3 T) perovskite phases. The\nmagnetization of MFSO perovskite was recorded in a larger\n\feld as the low-\feld magnetic response of the perovskite phase\nis more a\u000bected by minor amounts of magnetic impurities.\nIn that \feld, ZFC and FC curves closely coincide with each\nother. Lower panels: Temperature dependence of the heat\ncapacityCof MFSO in (c) ilmenite and (d) perovskite phases.Figure 3 (upper panels) shows the temperature Tde-\npendence of the magnetization M(plotted as M=H ) for\nboth MFSO compounds. While the ilmenite compound\ndisplays a ferrimagnetic transition near 270 K[6], per-\novskite MFSO (lower panel) exhibits a paramagnetic re-\nsponse near that temperature. On the other hand a peak\nis observed in the magnetization curves near 60 K, sug-\ngesting antiferromagnetic ordering below that tempera-\nture. Magnetic hysteresis measurements performed at\nlow temperatures ( T= 10 K, not shown) indicate a lin-\near response to magnetic \feld in perovskite MFSO, in\ncontrast to the ilmenite case.[6]\nThe inverse susceptibility of the perovskite material\ndoes not exactly follow a Curie-Weiss behavior above 100\nK; however considering a near-linear behavior of H=M\ndata measured at higher temperatures (300 K < T <\n500K, not shown) yields a Curie-Weiss temperature \u0012CW\n\u0018- 180 K. The respective magnetic transitions of the two\nMFSO materials are also observed in the heat capacity C\ncurves depicted in Fig. 3, at 268 K and 59 K respectively.\nOne oxide with ilmenite structure is the quasi two-\ndimensional antiferromagnet MnTiO 3[10]. In that\nstructure, the magnetic moments of Mn2+cations are\nantiferromagnetically coupled to each other within and\nin between the hexagonal (0001) planes. Interestingly,\nreplacing Mn2+by Fe2+as in FeTiO 3brings forth a\nferromagnetic coupling within the hexagonal planes,\nalthough adjacent planes remain antiferromagnetically\ncoupled[11]. The NiMnO 3and CoMnO 3ilmenites[12]\ninstead display ferromagnetic (ferrimagnetic) ordering\nabove room temperature. The microscopic magnetic\nstructure of NiMnO 3was investigated in detail using\nneutron di\u000braction[13]. It was found that the ferri-\nmagnetic state is composed of alternating Ni2+and\nMn4+hexagonal planes, yielding an interaction pattern\nbetween magnetic cations similar to that of hematite\nFe2O3. We could not perform neutron di\u000braction\nexperiments on our samples since less than 100 mg of\neach material were synthesized, and Yet, if we similarly\ntranspose the crystal and magnetic structure of NiMnO 3\nto the one of ilmenite MFSO, replacing in their respec-\ntive atomic positions Mn4+and Ni2+in NiMnO 3by\nMn2+and Fe3+/Sb5+in MFSO, we can propose the\nmagnetic structure presented in Fig. 4. This structure\nis also reminiscent of that of the Fe 2O3hematite, as\npredicted theoretically[6].\nLet us now consider the perovskite phase of\nMn 2FeSbO 6; there are rather few perovskites with\nsolely Mn cations on the perovskite A-site. Relatively\nlarge amounts of Mn could be doped onto the A-site\nunder pressure: 33 % in (In,Mn)MnO 3(P= 6 GPa,T\n= 1500oC,d= 40 mins), and 75 % in LaMn 3Cr4O12\n(P= 8-10 GPa, T= 1000oC,d= 30 mins), bringing\nforth antiferromagnetic monoclinic[14] resp. cubic[15]4\nFIG. 4. (Color online) Left panel: Predicted magnetic structure for (left) ilmenite and (right) perovskite MFSO; (a) side view\nand (b) top view. Mn and Fe atoms are indicated in red and blue colors respectively. The atoms, forming the primitive cell of\nMn2FeSbO 6, are indicated by numbers. The middle panel shows the distance dependence of interatomic exchange interactions\nwithin Mn and Fe sublattices (squares and triangles, respectively), and between Mn and Fe sublattices (circles); see inset for\nnotation. The interactions between other sites can be obtained by applying the symmetry operations of the space group P21=n.\nphases. It seems that only one perovskite compound\nwith solely A-site Mn cations has been reported so far:\nMnVO 3synthesized under pressure[16]. Akin to MFSO,\nthe structure of MnVO 3(MVO) changes from ilmenite\nto perovskite above a critical pressure of \u00184.5 GPa.\nIn the high-pressure conditions ( P > 4.5 GPa,T\u0018\n1000-1200oC,d\u001830-60 mins), an orthorhombic MVO\nphase is stabilized. In constrast to MFSO, the magnetic\nproperties of ilmenite and perovskite phases of MVO\nwere found to be quite similar, with a weak ferromag-\nnetism below 65-70 K reported in both cases[16]. \u0012CW\nwas found to decrease (in absolute value) from \u0018-430\nto -250 K between ilmenite and perovskite phases[16].\nA recent study instead reported for perovskite MVO\nsynthesized under pressure ( P= 8 GPa,T= 1100oC,\nd= 30 mins), antiferromagnetism below 46K, and \u0012CW\n\u0018-154K[17].\nWe investigate in more detail the magnetic con\fgura-\ntion of the MFSO compounds in their respective phases.\nAs discussed above, by analogy with the NiMnO 3il-\nmenite, we can predict the magnetic structure of MFSO\nilmenite (see left panel of Fig. 4). The structure\nmatches the results of earlier theoretical calculations, in\nwhich Fe and Mn cation were ferromagnetically coupled\nwithin their respective hexagonal planes, and Fe and Mn\ncations were antiferromagnetically coupled along the c-\ndirection[6]. In the perovskite case, we have elaborated\nanother theoretical model to \fnd the possible magnetic\nstructures. For these purposes, we construct an elec-\ntronic low-energy (Hubbard-type) model for the `mag-\nnetic' 3dbands of Mn and Fe, using results of \frst-\nprinciples electronic structure calculations and the ex-\nperimental parameters of the crystal structure. The ad-\nvantage of the model analysis is that it is relatively easyto consider many types of di\u000berent magnetic structures\nand, thus, determine the correct magnetic ground state,\nalong the same line as it was recently done for multi-\nferroic manganites [18]. More speci\fcally, we start with\nthe local-density approximation (LDA), and determine\nparameters such as crystal \feld, transfer integrals and\ne\u000bective Coulomb interaction in the 3 dbands. Since the\n3dbands are well separated from the rest of the spectrum,\nsuch a construction is rather straightforward (the details\ncan be found in Ref. [19], and the parameters are listed\nin Ref [20]). Then, we solve the obtained model in the\nmean-\feld Hartree-Fock (HF) approximation and derive\nparameters of interatomic exchange interactions, using\nthe method of in\fnitesimal spin rotations near an equi-\nlibrium magnetic state [19]. The behavior of these pa-\nrameters is depicted in the middle panel of Fig. 4, and the\nmagnetic model itself is de\fned as HS=\u0000P\nhijiJijei\u0001ej,\nwhere eiis the direction of the spin at the site i, and\nthe summation runs over the inequivalent pairs hiji. As\nexpected for the d5con\fguration of the Mn3+and Fe2+\nions, all interactions are antiferromagnetic (AFM). More-\nover, the structure of these interactions is very complex:\nthe interactions spread far beyond the nearest neighbors\nand compete with each other. This partly explains the\nrelatively large ratio of j\u0012CWjto the magnetic transition\ntemperature, and in comparison to the ilmenite phase.\nIn order to determine the magnetic structure, resulting\nfrom competing AFM interactions, we have performed\nextensive HF calculations of the low-energy model. First,\nwe searched for the lowest energy structure with collinear\narrangement of spins within the primitive cell. We have\nfound that this magnetic structure is \"\"###\" , where the\narrows describe the relative directions of spins at the site\n1-6 (see right panel of Fig. 4 for the notations). Then,\nwe considered possible modulations of this structure with5\nthe spin-spiral vector q. We have found that even without\nrelativistic spin-orbit (SO) interaction, the \"\"###\" struc-\nture is unstable with respect to the spin spiral alignment\nand the theoretical magnetic ground state is close to q=\n(0,1/2,0), in units of reciprocal lattice translations (the\nenergy gain is about 23 meV per Mn 2FeSbO 6). The SO\ninteraction speci\fes the spacial orientation of spins and\nalign them mainly in the abplane, as shown in Fig. 4.\nAlthough the magnetic spin-spiral alignment formally\nbreaks the inversion operation, without SO interaction,\nthe latter can always be combined with an appropriate\nspin rotation. Therefore, the electronic polarization\n(P) will identically vanish. In order to fully break the\ninversion symmetry, one should forbid the spin rotations.\nThis is done by including the SO interaction, which\nthus yields \fnite P. Using the Berry-phase formalism,\nadopted for the low-energy model [18], the latter can be\nestimated as P\u001810\u0016C/m2and is mainly parallel to the\naaxis.\nIn conclusion, we have investigated the magnetic\nproperties of two polymorphs of Mn 2FeSbO 6, namely\nthe ferrimagnetic ilmenite one and the high-pressure\nantiferromagnetic perovskite. The latter phase is a\nrare example of A2B0B00O3perovskite material with\nMn cations on the A-site of the perovskite structure\nand, since Fe and Sb cations order on the B-sublattice,\nmagnetic cations are present on both A- andB-sites.\nTheoretical calculations predict a complex magnetic\nstructure, and electronic polarization for the perovskite\nphase. 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Lett. 103, 047601 (2009)."    },    {        "title": "2209.03189v2.Effects_of_site_dilution_on_Compensation_in_Ising_Spin_1_2_trilayered_triangular_Ferrimagnets_with_non_equivalent_planes.pdf",        "content": "arXiv:2209.03189v2  [cond-mat.stat-mech]  18 Apr 2023Effects of site dilution on Compensation in Ising Spin-1/2\ntrilayered triangular Ferrimagnets with non-equivalent\nplanes\nSoham Chandra∗1\n1Department of Physics, Presidency University, 86/1 Colleg e Street, Kolkata -700 073, India\nAbstract\nUsing Monte Carlo simulations with the Metropolis algorith m, the magnetic and thermodynamic behaviours\nof a spin-1/2, trilayered ferrimagnetic system on triangular monolayers with quenched nonmagnetic impurities are\nstudied. Two differenttheoretical atoms, AandB, makeupthe ABAandAABtypesofdistinctconfigurations. Like\natoms (A-A and B-B) interact ferromagnetically, while unli ke atoms (A-B) interact antiferromagnetically. Only the\nA-layers are randomly site-diluted with dilution percenta ges ranging from 5% to 45%. Such diluted magnetic thin\nsystems exhibit magnetic compensation which depends sensitively on the concentration of impuriti es. The phase\ndiagram in the Hamiltonian parameter space related to the oc currence of magnetic compensation phenomenon and\nthe effect of site dilution is discussed in detail. Special at tention is given to the mathematical dependencies of\ncompensation temperature on the concentration of nonmagne tic impurities. Depending upon the concentration\nof nonmagnetic impurities, the compensation and critical p oints shift with the equilibrium magnetic behaviours\nchanging between distinct ferrimagnetic behaviours. For e ach combination of the coupling strengths, with values\nof the impurity concentration above a threshold, compensat ion appears where previously was absent. Suggested\nmathematical formulae show how threshold impurity concent ration relies on Hamiltonian parameters.\nKeywords: Spin-1/2 Ising triangular trilayer; quenched nonmagnetic impurities ; Metropolis Monte Carlo simula-\ntion; Compensation temperature; Threshold concentration of imp urities\n∗E-mail addresses: soham.rs@presiuniv.ac.in ; sohamc07@g mail.com\n11 Introduction\nAfter the discovery of ferrimagnetism in 1948 [1,2], a fer-\nrimagnet is often modelled as a combination of two or\nmore magnetic substructures, e.g., sublattices, sublayers\nor subsets of atoms. Among them, a certain class of lay-\nered ferrimagnets have generated considerable attention\nin recent times as they show compensation effect . Such\nthin magnetic systems, for which one dimension is greatly\nreduced than the other two, serve as a bridge between\nmonolayer (2D) and bulk (3D) versions of a magnetic ma-\nterial. Different magnetic responses to the variation of\ntemperature of component sublayers, when combined, lead\nto the appearance of compensation points, i.e., tempera-\ntures lower than the critical point for which magnetiza-\ntion of the total system is zero but the sublayers retain\nmagnetic ordering (see references [3–7]). Such layered fer-\nrimagnets with ferromagnetic and antiferromagnetic in-\nteractions have revealed interesting magnetic responses\nand phase diagrams. The layered ferrimagnets with inter-\nlayer antiferromagnetic coupling between adjacent ferro-\nmagnetic layers have been used in giant magnetoresistance\n(GMR) [10], magneto-optical recordings [11], the magne-\ntocaloric effect [12] and spintronics [13]. That is why both\ntheoretical and experimental investigations on such ferri-\nmagnets are important. The experimental realization of\nbilayer [14], trilayer [15, 16], and multilayer [17–22] sys-\ntems, with desired characteristics, has become a reality\nas thin film growth techniques, e.g. molecular-beam epi-\ntaxy (MBE) [23], metalorganic chemical vapour deposi-\ntion (MOCVD) [24], pulsed laser deposition (PLD) [25],\nand atomic layer deposition (ALD) [26,27] have been de-\nveloped and extensively used with time.\nExtensive search for novel, magnetically compensated\nmaterials exhibiting better performance is centered around\nthe manipulation and control of compensation points [28,\n29]. As a result, theoretical magnetic models are use-\nful for providing physical insights into the compensation\nphenomenon and describing experimental data. Now let\nus review a few of the popular theoretical techniques for\ntreating spin models. References [3, 30–36] demonstrate\nthe versatility of the mean-field analysis as a very suc-\ncessful semi-analytical method for establishing the ther-\nmodynamics of the magnetic systems. Furthermore, there\nare only a few analytical techniques that are exactly solv-\nable for different spin-systems, such as the generalized\nclassical transfer matrix method and decoration-iteration\nmapping [37–40, 65], Bethe ansatz-based quantum trans-\nfer matrix and nonlinear integral equations method [41,\n42], the Jordan-Wigner transformation [43, 44] etc. In\n[45], the dynamic magnetic properties of antiferromag-\nnetic/ferromagnetic YMnO 3/FM bilayer under a time-\ndependent magnetic field are studied by Monte Carlo sim-\nulation on a mixed-spin (5 /2,2,3/2) Ising model. By\nMonte Carlo Metropolis algorithm, in [46], the effect of\nnext nearest neighbour interactions on compensation tem-\nperature and phase transition is investigated in a trilay-\nered ferrimagnetic system on a square lattice. This techno-\nlogical aspect e.g. the magnetocaloric effect in the spin-1\nBlume-Capel model is theoretically studied in [47], using\nthe mean-field theory from the Bogoliubov inequality.Let us now look at some recent developments in diluted\nmagnetic systems. The effect of site dilution on compensa-\ntion and critical temperatures of a two-dimensional mixed\nspin-1/2 and spin-1 system has been studied using Monte\nCarlo simulations, in [48]. In [49], the effects of dilution in\na cylindrical magnetic nanowire system composed of fer-\nromagnetic core and shell layers were investigated using\neffective field theory (EFT) with correlations, with both\nferromagnetic and antiferromagnetic exchange couplings\nat the core-shell interface and in the antiferromagnetic\nnanowire case, a compensation point could be introduced\nby bond dilution at the surface. In [50], Monte Carlo\nsimulation with Wolff cluster algorithm is employed to in-\nvestigate the thermodynamic and magnetic properties of a\nsite diluted spin-1/2 Ising multilayered ferrimagnet. The\ninvestigated system is made up of non-equivalent planes\nwith quenched site diluted alternating planes having the\ndominant in-plane coupling. In [51], the effect of nonmag-\nnetic impurities and roughness on the finite temperature\nmagnetic properties of core-shell spherical nanoparticles\nwith anti-ferromagnetic interface coupling is investigated.\nIn [52], a Monte Carlo study of the magnetic properties\nof an Ising multilayer ferrimagnet is performed with two\nkinds of non-equivalent planes, one of which is site-diluted.\nIn [53], the magnetic properties of a diluted trilayered\ngraphene structure with non-equivalent planes with alter-\nnating spins 1 and 3 /2 are studied. In [54], the effect of site\ndilution on the magnetic properties of a mixed spin honey-\ncomb nano-lattice (with two sublattices, one with spin-3/2\nand the other with spin-5/2) is investigated using Monte\nCarlo simulations with the Metropolis algorithm.\nSo previous results in the literature indicate that the\nmagnetic properties of the thin magnetic systems with\nnonmagnetic impurities are interesting compared to their\npristine counterparts. Such studies provide better insight\ninto the physics behind real magnetic quasi-3D systems.\nAfter experimentally realizing artificial gauge potentials,\nit is now possible to engineer relevant spin interactions in\nquantum simulators [55] using ultracold bosonic quantum\ngases [56] in optical lattices on a triangular lattice geom-\netry [57]. This methodology is successfully implemented\nby a Penning trap apparatus with laser-cooled9Be+ions\n(∼300 spins), naturally forming a stable 2D triangular\nCoulomb crystal [58]. Here, each ion is a spin-1 /2 qubit\nand in [59], the authors used high-fidelity quantum control\nand a spin-dependent optical dipole force (ODF) to engi-\nneer a continuously tunable Ising-type spin-spin coupling.\nMotivated by these advances, an equilibrium Monte Carlo\nstudy on compensation has been performed on a spin-1 /2\ntrilayered triangular system [7]. But how the quenched\ndisorder e.g. site-dilution influences the compensation and\ncritical properties of the trilayered triangular Ising system\nis still unresolved. In this work, we would use a Monte\nCarlo approach to investigate a diluted trilayered Ising\nspin-1/2 ferrimagnet on a triangular lattice similar to the\none in [7]. We, in this article, would mainly focus to find\nanswers to the following:\n(a) How does the concentration of nonmagnetic impurities\nat the diluted planes affect the critical and compensation\nbehaviour?\n(b) How would the phase diagram change with the change\nin the concentration of impurity?\n2(c) Can we define any mathematical relationship between\nthe physical properties of the system and the concentra-\ntion of impurities?\nThe rest of the paper is organized as follows: the mag-\nnetic model is explained in Section 2. The strategy of\nsimulation for the trilayered system is presented in Sec-\ntion 3. The numerical results are presented and discussed\nin Section 4. The work is summarised in Section 5.\n2 Theoretical Model\nOur interest in this article is a ferrimagnetic trilayer, in\nwhich the spins are located at the sites of the triangular\ncrystalline lattice. The magnetic system is composed of in-\nequivalent parallel monolayers. In the ABA-type system,\nalternately stacked planes are populated by either the A\nor B type of theoretical atoms. Similarly in the AAB-type\nsystem, the top and mid-layer are populated by the A-\natoms and the bottom layer is populated by the B type\nof atoms. In both configurations, plane B, with the domi-\nnant in-plane coupling is unaffected but the A-planes are\nsite-diluted with the concentration of nonmagnetic atoms\nbeingρ. All magnetic atoms are assumed to have spin-\n1/2. The schematic view of the trilayer is presented in\nFigure 1.\nFigure 1: (Colour Online) Miniaturised versions (3 ×4×4)\nofsite-diluted : (A) ABA-type and (B) AAB-type triangu-\nlar trilayered Ising superlattices. Each of the sublattices\nof the ferrimagnetic systems are formed on triangular lat-\ntice. The actual simulation is carried out on a system with\nNsites= 3×100×100 .\nThe magnetic interactions are limited only between the\nnearest neighbours and are Ising-like. The nature of mag-\nnetic interactions are:(a) A-A →Ferromagnetic; (b) B-B\n→Ferromagnetic; (c) A-B →Anti-ferromagnetic. The\nHamiltonian for such a trilayered ferrimagnetic system, us-\ning nearest neighbour Ising mechanics [60], is (with all theSz’s arezcomponents of spin moments on lattice sites):\nHABA =−JAA[/summationdisplay\n<t,t′>(ξtSz\nt)(ξt′Sz\nt′)\n+/summationdisplay\n<b,b′>(ξbSz\nb)(ξb′Sz\nb′)]−JBB/summationdisplay\n<m,m′>Sz\nmSz\nm′\n−JAB[/summationdisplay\n<t,m>(ξtSz\nt)Sz\nm+/summationdisplay\n<m,b>Sz\nm(ξbSz\nb)] (1)\nHAAB =−JAA[/summationdisplay\n<t,t′>(ξtSz\nt)(ξt′Sz\nt′) +/summationdisplay\n<m,m′>(ξmSz\nm)(ξm′Sz\nm′)]\n−JBB/summationdisplay\n<b,b′>Sz\nbSz\nb′−JAA/summationdisplay\n<t,m>(ξtSz\nt)(ξmSz\nm)\n−JAB/summationdisplay\n<m,b>(ξmSz\nm)Sz\nb (2)\nwhereJAAis the in-plane coupling strength between\nnearest neighbours on the A-layers. On similar lines we\nhave, for the B-layer, JBB.JABis the inter-layer nearest\nneighbour coupling strengths between the A and B lay-\ners. According to the nature of the magnetic interactions:\nJAA>0 ,JBB>0 andJAB<0.< t,t′>,< m,m′>\nand< b,b′>are nearest neighbour summation indices\nfor the top, mid and bottom layers respectively. Simi-\nlarly< t,m > and< m,b > stand for summations over\nnearest-neighbor pairs in vertically adjacent layers. We\nwould take JBBto be the dominant coupling strength\nand use the ferromagnetic coupling ratio JAA/JBBand\nthe antiferromagnetic coupling ratio JAB/JBBto be the\ncontrolling parameters. The configurational averages of\nthe occupation probability of A-atoms, for the ABA vari-\nant is: ¯ξt=¯ξb= (1−ρ) and for the AAB variant is:\n¯ξt=¯ξm= (1−ρ), for a given concentration of nonmag-\nnetic atoms, ρ. There is no out-of-plane interaction be-\ntween the top and bottom layers. We will use periodic\nboundary conditions in-plane and open boundary condi-\ntions along the vertical.\n3 Simulation Protocol\nTo study the magneto-thermal behaviours of the two dis-\ntinct site-diluted Ising trilayered configurations of Figure\n1, we have used Monte Carlo simulations with Metropo-\nlis single-spin flip algorithm [61, 62]. Each of the three\nplanes has a linear dimension of 100 sites. So the total\nsites are Nsites = 3×100×100. From the discussions\nin Section B, we find the size of the lattice is statisti-\ncally reliable for our purpose. The zcomponents of spin\nprojections at the site, i, isSz\ni(Sz\ni=±1/2), only partic-\nipate in the Ising interactions. The system was initiated\nat a high-temperature paramagnetic phase, with a frac-\ntionρof randomly selected sites on the A-layers being\nsite-diluted. From the rest, half of the spin projections\nare inSz\ni= +1/2 and the other half is in Sz\ni=−1/2.\nThe B-layer is not diluted i.e. it has an equal population\nofSz\ni= +1/2 andSz\ni=−1/2. Now, a trial configura-\ntion is constructed by reorienting a randomly chosen single\nspin. This trial configuration is accepted with probability,\np=min{1,e−β∆E}; where ∆ Eis the energy difference be-\ntween the trial and the current configuration. Similar 3 L2\nrandom single-spin updates constitute one Monte Carlo\n3sweep (MCS) of the entire system and this one MCS is\nthe unit of time in this study.\nAt every temperature step, the system goes through\n105MCS. At any temperature step, the last configuration\nof the system at the just previous temperature acts as the\nstarting configuration. For equilibration, we shall discard\nthe first 2 ×104MCS. From the next 8 ×104MCS, we\nwould collect data after every 800 MCS (to minimise the\nauto-correlation effect) for 100 uncorrelated configurations\nfor relevant physical quantities. The temperatures of the\nsystems are measured in the units of JBB/kB. So the tem-\nperatures reported in this article are effectively dimension-\nless. From now onwards, temperature and dimensionless\ntemperature would be used and understood interchange-\nably. Periodic boundary conditions are used in-plane (i.e.\nalong the x and y axes) and open boundary conditions are\nused along the vertical (i.e. z axis). The in-plane coupling\nstrength of the B-layer, JBBis considered to be the most\ndominant one and is set to unity (and also the unit of en-\nergy scale). Other two coupling strengths, namely, JAA\nandJABare measured with respect to JBBi.e. seven dis-\ntinct values of JAA/JBBand|JAB/JBB|are considered,\nfrom 0.04 to 1.0 with an interval of 0 .16. For each distinct\ncombination of coupling strengths, the time/ensemble av-\nerages of the following quantities are calculated at each\nof the temperature steps ( T), in the following manner for\nboth the configurations ( ABA,AAB ) identically:\n(1) Sublattice magnetizations Mqi(T) for the indi-\nvidual layers, are identically calculated after equilibration,\nat thei-th MCS, by:\nMqi(T) =1\nL2L/summationdisplay\nx,y=1/parenleftbig\nSz\nqi(T)/parenrightbig\nxy(3)\nThe sum extends over all sites in each monolayer with x\nandybeing the co-ordinates of a spin on the q-th sublayer\nand runs from 1 to L(= 100, in this study). Next, the\ntime (equivalently, ensemble) average (in angular braces)\nis obtained from the Nuncorrelated configurations by:\n/an}bracketle{tMq(T)/an}bracketri}ht=1\nNN/summationdisplay\ni=1Mqi(T) (4)\nIn the above equations, qis to be replaced by t,mor,b\nfor top, mid or bottom layers respectively.\n(2) Time averaged total magnetisation ,Mtot(T),\nof the trilayer serves as the order parameter and at tem-\nperature, T, it is defined as:\nMtot(T) =1\n3(/an}bracketle{tMt(T)/an}bracketri}ht+/an}bracketle{tMm(T)/an}bracketri}ht+/an}bracketle{tMb(T)/an}bracketri}ht) (5)\n(3) Fluctuation of the order parameter , ∆M(T),\nat temperature Tis obtained after equilibration as:\n∆M(T) =/radicaltp/radicalvertex/radicalvertex/radicalbt1\nNN/summationdisplay\ni=1[Mtot,i(T)−Mtot(T)]2(6)\nwhereMtot,i(T) is the total magnetisation of the system\nafter the i-th uncorrelated MCS.(4) Canonical average of associative energy per\nsite,E(T), at temperature Tis obtained after equilibra-\ntion as:\nE(T) =/an}bracketle{tH(T)/an}bracketri}ht\n3L2(7)\nfrom the Nuncorrelated MCS. Equations 1 and 2 are used\nhere.\n(5) The fluctuation of the associative energy per\nsite, ∆E(T), at temperature Tis obtained after equilibra-\ntion as:\n∆E(T) =/radicaltp/radicalvertex/radicalvertex/radicalbt1\nNN/summationdisplay\ni=1[Ei(T)−E(T)]2(8)\nwhereEi(T) is the associative energy per lattice site of\nthe system after the i-th uncorrelated MCS.\n(6) The Binder cumulant ,UL(T) for a given Lis\ndefined as [62]:\nUL(T) = 1−/an}bracketle{tM4\ntot(T)/an}bracketri}htL\n3/an}bracketle{tM2\ntot(T)/an}bracketri}ht2\nL(9)\nat dimensionless temperature T, after equilibration. We\nshall use the position of minima ofdUL\ndTabout the pseudo-\ncritical point (obtained from the plot of ∆ EversusT) to\narrive at the value of critical temperature [63]. Only a\nsingle system size e.g. L= 100 in this study is used for\nthis purpose.\nAt each temperature step, averaging is performed over\n10 different sample realizations (same macroscopic but\ndifferent microscopic arrangements) for every mentioned\nquantity. The estimate of error is obtained via Jackknife\nmethod [64] and are often smaller than the point markers\naway from the critical point.\n4 Results\n4.1 Thermodynamic response\nIt would be useful to first examine the magnetic response\nof the diluted configurations. In Figure 2, a comparison\nhas been made between the pure ( ρ= 0) case (Courtesy\n[7]) and with ρ= 0.20 for both the configurations, ABA\nand AAB. After site-dilution, the observations are: (a) a\nprominent shift in the compensation temperatures and rel-\natively small (e.g. in Figure 2(a) with ABA :JAA/JBB=\n0.68 andJAB/JBB=−0.04 and in Figure 2(b) with ABA :\nJAB/JBB=−0.52 andJAA/JBB= 0.04) or undetectable\n(within the errorbars) shift in the critical temperatures\n(see Figures 10(a) and (b)), keeping the magnetization\nprofile same; (b) visible shift in critical temperatures lead-\ning to the creation of a compensation point, resulting in a\ndifferent magnetization profile ((e.g. in Figure 2(a) with\nABA :JAA/JBB= 0.68 andJAB/JBB=−0.68 and in Fig-\nure 2(b) with ABA :JAB/JBB=−0.52 andJAA/JBB=\n0.68)). A similar observation is valid for AAB configura-\ntion as well [Figures 2(c) and (d) and Figures 10(c) and\n(d)].\nFor both ABA and AAB types, to probe further with\nfixed combination of coupling strengths, the plots for the\n4-0.2-0.1 0 0.1Mtot (µB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00ABA\nJAA/JBB=0.68\nρ=0.00\n-0.2-0.1 0 0.1\n 0  0.5  1  1.5  2  2.5\nT (JBB/kB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00ABA\nJAA/JBB=0.68\nρ=0.20 (a)-0.2-0.1 0 0.1Mtot (µB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00ABA\nJAB/JBB=-0.52\nρ=0.00\n-0.2-0.1 0 0.1\n 0  0.5  1  1.5  2  2.5\nT (JBB/kB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00ABA\nJAB/JBB=-0.52\nρ=0.20 (b)\n-0.2-0.1 0 0.1Mtot (µB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00AAB\nJAA/JBB=0.68\nρ=0.00\n-0.2-0.1 0 0.1\n 0  0.5  1  1.5  2  2.5\nT (JBB/kB)JAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00AAB\nJAA/JBB=0.68\nρ=0.20 (c)-0.2-0.1 0 0.1Mtot (µB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00AAB\nJAB/JBB=-0.52\nρ=0.00\n-0.2-0.1 0 0.1\n 0  0.5  1  1.5  2  2.5\nT (JBB/kB)JAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00AAB\nJAB/JBB=-0.52\nρ=0.20 (d)\nFigure 2: (Colour Online) Plots of Order parameter, Mtotversus dimensionless temperature, T, for the pure and\nwith concentration of nonmagnetic impurities, ρ= 0.20, for: (a) ABA: JAA/JBB= 0.68 and varying JAB/JBB;\n(b) ABA: JAB/JBB=−0.52 and varying JAA/JBB; (c) AAB: JAA/JBB= 0.68 and varying JAB/JBB; (d) AAB:\nJAB/JBB=−0.52 and varying JAA/JBB. The impurity-driven creation of compensation points and shift of c om-\npensation temperatures towards the lower temperature end are visible in these cases.\ntotal magnetization Mtotas a function of temperature, T,\nare shown in Figure 3 with the concentration of impuri-\nties,ρ, acting as the parameter. We observe on increas-\ningρ, various types of magnetization profiles come into\nexistence. According to the classification schemes of mag-\nnetization profiles, due to Neel [1] and Strecka [65]: (a)\nSmall-to-undetectable shift in critical temperatures along-\nwith larger shifts in compensation temperature leads to\nthe same magnetization profile, N(Ref. Figure 3(a),(c));\n(b) visible shift in critical temperatures leading to the\ncreation of compensation point, changes the magnetiza-\ntion profiles (Ref. Figure 3(b),(d)) from QtoRtoN.\nThe most interesting facet is the creation of compensation\npoints in both configurations by increasing the concentra-\ntion of nonmagnetic impurities for specific combinations\nof coupling strengths.\nNow, for two specific combinations of coupling strengths,\nwe plot both the fluctuations, of the order parameter and\nthe associative energy per site, versus the dimensionless\ntemperature in Figures 4 and 5 respectively. The cusp-likedivergence is always associated with the critical point for\nall the impurity concentrations for both fluctuations. Also:\n(a) For the coupling combination which had compensation\neven in the pure case, as we approach the compensation\npoint we find the hump-like smeared peaks about it. With\nthe increase in the impurity concentration, the dilution of\nbonds leads to the gradual flattening of both functions.\nBut looking at the position of the local maximum, we un-\nderstand the ordering temperature of the A-layers gradu-\nally shift towards the lower end with the increase in the im-\npurity concentration. The position of the critical point is\nalmost unaffected (Refer to Figures 4(a),(c) and 5(a),(c)).\n(b) For the higher coupling combination, we could visu-\nally detect the maxima about the critical point shifting\ntowards the left until a compensation point emerges. Af-\nter that the behaviour is almost similar to (a). But about\nthe compensation point, the hump-like appearance almost\nflattens out (Refer to Figures 4(b),(d) and 5(b),(d)).\n5-0.2-0.1 0 0.1\n 0 0.5  1 1.5  2 2.5Mtot (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45ABA\nType: N(a)\nJAA/JBB=0.20\nJAB/JBB=-0.20\n-0.2-0.1 0 0.1\n 0 0.5  1 1.5  2 2.5Mtot (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45ABA\nTypes: Q,R,N(b)\nJAA/JBB=0.84\nJAB/JBB=-0.84\n-0.2-0.1 0 0.1\n 0 0.5  1 1.5  2 2.5Mtot (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45\nAAB\nType: N(c)\nJAA/JBB=0.20\nJAB/JBB=-0.20\n-0.2-0.1 0 0.1\n 0 0.5  1 1.5  2 2.5Mtot (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45AABTypes: Q,R,N(d)\nJAA/JBB=0.84\nJAB/JBB=-0.84\nFigure 3: (Colour Online) Plots of Order parameter, Mtotversus dimensionless temperature, T, with variable\nconcentration of nonmagnetic impurities, ρ, for: (a) ABA: JAA/JBB= 0.20 andJAB/JBB=−0.20; (b) ABA:\nJAA/JBB= 0.84 andJAB/JBB=−0.84; (c) AAB: JAA/JBB= 0.20 andJAB/JBB=−0.20; (d) AAB: JAA/JBB=\n0.84 andJAB/JBB=−0.84 . Shift of the Compensation temperatures towards the lower te mperature end are visible\nwith an increase in site-dilution. The impurity-driven creation of comp ensation point is also witnessed for the strongest\ncombination of coupling strengths.\n4.2 Effect of impurity\nAfter discussing the general results of site-dilution, we\nwould quantitatively discuss the effects of spin-0 impuri-\nties in this subsection. Various compensation behaviours\nand magnetization profiles have technological importance\n[66,67]. So it is important to find out the underlying sys-\ntematics due to the site-dilution.\nFigure 6 provides us with a general overview of what\nhappens when we introduce the quenched disorder in the\nform of spin-0 impurities. For both configurations, as we\ngo on increasing the concentration of spin-0 atoms, non-\ncompensating magnetization profiles in the higher cou-\npling combination regimes start to show compensation.\nThis feature is pretty robust as keeping either of JAA/JBB\nandJAB/JBBfixed and changing the other leads to the\nsame scenario. For example, in Figure 6(a), the ABA\nconfiguration with JAA/JBB= 0.68 had compensating\nmagnetization till about JAB/JBB=−0.36 . But as we\nreachedρ= 0.15, we witnessed compensating behaviours\nfor all the antiferromagnetic coupling ratios upto JAB/JBB=\n−1.00. For all the other cases we witness similar kinds of\nbehaviours, but with a different critical value of dilution .4.2.1 Phase Diagrams\nSo, the introduction of spin-0 impurities in the A-layers\nhas a very distinct effect on the creation of compensation\npoints. In the pure case, where compensation is absent\nfor some parameter values, we find compensation happen-\ning in a certain diluted configuration for the same combi-\nnation of parameters. What it means is the area of no-\ncompensation shrinks as we increase the concentration of\nnonmagnetic impurities in the way described above. This\nis an important observation as after a critical concentra-\ntion of spin-0 atoms, we witness all the combinations of\ncoupling strengths (within the range of this study) to have\ncompensation present.\nThe procedure of detecting the critical temperature for\na specific combination of coupling strengths is outlined in\nFigures 7(a) and (b). From the initial temperature series\nof Binder’s magnetization cumulant, UL, for the system\nsize, 3L2= 3×100×100, we would plot the tempera-\nture derivative of ULi.e.dUL\ndTversus the dimensionless\ntemperature. At the pseudo-critical point,dUL\ndTattains\nminima, when the temperature is increased as in Figure\n7(b) [63]. Next up, to draw the phase diagram in parame-\nter space, we need to detect the combinations of coupling\n6 0 0.004 0.008 0.012 0.016\n 0 0.5  1 1.5  2 2.5∆M (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(a)\nABA\nJAA/JBB=0.20\nJAB/JBB=-0.20\n 0 0.002 0.004 0.006 0.008\n 0 0.5  1 1.5  2 2.5∆M (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(b)\nABA\nJAA/JBB=0.84\nJAB/JBB=-0.84\n 0 0.004 0.008 0.012 0.016\n 0 0.5  1 1.5  2 2.5∆M (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(c)\nAAB\nJAA/JBB=0.20\nJAB/JBB=-0.20\n 0 0.002 0.004 0.006 0.008 0.01\n 0 0.5  1 1.5  2 2.5∆M (µB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(d)\nAAB\nJAA/JBB=0.84\nJAB/JBB=-0.84\nFigure 4: (Colour Online) Plots of fluctuation of the order paramete r, ∆M, versus dimensionless temperature, T,\nwith variable concentration of nonmagnetic impurities, ρ, for: (a) ABA: JAA/JBB= 0.20 andJAB/JBB=−0.20;\n(b) ABA: JAA/JBB= 0.84 andJAB/JBB=−0.84; (c) AAB: JAA/JBB= 0.20 andJAB/JBB=−0.20; (d) AAB:\nJAA/JBB= 0.84 andJAB/JBB=−0.84 .\nstrengths across the entire parameter space. Two exam-\nples are shown in Figures 7(c) and (d). After keeping ei-\nther the ferromagnetic or anti-ferromagnetic coupling ra-\ntio fixed, we plot the compensation temperature, Tcomp\nand critical temperature, Tcritas a function of the other\nvariable interaction strength. At the lower end, for fixed\nferromagnetic strength, we would linearly extrapolate the\ncurve for Tcomp to find out the intersection with the curve\nforTcrit. The x-coordinate of the intersecting point is the\nminimum value of antiferromagnetic strength where com-\npensation ceases to exist [Refer to Figure 7(c)]. For fixed\nanti-ferromagnetic strength, the intersection finding is to\nbe done at the higher end to find out the maximum value\nof ferromagnetic strength [Refer to Figure 7(d)]. The up-\nper bounds of linear interpolation procedure provide the\nerrors associated with the intersections [68].\nIn Figure 8, we plot the phase diagrams in the pa-\nrameter space ( JAB/JBB×JAA/JBB) for both the ABA\nand AAB configurations. The gradual diminishing of the\nA-marked areas (absence of compensation) as we increase\nthe concentration of dilution leads us to mathematically\nfit theA(ρ)/Atotversusρdata in Figure 9. To find out the\nvalues of A(ρ), for different ρ, the integration is performed\nby the Monte Carlo method [69]. In a similar situation,\nthis method is previously applied in [8,9] where technical\ndetails can be revisited.\nFor ABA, the linear fit is adequate with the fittingpolynomial:\nA(ρ)\nAtot=a1ρ+b1 (10)\nwhere,a1=−1.266±0.025 and b1= 0.319±0.004 . The\ncritical concentration of ABA configuration is: ρc,ABA =\n0.252±0.002 .\nFor AAB, the quadratic fit is adequate with the fitting\npolynomial:\nA(ρ)\nAtot=a2ρ2+b2ρ+c2 (11)\nwhere,a2=−1.381±0.130,b2=−0.990±0.034 and\nc2= 0.338±0.002. The critical concentration of AAB\nconfiguration is: ρc,AAB = 0.253±0.009 .\n4.2.2 Systematics\nIn this section, we would try to theoretically character-\nize the diluted ferrimagnetic systems. In Figure 10, we\nwitness the variation of critical and compensation tem-\nperatures with the concentration of spin-0 atoms for a few\ncombinations of coupling strengths. For both the con-\nfigurations, ABA and AAB, we observe the compensation\ntemperatures to vary in a nonlinear fashion as we increase\nsite-dilution. Alongside, the change in critical tempera-\ntures is limited to within ∼23% in ABA and ∼26% in\nAAB till 45% dilution. These features make such mag-\nnetic heterostructures very interesting. Next, we would\n7 0 0.001 0.002 0.003 0.004 0.005\n 0 0.5  1 1.5  2 2.5∆E (JBB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(a)\nABA\nJAA/JBB=0.20\nJAB/JBB=-0.20\n 0 0.002 0.004 0.006 0.008 0.01\n 0 0.5  1 1.5  2 2.5∆E (JBB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(b)\nABA\nJAA/JBB=0.84\nJAB/JBB=-0.84\n 0 0.001 0.002 0.003 0.004 0.005\n 0 0.5  1 1.5  2 2.5∆E (JBB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(c)\nAAB\nJAA/JBB=0.20\nJAB/JBB=-0.20\n 0 0.002 0.004 0.006 0.008 0.01\n 0 0.5  1 1.5  2 2.5∆E (JBB)\nT (JBB/kB)ρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45(d)\nAAB\nJAA/JBB=0.84\nJAB/JBB=-0.84\nFigure 5: (Colour Online) Plots of fluctuation of the associative ener gy per site, ∆ E, versus dimensionless temperature,\nT, with variable concentration of nonmagnetic impurities, ρ, for: (a) ABA: JAA/JBB= 0.20 andJAB/JBB=−0.20;\n(b) ABA: JAA/JBB= 0.84 andJAB/JBB=−0.84; (c) AAB: JAA/JBB= 0.20 andJAB/JBB=−0.20; (d) AAB:\nJAA/JBB= 0.84 andJAB/JBB=−0.84 .\nlike to model the variation of the compensation tempera-\ntures as the N-type magnetization profiles (having a com-\npensation point) find many important technological appli-\ncations [66,67].\nNow to model the variation of compensation temper-\natures with concentration of nonmagnetic impurities, a\nlinear polynomial may be employed. The mathematical\nforms follow:\nFor ABA:\nTcomp,ABA (ρ,JAA/JBB,JAB/JBB) =p1ρ+r1(12)\nFor AAB:\nTcomp,AAB (ρ,JAA/JBB,JAB/JBB) =p2ρ+r2(13)\nThe explicit dependence of the coupling strengths is con-\ntained in the coefficients at the right-hand side of the set\nof equations. That means, pi≡pi(JAA/JBB,JAB/JBB)\n;ri≡ri(JAA/JBB,JAB/JBB) withi∈ {1,2}. These\nparticular choices of fitting formulae provide us with re-\nduced chisquare ,χ2/nDOF∼1 and the asymptotic errors\nin the fitting parameters are mostly confined below 10%\n. So, this choice of mathematical forms is reliable for fur-\nther calculations. A few selective examples are provided\nin Figure 11 for both configurations. A discussion on the\nfitting parameters is necessary [70] and in Appendix A,\nwe show the behaviours of the fitting parameters for both\nconfigurations.Now, we would find out the threshold value of the con-\ncentration of nonmagnetic impurities for any given combi-\nnation of coupling strengths, above which we would wit-\nness compensation. First, in Figure 12, we would witness\nhow the threshold impurity concentrations vary with two\nHamiltonian parameters, JAA/JBBandJAB/JBB. It is\nevident that as the magnitude of either of the Coupling\nratios increases, the values of the threshold concentration\nof nonmagnetic impurities, ρT, increase. This motivates\nus to find out how ρTbehaves as a function of JAA/JBB\nandJAB/JBB. Essentially, for each configuration, we will\nthen have two 2D plots.\nFor the ABA configuration, from Figure 13, we see a\nsystematic variation of ρT. A mathematical fit now can\nbe attempted:\nΨ1(JAA/JBB,JAB/JBB) =a1/radicalbig\nJAA/JBB+c1(14)\nfor fixed JAB/JBB[Refer to Figure 13(a)]. And,\nΨ2(JAA/JBB,JAB/JBB) =a2/radicalbig\n|JAB/JBB|+c2(15)\nfor fixed JAA/JBB[Refer to Figure 13(b)]. Along with,\nthe coefficients are functions of coupling ratios e.g. a1≡\na1(JAB/JBB),c1≡c1(JAB/JBB) anda2≡a2(JAA/JBB),\nc2≡c2(JAA/JBB) .\nFor the AAB configuration, from Figure 14, we see a\nsimilar variation of ρT. Thus similar mathematical forms\n8 0 0.25 0.5 0.75 1\n-1 -0.5  0  0.5Tcomp (JBB/kB)\nJAB/JBBρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45\nABA(a)\nJAA/JBB=0.68\n 0 0.25 0.5 0.75 1\n 0  0.5  1  1.5Tcomp (JBB/kB)\nJAA/JBBρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45\nABA(b)\nJAB/JBB=-0.84\n 0 0.25 0.5 0.75 1\n-1 -0.5  0  0.5Tcomp (JBB/kB)\nJAB/JBBρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45\nAAB(c)\nJAA/JBB=0.68\n 0 0.25 0.5 0.75 1\n 0  0.5  1  1.5Tcomp (JBB/kB)\nJAA/JBBρ=0.00\nρ=0.05\nρ=0.10\nρ=0.15\nρ=0.20\nρ=0.25\nρ=0.30\nρ=0.35\nρ=0.40\nρ=0.45AAB(d)\nJAB/JBB=-0.84\nFigure 6: (Colour Online) Plots for, ABA stacking of (a) compensatio n temperature, Tcomp, versusJAB/JBB, (b)\ncompensation temperature, Tcomp, versus JAA/JBB, and AAB stacking of (c) compensation temperature, Tcomp,\nversusJAB/JBB, (d) compensation temperature, Tcomp, versusJAA/JBB, with variable concentration of nonmagnetic\nimpurities, ρ.\ncan be proposed:\nΦ1(JAA/JBB,JAB/JBB) =a3/radicalbig\nJAA/JBB+c3(16)\nfor fixed JAB/JBB[Refer to Figure 14(a)]. And,\nΦ2(JAA/JBB,JAB/JBB) =a4/radicalbig\n|JAB/JBB|+c4(17)\nfor fixed JAA/JBB[Refer to Figure 14(b)]. Along with,\nthe coefficients are functions of coupling ratios e.g. a3≡\na3(JAB/JBB),c3≡c3(JAB/JBB) anda4≡a4(JAA/JBB),\nc4≡c4(JAA/JBB) .\nThe value of threshold concentration of spin-0 impu-\nrities at any of the intersecting points on the Hamilto-\nnian parameter space can be obtained through the geo-\nmetric mean of the two perpendicular approaches via the\nfunctions presented in Equations 14 to 17: one along the\nJAA/JBBaxis and the other along the JAB/JBBaxis.\n5 Summary\nIn this work, we have investigated the role played by site\ndilution in creating compensation points in a ferrimag-\nnetic spin-1/2 Ising trilayer on triangular monolayers com-\npared to its pristine counterpart, using Metropolis Monte\nCarlo simulation. The ABA and AAB configurations have\nnon-equivalent planes in the sense that only A-layers arerandomly site-diluted but the B-layer, with dominant in-\nplane coupling , is pristine. While discussing the magneto-\nthermal behaviour, we witnessed that for a fixed concen-\ntration of spin-0 impurities, the compensation point shifts\ntowards the critical point and ultimately merges with it, as\nwe increase either of the coupling ratios. For a fixed combi-\nnation of coupling ratios, an increase in the concentration\nof the diluted sites leads to the shift of both, compensation\nand critical temperatures towards the lower temperature\nends. Thermal variation of fluctuations of magnetisation\nand associative energy also supports the previous observa-\ntion. The variation of Compensation temperature with the\nconcentration of nonmagnetic atoms can be modelled by a\nlinear variation. Across the entire range of concentration\nof nonmagnetic impurities, we witness continuous phase\ntransitions (second order phase transitions) across the crit-\nical points for all the combinations of coupling strengths.\nStill, the most interesting and important observation is\nthe impurity-driven creation of compensation points for\ncertain combinations of coupling strengths in site-diluted\nsystems. The threshold concentration of nonmagnetic im-\npurities, ρT, above which a compensation point is created\nfor a certain combination of coupling ratios in a diluted\ntrilayered system, varies in a parabolic manner with the\ncoupling ratios for both the ABA and AAB configurations.\nThus the fitted mathematical formulae characterise the\nsystem and provide us with a complete description of the\n9-0.25 0 0.25 0.5 0.75 1\n 0.7  0.8  0.9  1  1.1UL\nT (JBB/kB)UL (L=100)(a) ABA\nJAA/JBB=0.52\nJAB/JBB=-0.04\nρ=0.00\n 0.5 0.75 1 1.25\n−1 −0.75 −0.5 −0.25  0Tcomp, Tcrit\nJAB/JBBTcompTcrit(c)\nABA\nρ=0.00JAA/JBB=0.52\n 0.5 0.75 1 1.25\n−1 −0.75 −0.5 −0.25  0Tcomp, Tcrit\nJAB/JBBTcompTcrit(c)\nABA\nρ=0.00JAA/JBB=0.52\nJAB/JBB=−0.826 ± 0.006\n 0.5 0.75 1 1.25\n−1 −0.75 −0.5 −0.25  0Tcomp, Tcrit\nJAB/JBBTcompTcrit(c)\nABA\nρ=0.00JAA/JBB=0.52\nJAB/JBB=−0.826 ± 0.006\n 0.5 0.75 1 1.25\n−1 −0.75 −0.5 −0.25  0Tcomp, Tcrit\nJAB/JBBTcompTcrit(c)\nABA\nρ=0.00JAA/JBB=0.52\nJAB/JBB=−0.826 ± 0.006\n-60-40-20 0 20\n 0.7  0.8  0.9  1  1.1dUL/dT\nT (JBB/kB)dUL/dT (L=100)\n(b) ABA\nJAA/JBB=0.52JAB/JBB=-0.04\nρ=0.00\n 0.25 0.5 0.75 1 1.25\n 0  0.25  0.5  0.75  1Tcomp, Tcrit\nJAA/JBBTcompTcrit (d)\nABA\nρ=0.00JAB/JBB=−0.52\n 0.25 0.5 0.75 1 1.25\n 0  0.25  0.5  0.75  1Tcomp, Tcrit\nJAA/JBBTcompTcrit (d)\nABA\nρ=0.00JAB/JBB=−0.52\nJAA/JBB= 0.667\n± 0.008\n 0.25 0.5 0.75 1 1.25\n 0  0.25  0.5  0.75  1Tcomp, Tcrit\nJAA/JBBTcompTcrit (d)\nABA\nρ=0.00JAB/JBB=−0.52\nJAA/JBB= 0.667\n± 0.008\n 0.25 0.5 0.75 1 1.25\n 0  0.25  0.5  0.75  1Tcomp, Tcrit\nJAA/JBBTcompTcrit (d)\nABA\nρ=0.00JAB/JBB=−0.52\nJAA/JBB= 0.667\n± 0.008\nFigure 7: (Colour Online) For a pure ABA configuration: (a) variation of Binder’s Magnetization Cumulant and (b)\nvariation of slope of Binder’s Magnetization Cumulant versus dimensio nless temperature, with JAA/JBB= 0.52 and\nJAB/JBB=−0.04 ;and, dimensionless critical temperature Tcritand compensation temperature Tcomp as functions\nof (c)JAB/JBBwith fixed JAA/JBB= 0.52 (d)JAA/JBBwith fixed JAB/JBB=−0.52. The intersections of\nvertical dashed lines with the horizontal axis, mark the value of eith erJAB/JBBorJAA/JBB, below/above which no\ncompensation is detected. Where the errorbars are not visible, th ey are smaller than the point markers.\n-1.00-0.75-0.50-0.250.00\n0.00 0.25 0.50 0.75 1.00JAB/JBB\nJAA/JBBρ=0.25\nρ=0.20\nρ=0.15\nρ=0.10\nρ=0.05\nρ=0.00ABA\nP: Compensation is PRESENT\nA: Compensation is ABSENT\nP\nA-1.00-0.75-0.50-0.250.00\n0.00 0.25 0.50 0.75 1.00JAB/JBB\nJAA/JBBρ=0.25\nρ=0.20\nρ=0.15\nρ=0.10\nρ=0.05\nρ=0.00AAB\nP: Compensation is PRESENT\nA: Compensation is ABSENT\nP\nA\nFigure 8: (Colour Online) Phase diagrams in the ( JAB/JBB×JAA/JBB) plane for: (a) ABA and (b) AAB configu-\nrations with concentration of nonmagnetic atoms acting as a param eter. The impurity-driven changes in the relative\nareas in the Phase diagram are clearly visible. The blue ends of the cur ves are obtained via linear extrapolation.\ndiluted trilayered magnetic systems. These results may\ncontribute to the prospective research of diluted magnetic\nlayered materials with non-equivalent planes with several\ndifferent sublattice geometries.Acknowledgements\nThe author gratefully acknowledges financial assistance\nfrom the University Grants Commission, India in the form\n10 0 0.1 0.2 0.3 0.4\n 0 0.05 0.1 0.15 0.2 0.25 0.3A(ρ)/Atot\nρMC data\nLinear fit(a)\nABA\nf1(ρ)=a1ρ + b1 \na1= −1.266 ± 0.025 \nb1= 0.319 ± 0.004 ρc, ABA\n= 0.252 ± 0.002 \n 0 0.1 0.2 0.3 0.4\n 0 0.05 0.1 0.15 0.2 0.25 0.3A(ρ)/Atot\nρMC data\nQuadratic fit(b)\nAAB\nf2(ρ)=a2ρ2 + b2ρ + c2\na2= −1.381 ± 0.130 \nb2= −0.990 ± 0.034 \nc2= 0.338 ± 0.002 ρc, AAB\n= 0.253 ± 0.009 \nFigure 9: (Colour Online) Plots of the fractional area of the compen sating region in the ( JAB/JBB×JAA/JBB)\nplane,A(ρ)/Atotversus concentration of the nonmagnetic impurities, ρfor the: (a) ABA and (b) AAB configurations\nwith a faithful fitting function.\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 ABA (a)\nJAA/JBB=0.52\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 ABA (a)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 ABA (a)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 ABA (a)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 ABA (b)\nJAB/JBB=-0.52\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 ABA (b)\nJAB/JBB=-0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 ABA (b)\nJAB/JBB=-0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 ABA (b)\nJAB/JBB=-0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 AAB (c)\nJAA/JBB=0.52\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 AAB (c)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 AAB (c)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAB/JBB=-0.04\nJAB/JBB=-0.52\nJAB/JBB=-1.00 AAB (c)\nJAA/JBB=0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 AAB (d)\nJAB/JBB=-0.52\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 AAB (d)\nJAB/JBB=-0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 AAB (d)\nJAB/JBB=-0.52TcritTcomp\n 0 0.25 0.5 0.75 1 1.25 1.5\n 0  0.15  0.3  0.45Tcomp, Tcrit (JBB/kB)\nρJAA/JBB=0.04\nJAA/JBB=0.52\nJAA/JBB=1.00 AAB (d)\nJAB/JBB=-0.52TcritTcomp\nFigure 10: (Colour Online) Variation of Critical and Compensation tem peratures with an increase in the concentration\nof nonmagnetic impurities for a few representative cases: for an A BA configuration in (a) and (b) and for an AAB\nconfiguration in (c) and (d). Where the errorbars are not visible, t hey are smaller than the point markers.\nof a Research fellowship and extends his thanks to Dr.\nDebabrata Ghorai and Dr. Tamaghna Maitra for provid-\ning the computational facilities. Insightful comments and\nsuggestions by the anonymous referees are also gratefully\nacknowledged.References\n1. N´ eel M. L., Ann. de Phys. 12, 137 (1948).\n2. Cullity B. D. and Graham C. D., Introduction to\nMagnetic Materials, 2nd edn. (Wiley, New York,\n2008) .\n11 0 0.25 0.5 0.75 1\n 0  0.15  0.3  0.45Tcomp\nρJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(a) ABA\nJAA/JBB=0.68\nF(ρ)= p1ρ + r1 0 0.25 0.5 0.75 1\n 0  0.15  0.3  0.45Tcomp\nρJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00(b) ABA\nJAB/JBB=-0.68\nF(ρ)= p1ρ + r1\n 0 0.25 0.5 0.75 1\n 0  0.15  0.3  0.45Tcomp\nρJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(c) AAB\nJAA/JBB=0.68\nG(ρ)= p2ρ + r2 0 0.25 0.5 0.75 1\n 0  0.15  0.3  0.45Tcomp\nρJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00(d) AAB\nJAB/JBB=-0.68\nG(ρ)= p2ρ + r2\nFigure 11: (Colour Online) Fitting of Compensation temperatures wit h the increase in the concentration of nonmag-\nnetic impurities for a few representative cases: for an ABA configu ration in (a) and (b) and for an AAB configuration\nin (c) and (d). Where the errorbars are not visible, they are smaller than the point markers.\n(a)\n (b)\nFigure 12: (Colour Online) Plots of threshold concentration of spin- 0 impurities versus Hamiltonian parameters,\nJAA/JBBandJAB/JBB: for an ABA configuration in (a) and for an AAB configuration in (b) .\n3. Diaz I. J. L. and Branco N. S., Phys. B: Condens.\nMatter529, 73 (2018) .\n4. Diaz I. J. L. and Branco N. S., Phys. A: Stat. Mech.\nAppl.540, 123014 (2020).\n5. Chandra S. and Acharyya M., AIP Conf. Proc.2220 , 130037 (2020) .\n6. Chandra S., Eur. Phys. J. B 94, 13 (2021) .\n7. Chandra S., J. Phys. Chem. Solids 156, 110165\n(2021) .\n8. Chandra S., Phys. Rev. E 104, 064126 (2021) .\n12(a) 0 0.1 0.2 0.3 0.4\n 0  0.25  0.5  0.75  1ρT\nJAA/JBBJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(a) ABA\nρT = a1[JAA/JBB]1/2 + c1\n(b) 0 0.1 0.2 0.3 0.4\n-1 -0.75 -0.5 -0.25  0ρT\nJAB/JBBJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(b) ABA\nρT = a2|JAB/JBB|1/2 + c2\nFigure 13: (Colour Online) Plots of threshold concentration of spin- 0 impurities versus Hamiltonian parameters: (a)\nρTversusJAA/JBBand (b)ρTversusJAB/JBB; for an ABA configuration.\n(a) 0 0.1 0.2 0.3 0.4\n 0  0.25  0.5  0.75  1ρT\nJAA/JBBJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(c) AAB\nρT = a3[JAA/JBB]1/2 + c3\n(b) 0 0.1 0.2 0.3 0.4\n-1 -0.75 -0.5 -0.25  0ρT\nJAB/JBBJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(d) AAB\nρT = a4|JAB/JBB|1/2 + c4\nFigure 14: (Colour Online) Plots of threshold concentration of spin- 0 impurities versus Hamiltonian parameters: (a)\nρTversusJAA/JBBand (b)ρTversusJAB/JBB; for an AAB configuration.\n9. 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After finding a reliable\nfitting function, if the set of parameters turns out to be\nwell-behaved, the description then becomes indeed a good\none.\nFor the ABA type of trilayered stacking, in Figure 15,\nwe see the parameters p1andr1can be modelled by sim-\nple functions: (a) exponential and (b) linear. For the AAB\ntype of trilayered stacking, in Figure 16, the parameters\np2andr2can be similarly modelled by: (a) exponen-\ntial and (b) quadratic functions. All these choices are\nnot unique [70] but they provide us with a very simple\nyet complete way into the behaviours of the compensation\ntemperatures for both configurations. The fitting formu-\nlae are all mentioned within the figures.\nB On Compensation point\nNow the pertinent question for similar types of studies is\nwhether the compensation temperature is dependent on\nsystem size. We remind ourselves that the compensation\npoint is that temperature below the critical point where\nthe bulk magnetization of the system becomes zero i.e.\nTcomp< Tcritand Compensation is not related to criti-\ncality [3,4]. For the pristine counterpart of the system in\nthis study it was shown in [7] that for L≥70, the com-\npensation temperature becomes independent of the size of\nthe system. To recall how to detect the value of Tcomp in\nappropriate cases, we may look into Figure 2. The inter-\nsection of Mtot(T) with the x-axis (temperature axis/zero\nmagnetisation line), below the critical point, is the Com-\npensation point. While determining the point of intersec-\ntion, we would approximate the part of the magnetization\ncurve between the two neighbouring points on either side\nof the x-axis by a straight line [6,7].\nIn this study, we have varied the fraction of diluted\nsites from 0 (pure sample) to 0 .45 . So one would natu-\nrally ask how the compensation point behaves as a func-\ntion of linear system size, L, under the influence of growing\nnonmagnetic impurities. Because of the limited computa-\ntional resources available, only a few random combinations\nof coupling strengths were tested in this regard. For both\nthe ABA and AAB configurations, we present here the\nresults for JAA/JBB= 0.52 andJAB/JBB=−0.52 with\nfour selective ratios of site-dilution ρ={0.00,0.15,0.30,0.45}.\nThis specific combination of the coupling ratios resides al-\nmost exactly at the middle of the investigated spectrum\nof theJ’s and the equispaced values of ρenable us to-\nwards a systematic conclusion. From Figures 17 and 18,\nwe could see fluctuations in the values of compensation\ntemperature for smaller system sizes, upto L/lessorsimilar64, when\nwe introduce and increase the dilution percentage in steps\nupto 45%. But after L > 64, the fluctuations subside\nwith the value of the compensation temperature becom-\ning almost immune to the system size. For very few other\nrandomly checked combinations of coupling strengths, thesame feature is present. So, within the scope of available\nlimited computational resources, we may only use the val-\nues of compensation temperatures for linear system sizes\nL= 100 without compromising much on the accuracy for\nthe entire 490 different combinations of coupling strengths\nand dilution percentage. The discussion here may validate\nthe compromise and choice of the system size used in this\nstudy.\n15-3.75-3-2.25-1.5-0.75 0\n 0  0.25  0.5  0.75  1p1\nJAA/JBBJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00 (a) ABAF1(JAA/JBB , JAB/JBB)= -a1eJAA/JBB + c1\n-3.75-3-2.25-1.5-0.75 0\n-1 -0.75 -0.5 -0.25  0p1\nJAB/JBBJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00 (b) ABAF2(JAA/JBB , JAB/JBB) = v1(JAB/JBB) - w1\n 0 0.5 1 1.5 2\n 0  0.25  0.5  0.75  1r1\nJAA/JBBJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(c) ABA\nF3(JAA/JBB , JAB/JBB)= a2eb2(JAA/JBB) - c2 0 0.5 1 1.5 2\n-1 -0.75 -0.5 -0.25  0r1\nJAB/JBBJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00(d) ABA\nF4(JAA/JBB , JAB/JBB) = -v2(JAB/JBB) + w2\nFigure 15: (Colour Online) Behaviours of the fitting parameters of d iluted ABA trilayered triangular stacking. Where\nthe errorbars are not visible, they are smaller than the point marke rs.\n-3.75-3-2.25-1.5-0.75 0\n 0  0.25  0.5  0.75  1p2\nJAA/JBBJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00 (a) AABG1(JAA/JBB , JAB/JBB)= -a3eJAA/JBB + c3\n-3.75-3-2.25-1.5-0.75 0\n-1 -0.75 -0.5 -0.25  0p2\nJAB/JBBJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00(b) AABG2(JAA/JBB , JAB/JBB) = u3(JAB/JBB)2 + v3(JAB/JBB)\n- w3\n 0 0.5 1 1.5 2\n 0  0.25  0.5  0.75  1r2\nJAA/JBBJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00(c) AAB\nG3(JAA/JBB , JAB/JBB)= a4eb4(JAA/JBB) - c4 0 0.75 1.5 2.25 3\n-1 -0.75 -0.5 -0.25  0r1\nJAB/JBBJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00(d) AAB\nG4(JAA/JBB , JAB/JBB) = -u4(JAB/JBB)2 - v4(JAB/JBB)\n + w4\nFigure 16: (Colour Online) Behaviours of the fitting parameters of d iluted AAB trilayered triangular stacking. Where\nthe errorbars are not visible, they are smaller than the point marke rs.\n16 0.74 0.78 0.82 0.86 0.9 0.94\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcompABA\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.00  0.6 0.64 0.68 0.72 0.76\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcomp ABA\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.15\n 0.38 0.42 0.46 0.5 0.54 0.58\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcomp ABA\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.30  0.18 0.22 0.26 0.3 0.34\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcomp\nABA\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.45\nFigure 17: (Colour Online) Compensation temperature versus linear system size of a diluted ABA trilayered triangular\nstacking. The reported value of compensation temperature ( L= 100) saturates about L >64.\n 0.7 0.74 0.78 0.82 0.86 0.9\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcompAAB\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.00\n 0.55 0.59 0.63 0.67 0.71 0.75\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcompAAB\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.15\n 0.36 0.4 0.44 0.48 0.52 0.56\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcomp AAB\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.30\n 0.14 0.18 0.22 0.26 0.3 0.34\n 20  50  80  110Tcomp(L)\nLTcomp(L=100)\nTcomp AAB\nJAA/JBB = 0.52\nJAB/JBB = -0.52\nρ = 0.45\nFigure 18: (Colour Online) Compensation temperature versus linear system size of a diluted AAB trilayered triangular\nstacking. The reported value of compensation temperature ( L= 100) saturates about L >64.\n17"    },    {        "title": "2009.10222v1.Magneto_Elastic_Coupling_to_Coherent_Acoustic_Phonon_Modes_in_Ferrimagnetic_Insulator_GdTiO__3_.pdf",        "content": "arXiv:2009.10222v1  [cond-mat.str-el]  21 Sep 2020Magneto-Elastic Coupling to Coherent Acoustic Phonon\nModes in Ferrimagnetic Insulator GdTiO 3\nD. Lovinger,1E. Zoghlin,2P. Kissin,1G. Ahn,3K. Ahadi,2P. Kim,1M.\nPoore,1S. Stemmer,2S. J. Moon,3S. D. Wilson,2and R. D. Averitt1\n1Department of Physics, University of California, San Diego , La Jolla, CA, 92093\n2Materials Department, University of California, Santa Bar bara, CA, 93106\n3Department of Physics, Hanyang University, Seoul, South Ko rea, 04763\n(Dated: September 23, 2020)\nIn this work we investigate single crystal GdTiO 3, a promising candidate material for Floquet en-\ngineering and magnetic control, using ultrafast optical pu mp-probe reflectivity and magneto-optical\nKerr spectroscopy. GdTiO 3is a Mott-Hubbard insulator with a ferrimagnetic and orbita lly ordered\nground state ( TC= 32 K). We observe multiple signatures of the magnetic phase transition in\nthe photoinduced reflectivity signal, in response to above b and-gap 660 nm excitation. Magnetic\ndynamics measured via Kerr spectroscopy reveal optical per turbation of the ferrimagnetic order\non spin-lattice coupling timescales, highlighting the com petition between the Gd3+and Ti3+mag-\nnetic sub-lattices. Furthermore, a strong coherent oscill ation is present in the reflection and Kerr\ndynamics, attributable to an acoustic strain wave launched by the pump pulse. The amplitude of\nthis acoustic mode is highly dependent on the magnetic order of the system, growing sharply in\nmagnitude at TC, indicative of strong magneto-elastic coupling. The drivi ng mechanism, involving\nstrain-induced modification of the magnetic exchange inter action, implies an indirect method of\ncoupling light to the magnetic degrees of freedom and emphas izes the potential of GdTiO 3as a\ntunable quantum material.\nI. INTRODUCTION\nTherare-earthtitanates (unit formulaRTiO 3, whereR\nis a rare-earth ion) are a class of complex materials with\nstrongly correlated spin, orbital, and lattice degrees of\nfreedom. They are3 d1compounds with a single d-orbital\nelectron occupying the Ti3+t2gorbital, whose degener-\nacy is broken by strong crystal field splitting [1]. This\npresents an opportunity to study a strongly correlated\nsystem in relative simplicity, which nonetheless exhibits\nrich physics and interesting properties. The perovskiteti-\ntanates, for example, are Mott-Hubbard (MH) insulators\nwith interconnected orbital and spin order [2–6]. Of par-\nticular interest is the complex magnetic phase diagram\nfor this class of materials, with a magnetic ground state\nthat varies from ferrimagnetic to antiferromagnetic as a\nfunction of the rare-earthion size and subsequent change\nin Ti-O-Ti bond angle [1, 7]. Various theories have at-\ntempted to explain the magnetic order in titanates [5, 8–\n10], all of which highlight the need to consider the roles\nof structure and electronic correlation to understand the\ncomplexity embodied in the magnetic phase diagram.\nCommon to all descriptions of magnetism in the per-\novskite titanates is the defining role of the lattice and\nits distortion. It has been argued, for example, that the\ndegree of GdFeO 3distortion and changes to the Ti-O-\nTi bond angle directly modify the exchange interaction\nwhich, in turn, determines the magnetic order [7, 11].\nMore recent results emphasize the importance of orbital\norder in determining the magnetic order. In particular,\nthe direct coupling between the orbital order and lat-\ntice, ratherthantheorthorhombicdistortion, contributes\nmost strongly to the ground state [12–14]. Whether itis particular structural distortions or more generalized\nJahn-Teller distortions, and regardless of the role of or-\nbital ordering, it is clear that magnetic order in titanates\nis highly dependent on the lattice.\nIn this work we study GdTiO 3(GTO), a titanate with\nan orthorhombic perovskite-type unit cell and relatively\nlarge GdFeO 3-type distortion. GTO lies just within the\nferromagnetic (FM) region of the phase diagram. The\nproximity to the FM-AFM transition makes GTO partic-\nularly sensitive to the effect of structural changes on the\nmagnetism [12]. Below the critical temperature TC=\n32 K it is ferrimagnetically (fM) ordered: the Ti3+spins\nare aligned ferromagnetically along the c-axis and cou-\npled antiferromagnetically to the Gd sublattice [3, 8, 12].\nThe magnetism saturates at 6 µB(7µBGd – 1µBTi) in\na relatively small field of ∼0.1 T, with no discernable hys-\nteresis [15]. The magnetocrystalline anisotropy is small,\nwith the a-axis as the hard magnetization axis and the\nb-cplane nearly isotropic. The fM order is accompanied\nand mediated by ( yz, zx, yz, zx )-type orbital order, a re-\nsult of inter-atomic hybridization between the t 2gand eg\norbitals [1].\nThepresentworkonGTOis motivated notonlyby the\nrelative simplicity of the system and rich interconnected\norder, but also the potential for Floquet engineering and\nultrafast control of magnetism. Liu et al.explored the\nMott insulating titanates as a candidate for tuning the\nspin-orbital Floquet Hamiltonian and subsequent modi-\nfication of the spin exchange interaction using light [16].\nMeanwhile, Khalsa et al.suggest direct excitation of a\nGTO mid-IR active phonon mode to transiently mod-\nify the exchange interaction and switch the ground state\nfrom FM to AFM on ultrafast timescales [17]. A similar2\n0 2 4 6 \nPhoton Energy (eV )0123451()( -1 cm -1 )\n10 \n50 \n150\n3001 2 3 \nPhoton Energy (eV) 120160200240\n0 2 4 6 \nPhoton Energy (eV )0123456n( )1 2 3 \nPhoton Energy (eV) 2.12.22.310 3(a) (b)\nTi 1 t2g  LHBTi 2 t2g UHB\nO 2pEFU\n1.88 eV(c)\nT (K) \n10 \n50 \n150\n300T (K) 1.88 eV\nFIG. 1. (a) Optical conductivity and (b) index of refraction of GTO/LSAT thin film as a function of photon energy and\ntemperature. Red arrows indicate the 1.88 eV pump/probe ene rgy. The weak feature in σ1at 2 eV corresponds to the MH\ngap, while the steep feature near 5 eV arises from O 2pto Ti3dand Gd 4fcharge transfer transitions. (c) Depiction of 1.88 eV\nlaser excitation, corresponding to intersite Ti 3d-3dtransition across the MH gap.\nexperiment utilizing phononic control has been proposed\nby Guet al.in other titanates [18].\nWhile the conditions of our experiment lie outside\nthe regimes discussed above, we do observe strong cou-\npling between light, the lattice, and the sample mag-\nnetism. Time-resolved pump-probe and magneto-optical\nKerr effect (MOKE) measurements tuned to ∼1.88 eV,\njust above the bandgap, allow us to measure the evo-\nlution of photoexcited states on femtosecond – picosec-\nond timescales. We observe multiple signatures of the\nmagnetic phase transition in the photoinduced reflectiv-\nity signal, as well as optical perturbation of the fM order\non spin-lattice coupling timescales in the MOKE signal.\nIn addition, an acoustic phonon mode is present in both\nsignals, whose amplitude is highly coupled to the mag-\nnetic order. This implies strong magneto-elasticcoupling\nthrough transient, strain-induced modification of the ex-\nchange interaction, connecting the lattice and magnetic\ndegrees of freedom and indicating that the exchange in-\nteraction is tunable on ultrafast timescales.\nII. METHODS\nSingle crystaland thin film samplesofGdTiO 3werein-\nvestigated. The photoinduced reflectivity signal in both\nis extremely similar and the following work, except for\nthe measurement of the optical constants, was performed\non a single crystal sample. For comparison, the thin\nfilm time-resolved reflectivity data is presented in SM\nI [19]. GdTiO 3thin films ( ∼20 nm) were grown on\na (001)(La 0.3Sr0.7)(Al0.65Ta0.35)O3(LSAT) substrate by\nhybrid molecular beam epitaxy [20]. GdTiO 3bulk sin-\ngle crystals were grown by high pressure laser floating\nzone method [21]. A small fraction of the crystal rodwas cut and polished to optical quality, with bc-axis in\nplane and a-axis out of plane. Powder X-ray diffraction\nmeasurements indicate extremely high quality crystals\nwith no notable impurity peaks and lattice parameters\nat 5.393, 5.691, 7.664 ˚A fora,b,c-axis [21], well matched\nto literature values [12]. Magnetization measurements in-\ndicate no visible hysteresis and a saturation moment of\n6µB/FU.\nTo determine the optical conductivity and index of re-\nfraction, frequency-dependent reflectivity spectra R(ω)\nin the photon energy region between 3 meV and 85\nmeV were measured by using a Bruker VERTEX 70v\nFourier transform spectrometer. The GdTiO 3thin film\nwas mounted in a continuous liquid helium flow cryostat.\nWe used two spectroscopic ellipsometers (IR-VASE Mark\nII and M-2000, J. A. Woollam Co.) for obtaining the\ncomplex dielectric constants ǫ(ω) =ǫ1(ω) +iǫ2(ω) in\nthe energy range from 60 meV to 0.75 eV and 0.75 eV\nto 6.4 eV, respectively. The optical conductivity of the\nGdTiO 3film was obtained by two-layermodel fit employ-\ning Drude-Lorentz oscillators for optical response of each\nlayer [22].\nUltrafast optical pump-probe reflectivity measure-\nments (∆R/R) are performed using a 1040 nm 200 kHz\nSpectra-Physics Spirit Yb-based hybrid-fiber laser cou-\npled to a non-colinear optical parametric amplifier. The\namplifierproduces ∼20fs pulsescenteredat660nm (1.88\neV), which are split, cross-polarized (pump s-polarized,\nprobep), and used asdegeneratepump and probebeams.\nThe pump is aligned along the b-axis of the GTO crystal.\nThis excitation corresponds to an intersite Ti 3d–3d\ntransition across the Mott-Hubbard gap, shown in Fig.\n1(c). A moderate pump fluence of ∼100µJ/cm2is used\nto minimize sample heating ( ∼4 K at 10 K), ensuring we\nare in the linear excitation regime.3\nTime-resolved magneto-optical Kerr spectroscopy is\nused to probe the magnetization dynamics. The same op-\ntical system described above is used here, including laser\nenergy, fluence, and magneto-optical cryostat (Quan-\ntum Design OptiCool). The photoinduced Kerr rotation\n(∆θK) is measured using balanced photodiodes in the po-\nlar Kerr geometry at near-normalincidence, in a continu-\nously variable external magnetic field (0 – 7 T), with the\npump polarized along the b-axis and Kerr probe polar-\nized along the c-axis of the crystal. The magnetic field is\napplied normal to the sample surface, along the a-axis of\nthe crystal, resulting in a Kerr signal proportional to the\nout-of-plane z-component of the photoinduced change in\nmagnetization ∆ Mz. In order to eliminate non-magnetic\ncontributions to the signal and ensure we are measur-\ning genuine spin dynamics, we take the difference of the\nKerrsignalatvariouspositive andnegativeapplied fields:\n∆θK= ∆θ(+M)−∆θ(−M) (see SM IV for details)\n[19, 23–27].\nIII. EXPERIMENTAL RESULTS\nThe temperature-dependent optical conductivity of\nthin-film GdTiO 3is shown in Fig. 1(a) for photon ener-\ngies ranging from 3 meV to 6.5 eV, and for temperatures\nfrom 10 K to 300 K. A weak feature is present, centered\nat 2 eV, corresponding to the Mott-Hubbard gap. Apart\nfrom weak thermal broadening with increasing temper-\nature, the peak at 2 eV is nearly temperature indepen-\ndent. While early studies of GTO measured a MH gap\nof 0.2 – 0.7 eV [28], more recent photoluminescence and\nDFT/DFT+U results place the gap closer to 1.8 – 2 eV\n[29]. Thesmallpeakintheopticalconductivityspectrum\nat 2 eV measured here supports these recent findings. At\nmuch higher energies we observe a significant increase in\nthe optical conductivity. The features near 5 eV corre-\nspond to O 2pto Ti3dand Gd 4ftransitions. Fig. 1(b)\nshows the index of refraction in the same energy range,\nwhich remains relatively constant as a function of tem-\nperature.\nThe time-dependent photoinduced change in reflectiv-\nity ∆R/Rfor a GdTiO 3single crystal is shown in Fig.\n2(a), for all measured temperatures between 10 – 295\nK (legend on Fig. 2(b)). The black lines represent ex-\nponential fits to the data as described below. The pho-\ntoinduced change in ∆ R/Ris positive; following laser\nexcitation a non-equilibrium electron population is estab-\nlished in ∼500 fs, which then exchanges energy and equi-\nlibrates with the spin and lattice subsystems throughvar-\nious pathways, each with a characteristic timescale. This\nis visible as the slower, multi-component exponential re-\nlaxation. As the temperature is decreased from 295 K\nthe signal amplitude increases, recovery dynamics slow,\nandtwoadditionalfeaturesemerge. The firstisa delayed\nrise time, correspondingto a further departure from equi-\nlibrium in the first ∼15 ps, emerging below T= 100 K.0 200 400 600 800 1000 \nTime (ps) 010 20 R/R 10 -3 \n0 5 10 15 \nTime (ps) 0510 R/R 10 -3 0 500 1000\nTime (ps) 0510 15 20 R/R 10 -3 \n10 K\nslow e-ph \ns-l OO \n100 K0 100 200 300\nTemperature (K) 0510 15 20 25 Peak R/R (500 ps) 10 -3 \n295\n200\n100\n80 \n60 \n45 \n40 \n35 \n30 \n25 \n20 \n10 (a) (b)\n(c)T (K) \nFIG.2. (a)Photoinduceddifferentialreflectivitysignal∆ R/R\nat all measured temperatures, from 10 K to 295 K, taken on\na single crystal GdTiO 3sample. The black curves are expo-\nnential fits to the data, of the form given in Eq. 1. The\nblue star indicates the data curve taken at TC. (b) ∆R/R\nvalues at 500 ps, an approximation of the peak signal at all\ntemperatures. The red line is a power law fit, commonly seen\nin systems undergoing a second-order magnetic phase transi -\ntion. (c) Representative pump-probe scans at 10 K and 100\nK, indicating the various timescales involved in the recove ry\nprocess (see Eq. 1).\nSecond, there is a crossover point visible at delay times\nof∼200 ps where recovery dynamics flatten and reverse\ndirection to become an additional rise time. This occurs\nprecisely as the ferrimagnetic ordering temperature TC\n= 32 K is crossed (marked by a blue star), indicating\nthat magnetization dynamics manifest in the differential\nreflectivity signal.\nTo further investigate the temperature dependence of\nthe reflectivity signal, we plot the peak signal amplitude\n(at 500 ps) in Fig 2(b). The behavior here is distinctive,\nnot uncommon in materials undergoing a second-order\nmagnetic phase transition. The red curve represents a\npower-law fit to the data, of the form A=A0t−w, where\nwis the critical exponent and tis the reduced tempera-\nturet=|T−TC|\nTC. The value of wdepends upon the sym-\nmetry and universality class of the magnetic transition.\nOurfit producesacriticalexponent w= 1.28±0.02. This\nvery nearly matches the critical behavior predicted by\ndynamical scaling theory for the 3D Ising model, which\nyieldsw≈1.32[30–32](furtherdetailedinSMII)[19,30–\n35]. Whilethisisanindirectmethodofmeasuringcritical\ndynamics and is not intended to be a rigorous analysis,4\n050 100 150   s-l  (ps)   \n-6 -4 -2 0\nAmplitude s-l 10 -3\n 50\n 60\n 80\n100\n150\n250 10\n 20\n 25\n 30\n 35\n 400 100 200 300 \nTemperature (K) \n0\nTime (ps) -3 -2 -1 01R/R Residual 10 -4 \n0 50 100 150\nTime (ps) -2 -1 01R/R Res. 10 -4 (a)\n(b)\n10 20 30 40 50 T (K) \nFIG. 3. (a) Time constant (black) and amplitude (red) of the\nspin-lattice coupling term, extracted from exponential fit s to\nthe ∆R/Rdata. The vertical gray section indicates the fM\ntransition region TC= 32 K, where the lifetime is too long\nto measure (see 30 K curve marked by a blue star in Fig.\n2(a)). The reddashed line depicts zero amplitudeand clarifi es\nthe crossover region. (b) Coherent acoustic phonon respons e,\nisolated by subtracting the exponential fits from the ∆ R/R\ndata. The inset shows the ∆ R/Rresidual to a longer delay\ntime of 150 ps, where higher frequency components emerge.\nit is clear that the peak reflectivity follows power-law be-\nhavior as expected at a magnetic phase transition. The\nresults suggests that there is indeed a magnetic contribu-\ntion in the ∆ R/Rsignal. Additionally, the qualitative\nform of the peak amplitude vs temperature follows that\nof the temperature dependent magnetic susceptibility in\nbulk GTO [15], and the magnetization M in films [21, 36].\nWhile by no means conclusive, the universal scaling be-\nhavior and agreement with thermal magnetization does\nstrongly suggest that the ∆ R/Rsignal measured, par-\nticularly at longer times (500+ ps), is sensitive to spin\ndynamics.\nTosubstantiatetheseclaims, wequantitativelyanalyze\nthe full time-dependent response. Below 100 K, the dy-\nnamics can be fit by a sum of four exponentials with a\nconstant offset, of the form:∆R/R(t) =Ae−phe−t/τe−ph+AOOe−t/τOO+\nAs−le−t/τs−l+Aslowe−t/τslow+C,(1)\nshown as black lines in Fig. 2(a). Not listed is an ad-\nditional error function term, which describes the initial\nstep-like rise dynamics at t= 0. A visual representation\nofthevarioustimescalesisshowninFig. 2(c)fortwotem-\nperatures. After excitation the dynamics follow a general\ntrend; there is a very fast initial recovery, τe−phon the\norder of ∼500 fs, followed by an intermediate term τOO\non the order of 2 – 8 ps, both of which are clearly visible\nin the inset of Fig. 2(c). Note that τOOis an additional\nrise time which vanishes at higher temperatures, the full\ndynamics fitting to only 3 exponentials (i.e. above TC).\nThis is followed by a slower term τs−lon the order of\n100’s of picoseconds, and a final much slower recovery\nτslow. A careful inspection of the reflectivity data also in-\ndicates the presence of small oscillations about the black\nfitted curves, which we discuss below.\nThese measured timescales are well separated and can\nbe attributed to distinct physical processes. The initial\npump pulse excites an intersite Ti 3d-3dtransition. This\ndirectly creates a population of hot carriers which ther-\nmalize via electron-electron (e-e) scattering, then subse-\nquently exchange energy with the lattice, orbital, and\nspin degreesof freedom. We focus on the spin-lattice cou-\npling process here, with a full discussion of the remaining\nprocesses and time constants in SM III [1, 11, 19, 37–45].\nThe most relevant component of the ∆ R/Rsignal\nis the third fitted exponential, τs−l, attributed to spin-\nlattice coupling and shown in Fig. 3(a). This term has a\ncharacteristic timescale of 10 – 140 ps, excluding the re-\ngion at the magnetic phase transition temperature TC=\n32K where the lifetime growstoo long to accuratelymea-\nsure. This critical region is visible as a flattening of the\n∆R/R recoveryat 30 K, indicated by the blue star in Fig.\n2(a). Before the onset of this third recovery term τs−l,\nthe ∆R/R signal reveals dynamics indicative of electron-\nlattice equilibration. It follows that this longer lifetime\nis related to equilibration of the spin subsystem with the\nlattice. The time constant measured, on the order of 100\nps in the magnetic phase, is consistent with spin-lattice\ncoupling in other magnetic insulators [46–48]. The char-\nacteristic time is relatively constant in the paramagnetic\nphase until 150 −200 K, where it begins to slowly in-\ncrease. This corresponds to the onset temperature ( ∼180\nK) of spin-spin coupling between the Gd3+and Ti3+ions\n[8]. Closer to 100 K τs−lfurther increases, indicating the\nonset of short-range fM spin correlations. This is also\napparent in the increase in amplitude at this tempera-\nture. Finally, as TCis crossed (dark gray region) we see\nevidence of the second-order ferrimagnetic phase transi-\ntionasthetimeconstantdivergesandamplitudeswitches\nsign. The now-negative amplitude implies an additional\nrise time in the signal; as energy is transferred to spins\nand the ferrimagnetic order is disrupted, the system is5\n-2 0246 (rad) 10 -4 \n-2 0246\n0 500 1000 \nTime (ps)-2 0246 (rad) 10 -4 \n0 500 1000\nTime (ps) 0246810 10 -4 \n 10\n 20\n 25\n 30\n 35\n 40\n 50\n 60\n 80\n25010 -4 H = 0.1 T 0.25 T\n0.5 T1 TT (K) \nFIG. 4. Time-resolved Kerr dynamics at various magnetic\nfields, recorded as the difference between the MOKE signal\nin opposing field directions ∆ θ= ∆θ(+H)−∆θ(−H). This\nis a measure of the photoinduced change in the out-of-plane\nmagnetization ∆ Mz. Red arrows indicate the crossover to\nnegative values of ∆ θ.\nbrought further out of equilibrium. In the paramagnetic\nphase there is no long-range spin order to disrupt, such\nthat spin-lattice thermalization manifests as a simple re-\ncovery to equilibrium. The critical behavior, amplitude\nreversal, timescale, and temperature dependence of the\nτs−lcomponent all suggest that we are measuring spin-\nlattice coupling on a timescale of ∼100 ps, and that it is\nhighly sensitive to the onset of magnetic order.\nThe final interesting feature of the ∆ R/Rdata is a\nslow coherent oscillation, prominent at early times. By\nsubtracting the exponential fits at each temperature we\ncan extract the oscillatory component, plotted in Fig.\n3(b). The result is peculiar – we observe a low-frequency\nphonon mode which grows in amplitude and becomes\nchirped, slowing down and redshifting as it propagates.\nThe oscillationperiod (on the orderof20ps), suggestsan\nacoustic strain wave launched by the pump pulse which\npropagates through the crystal [49]. The probe beam re-\nflected from the sample surface interferes with a portion\nreflected from the strain wave boundary, resulting in an\noscillatory signal. The temperature dependence of this\nmode is striking – the amplitude is relatively constant at\nhigh temperatures, then grows sharply precisely at the\nfM phase transition temperature. Though it appears to\nbe an acoustic mode, it is also clearly coupled to the\nmagnetic order. This suggests strong magneto-acoustic\ncoupling, tying the dynamics of the magnetic subsystem\nto the transiently strained lattice.\nTo gain further insight into the magnetization dynam-\nics and the influence upon acoustic phonon propaga-tion, we utilize time-resolved magneto-optical Kerr effect\n(MOKE) spectroscopy. Fig. 4presents the photoinduced\nKerr rotation ∆ θ, proportional to the change in out-of-\nplane (a-axis) magnetization ∆ Mz, for all temperatures\nand four fields between 0.1 – 1 T. For details of the anal-\nysis, see SM IV [19, 23–27]. Additional static Kerr rota-\ntion measurements are presented in SM V [19]. At lower\nfield strengths, the Kerr signal reveals a quick rise in\nthe photoinduced out-of-plane magnetization ∆ Mz, fol-\nlowed by a reduction and change in sign of ∆ Mz. This\ncan be interpreted asa pump-induced increaseand subse-\nquent decrease in the net out-of-plane magnetic moment,\nbut not necessarily a reversal of the total magnetic mo-\nment. Therearetwoprimarycomponentsto the Kerrsig-\nnal, one positive (growing in ∼100 ps), and one negative\n(growing in slower, ∼100 – 500 ps). These dynamics are\nslow and long-lived, as expected in magnetic insulators\nlikeGTOduetothelocalizednatureofquasiparticles[48].\nTo describethe temperature dependence ofthe signal, we\nfocus on lower field strengths H= 0.1−0.5 T. At high\ntemperature, in the paramagnetic phase, the Kerr signal\nis weak and indicates the lack of long-range magnetic or-\nder. As the temperature is lowered there is an increase in\nthe photoinduced rotation, with a clear negative signal\nemerging below TC. This negative component is largest\nand appears at earlier delays right at the transition tem-\nperature ( TC= 32 K). With decreasingtemperature, the\ncrossover to negative values of ∆θoccurs at later times.\nWell below TC, in the strongly ordered phase, the signal\nremains positive at all time delays.\nWe also observe a significant field dependence in the\ndata. The maximum signal amplitude at all tempera-\ntures increaseswith increasing field. In addition, the neg-\nativeamplitude componentismostpronouncedat0.25T,\ndecreasing in amplitude at higher fields and vanishing en-\ntirely by 1 T. At this high field, we note that the photoin-\nduced magnetization dynamics look qualitatively similar\nto the photoinduced reflectivity signal ∆ R/Rshown in\nFig.2(a). In the ∆ R/Rdata, the measured signal is\nprimarily the result of Ti sublattice dynamics due to the\n1.9 eV intersite Ti-Ti excitation, and is dominated by Ti\nspin dynamics: the spin-lattice and spin relaxationterms\n(τs−landτslow). It follows that the MOKE signal mea-\nsured at 1 T is primarily a measure of Ti spin dynamics\ndue to its similarity with the ∆ R/Rsignal. Fits to the 1\nT MOKE data support this, yielding a component with\na timescale of 100 – 200 ps and a very similar tempera-\nture dependence when compared to τs−lextracted from\nthe ∆R/Rdata (see SM VI for details) [19]. As the field\nis lowered from 1 T, the magnetization dynamics must\nbe increasingly influenced by the Gd spins. The ferri-\nmagnetic nature of GTO, with two competing magnetic\nsublattices, is key to understanding the observed behav-\nior as we now discuss.6\n 50 \n 60 \n 80 \n100 \n150 \n250  10 \n 20 \n 25 \n 30 \n 35 \n 40 \nTime (ps)-2 -1 012 Residual (rad) 10 -5 Gd \nTi \n  MZMZ > 0 MZ < 0 \nMZ > > 0 \nMMZ << 00\nMMZ >>>> 0\nT (K) (a)\n(b)\n0 10 20 30 40 50 H = 0.1 T\nH = 1 Tt < s-lt ≥ s-l \nt = 0\nHz\na-axis\nFIG. 5. (a) Schematic depiction of the spin dynamics. Pho-\ntoexcitation directly perturbs the Ti spins, which fluctuat e\nand decrease their projection along the applied field H (para l-\nlel toa-axis, +z) int < τs−lps, increasing the MOKE signal.\nAt longer times: if H and/or magnetic order is weak, induced\nspin fluctuations and the AFM exchange coupling lowers the\nprojection of Gd spins along z. Conversely, if H is larger\nthan the exchange field, at 1 T, Gd does not reorient and no\nnegative component of the signal is observed. (b) Coherent\nacoustic phonon response, isolated by subtracting the expo -\nnential fits from the time-resolved Kerr data (at 1 T applied\nfield). The dynamics appear similar to the ∆ R/Rresidual,\nimplying a common origin which we attribute to a coherent\nstrain wave launched by the pump pulse. The appearance\nof this signal in the Kerr response indicates coherent acous tic\nphonon manipulation of the magnetic order, presumably from\nexchange modulation.\nIV. DISCUSSION\nGTO is ferrimagnetic, the Ti and Gd sublattices cou-\npled via an AFM exchange interaction. Gd spins have a\nsignificantly larger magnetic moment than Ti, 7 µBvs1µBrespectively [15]. Below TC, at zero field, the two\nsublattices are aligned into fM domains such that there is\nnomacroscopicmoment. As the applied field Halongthe\na-axis is increased, spins are rotated to form long-range\ncollinearfMorder,withtheGdsublatticealignedparallel\nto H and Ti anti-parallel. In a field of only 0.1 T satu-\nration is approached, with spins slightly canted from the\na-axis/H and a net magnetization of M≈5µB. With in-\ncreasing field, canting and spin fluctuations are reduced,\nincreasing the net moment along H. As we approach 1 T,\nspin fluctuations are minimized and the magnetization\nbecomes saturated at M≈6.0µB[15]. The interac-\ntion of these two competing magnetic sublattices after\nphotoexcitation will depend on the temperature and ap-\nplied field and is illustrated in Fig. 5(a). The 1.9 eV\npump pulse directly excites the Ti sublattice, increas-\ning Ti spin fluctuations on timescales t < τs−l. This\ncauses partial reorientation and a decrease in the projec-\ntion of Ti spins along the Gd moment, corresponding to\na rapid increase of ∆ Mzand a rise in the MOKE signal\n(i.e. the Ti sublattice magnetization oriented along -zis\ndecreased, leading to an overall increase in the net mag-\nnetization in the + zdirection due to the ferrimagnetic\norder). Various pathways exist which may perturb the\nTi spins on such timescales, including spin-orbit coupling\n[50, 51] (orbital order is disrupted in t <8 ps, see SM III)\n[1, 11, 19, 40–45], and exchange modification, discussed\nbelow. Subsequently, energy is transferredto the Gd sub-\nlattice through spin-lattice thermalization on a timescale\nt≥τs−l. The spin-lattice coupling timescale of 100 –\n200 ps measured from fits to the ∆ R/Rand MOKE data\ncorresponds to the timescale on which the MOKE signal\nchanges sign, indicating the delayed contribution of Gd\nspins to the signal. Such ultrafast magnetic sublattice\ndynamics, albeit with different mechanisms, have been\ndiscussed in a variety of materials, including those with\nsimilar rare-earth/transition metal correlations [52, 53].\nThe behavior that follows is field-dependent. At low\nfields, at times on the order of τs−l, the additional heat\ntransfer and the strong AFM exchange coupling between\nGd spins and partially-reorientedTi spins causesa reduc-\ntion in the Gd moment along the field direction. This is\nseen as the negative component, decreasing the signal\non spin-lattice timescales until the net ∆ Mzis negative.\nAt higher field strengths, the applied field locks Gd mo-\nments in place parallel to the field direction, minimizing\nfluctuations. After photoexcitation, the net magnetiza-\ntion only increases as the anti-parallel Ti spins fluctuate\nand partially reorient. This is true also at low tempera-\ntureswherethemagneticorderismorefirmlyestablished,\nand explains why ∆ Mzgoes negative only in the weakly\nordered state near TC.\nFinally, we cannot discount the possibility of direct\nphoto-induced modification of the exchange interactions.\nWhile the simplest explanation of the MOKE signal in-\nvolves only heating and spin-lattice coupling, the over-\nall heating is small (no more than ∼4 K at the lowest\ntemperatures at the fluence used). It is therefore not un-7\n024\n0 50 100\nFrequency (GHz) 012FFT Amplitude  (10 -7 )\n800 nm 10 \n 20 \n 25 \n 30 \n 35 \n 40 \n 50 \n 60 \n 80 \n100 \n295 T (K) \nFFT Amplitude  (10 -6 )\n0 50 100012\nFrequency (GHz) 012\n40 ps 60 ps\n0 50 100\nFrequency (GHz) TC = 32 K\n0 100 200 300 \nTemperature (K) 01FFT (norm.) 4  GHz\n20 GHz\n50 GHz\n80 GHz(a)\n(b)(c)\n(d) (e)660 nm\nFIG. 6. FFT amplitude of the ∆ R/Rresidual, taken at pump/probe wavelength of 660 nm (a) and 80 0 nm (b). The red dashed\nlines indicate the approximate peak positions of the 660 nm F FT. Note the redshift to lower frequency at higher wavelengt h.\n(c) The integrated FFT amplitudes at various frequencies as a function of temperature, normalized. (d) The FFT limited t o\nthe first 40 ps and (e) 60 ps of the ∆ R/Rdata. The higher frequency mode emerges only after 40 ps.\nreasonable to consider more direct electronic changes to\nthe system. The exchange interaction in the titanates is\nhighly dependent on the Ti-O-Ti bond angle and degree\nof GdFeO 3distortion, as well as the orbital order and\noccupation [7, 11]. GTO in particular lies on the cusp of\nthe AFM-FM phase boundary, making it especially sus-\nceptible to changes in these parameters. Photoexcitation\ndirectly disrupts the orbital occupation, which could af-\nfect the octahedral distortion and thus the spin exchange\ninteraction. This in turn would provide the drive for re-\norientation of Ti spins and change in M zat timescales\nt < τs−l, and for subsequent perturbation of Gd spins\nthrough exchange coupling with Ti. Further calculations\nof the energy scales of the Ti-Gd exchange field and cor-\nresponding timescales are required to confirm this.\nTo compare the magnetic dynamics to the ∆ R/Rre-\nsponse, we fit the MOKE data to a series of exponentials\nsimilar to Eq. 1 and subtract the fits. Once again, a\nslow coherent oscillation is revealed, shown in Fig. 5(b)\nfor the data taken at 1 T. The similarity of the oscilla-\ntory Kerr signal to the oscillation in ∆ R/Ris striking –\nboth phonon modes have the same frequency, same time-\ndependent redshift, and same temperature dependence,\nwith the amplitude growing rapidly at TC. We rule out\nthe possibility of a magnon – at lower fields there is no\nchange in the frequency of oscillation as we would ex-\npect from coherent spin precession (SM VII) [19]. The\namplitude is highly field-dependent however, becoming\nmuch smaller at lower fields. These observations suggestthat the oscillatory mode in the MOKE signal, neces-\nsarily a magnetic phenomenon due to the nature of the\nmeasurement technique, has the same origin as the os-\ncillatory mode in the ∆ R/Rsignal. This is consistent\nwith our interpretation of an acoustic strain wave with\nstrong magneto-elastic coupling. This mechanism has\nbeen studied in a variety of ferromagnetic systems, and\ninvolves elastic stress modifying the magnetic anisotropy,\nwhich exerts a torque on the spins and alters the net\nmagnetization [54–56].\nTo quantify the acoustic phonon response, we show in\nFig.6(a) the FFT of the full ∆ R/Rresidual, taken from\nFig.3(b) (inset). The oscillatory mode with ∼20 ps pe-\nriod featured in Fig. 3(b) appears as a strong peak at\n∼50 GHz. In this region of interest, it is apparent that\nthere are additional higher frequency modes in addition\nto the 50 GHz mode. The temperature dependence is\nalso clear; while the FFT amplitude is nearly constant\nat high temperatures, it grows rapidly upon approaching\nTC= 32 K and a higher frequency peak at ∼80 GHz\nemerges. This again suggests coupling to the magnetic\norder. Fig. 6(b) applies the same FFT analysis to data\ntaken at an increased pump/probe wavelengthof 800 nm.\nThe features are similar, but exhibit a clear redshift as\nindicated by the red dashed lines. This behavior is con-\nsistent with an acoustic strain wave since it arises (for\n∆R/R) from interference of the probe with itself. The\nphonon frequency is wavelength dependent, its form is8\ngiven by:\nf= 2nv/λ, (2)\nwherenis the index of refraction, vis the sound veloc-\nity, and λis the probe wavelength [49]. As we observe,\na higher probe wavelength results in a lower frequency\nacoustic phonon. Using the measured index of refraction\nnin Fig.1(b) wecanalsoestimatethe sound velocity. At\nthe lower frequency peak near 50 GHz we obtain a sound\nvelocity of 7 .2×103m/sand 7.7×103m/sfor a 660 and\n800 nm probe, respectively. This is a very reasonable\nrange for acoustic propagation in solid materials. These\nresults, and the fact that the oscillation frequency does\nnot depend on magnetic field, confirms our classification\nof the phonon mode as an acoustic strain wave.\nTomorecloselyexaminethelinktomagnetism,weplot\nthe integrated FFT amplitudes for all frequency peaks in\nFig.6(c). The normalized curves show a striking trend;\nthe amplitude is nearly constant at high temperatures,\nbut sharply increases at or very near to the magnetic or-\ndering transition. The temperature dependence of the\nFFT amplitudes follows the magnetic order parameter\nandisremarkablysimilartothedivergenceoneexpectsat\na second-order magnetic phase transition. This indicates\nthe presence of magneto-elastic coupling. The acoustic\nattenuation of sound waves near magnetic phase tran-\nsitions is well studied, and literature suggests that the\nattenuation follows power law behavior, similar to our re-\nsult [54]. In the vicinity of TC, energy density and spin\nfluctuations play the primary role in attenuation. This\nbehavior has been studied in a wide range of magneto-\nelasticallycoupledmaterials, including Ni [54], CoF 2[57],\nand MnF 2[58].\nA final interesting feature to note is shown in Fig. 6(d-\ne), comparing an FFT of the ∆R/Rdata limited to the\nfirst 40 ps (d) and to the first 60 ps (e) of the scan. This\nanalysis reveals that the high frequency component at\n80 GHz begins to emerge only after 40 ps, which is also\nvisible in the time-domain data (Fig. 3(b) inset). This\ntimescale is similar to the spin-lattice coupling timescale\nmeasured in both ∆ R/Rand MOKE, which ranges from\n∼50 – 150 ps. We have also discussed the spin dynam-\nics following photoexcitation, where Ti spins are imme-\ndiately perturbed and Gd follows after exchange path-\nway alterations and spin-lattice thermalization. Given\nthe similar timescales, we suggest that the emergence of\nthe 80 GHz mode indicates the onset of Gd spin dynam-\nics. Roughly 50 ps after photoexcitation the Gd spin sub-\nsystem begins thermalizing and fluctuating. This damps\nthe acoustic oscillation and changes the magnetic back-\nground. The nowhigherenergyofthe Gd spins altersthe\nspin-phonon and magnetostrictive interaction strengths,\nresulting in a change to the magnetically-coupled elas-\ntic parameters of the lattice and a subsequent shift in\nphonon frequency.\nA microscopic description of magneto-elastic coupling\ninvolves a transient modification of the exchange interac-\ntion. As the acoustic wave propagates it modulates thedistance between lattice sites and spins. This in turn pro-\nduces a periodic modification of the exchange interaction\nbetween neighboring spins, coupling the acoustic wave to\nthe magnetic order parameters. The result is an attenua-\ntion of the acoustic wave in the high-temperature phase\nwhere spin fluctuations are large, lessening as spin cor-\nrelations increase in the low temperature ordered phase.\nThe same mechanism decreases acoustic attenuation, in-\ncreasing the phonon amplitude, in an applied magnetic\nfield as observed in our MOKE signal. This has been\ndescribed by an approximate analytical theory [58–60],\nwhich generally predicts maximal acoustic damping at\nthe critical point and a MHz frequency shift in the or-\ndered phase. We observe that the damping is consis-\ntently large throughout the high temperature paramag-\nnetic phase, and we do not observe such a frequency shift\nwith temperature. In our experiment, however, a MHz\nfrequency shift is too small to be observed, and the dy-\nnamics at picosecond timescales are strongly coupled to\nout-of-equilibrium degrees of freedom that will affect the\nacousticwavepropagationandattenuationinotherunan-\nticipated ways.\nThephonon behaviorweobserveundoubtedly suggests\na strong coupling of the lattice to the magnetic order\nin GdTiO 3. Furthermore, the mechanism implies tran-\nsient exchange modification on an ultrafast timescale.\nThese conclusions are not without precedence. Ultra-\nfast magneto-elastic coupling has been demonstrated by\nBigotet al., for example, in Ni thin films [61], with exper-\niments going so far as to control the magnetic precession\nthrough acoustic pulses [62]. Kimel et al.have shown\noptical quenching of magnetic order through phonon-\nmagnon coupling in FeBO 3[48] and Nova et al.have\nshownthatMid-IRandTHzexcitationresonantwithspe-\ncific lattice modes is able to drive collective spin preces-\nsion [63]. Our work represents another potential method\nof using light to indirectly alter the magnetic degrees of\nfreedom on ultrafast timescales, through coupling to an\nacoustic phonon mode.\nV. CONCLUSION\nWe have used a multi-modal approach, consisting\nof time-resolved photoinduced reflectivity and magneto-\noptical Kerr (MOKE) spectroscopy, to study magneto-\nelastic coupling in the ferrimagnetic insulator GdTiO 3.\nWe observe multiple, clear signatures of the ferrimag-\nnetically ordered phase at TC= 32 K in both signals,\nand measure spin-lattice thermalization timescales τs−l\non the order of 100 picoseconds, as might be expected in\na magnetic insulator.\nFrom the MOKE signal we observe long-lived spin dy-\nnamics and optical perturbation of the ferrimagnetic or-\nder. This includes a change in sign of the photoinduced\nmagnetization on the same timescale as spin-lattice cou-\npling. The ferrimagnetic nature of GTO, with two mag-9\nnetic sublattices coupled antiferromagnetically, is respon-\nsible. Photoexcitation at 660 nm directly perturbs the\nTi moments, increasing fluctuations and causing a par-\ntial reorientation and decrease in the projection of Ti\nspins along the Gd moment. This is measured as an\nincrease in the MOKE signal. Heat is then transferred\ntothe Gdsubsystem throughspin-latticecoupling, which\nwhen combinedwith the AFM exchangeinteractionleads\nto a reduction of the Gd magnetic moment along the z-\ndirection, lowering the net magnetization. Modified ex-\nchange pathways likely also play a role in the delayed\nreorientation of Gd spins on these timescales. The data\nshows that (a) there is a delayed response of the Gd ions\ntotheopticalexcitationand(b)thatspin-latticecoupling\nand the AFM exchange interaction facilitates this.\nIn both the reflectivity and MOKE signals, a clear\ncoherent acoustic phonon is present. This strain wave\nlaunched by pump is intimately tied to the sample mag-\nnetism, with an amplitude that grows sharply at TCand\nclosely follows the magnetic order parameter. As the\nacoustic wave propagates it periodically alters the dis-\ntance between local spins, modifying the exchange inter-\naction. In this way, the lattice parameters are coupled to\nthe magnetic order, which causes an attenuation of the\nacoustic mode near and above TC, where spin fluctua-\ntions are large. This represents a laser-induced modifica-\ntion of the exchange interaction on ultrafast timescales\nthroughcouplingtoanacousticphononmode. Whilethe-\nory exists to describe magneto-elastic coupling, it is not\nparticularly well-suited to the experiment and timescalesmeasured here. A deeper theoretical understanding of\nthe mechanisms at work would be instrumental in quan-\ntifying our results and motivating further studies. This\nwork also suggests that more controlled excitation may\nbe of interest in transiently controlling the properties of\nmaterials. An experiment ofthis naturehas alreadybeen\nproposed to modify the exchange interaction in GTO, us-\ning a resonant mid-IR pulse to directly excite specific\nphonon modes [17]. The work performed here indicates\nthe potential for GTO, and likely other titanates, as tun-\nable magnetic materials, and highlights the need for fur-\nther investigations of this nature on the road to coherent\ncontrol of materials on ultrafast timescales.\nVI. ACKNOWLEDGEMENTS\nWe thank Leon Balents for helpful discussions and\nassistance with interpretation of the data. This work\nwas supported primarily by ARO Award W911NF-16-\n1-0361 and additional support was provided by the W\nM Keck Foundation (SDW). 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J.ˇSaf´ arik University, Park Angelinum 9, 041 54 Koˇ sice, Slov ak Republic\nAbstract\nCritical and compensation properties of a mixed spin-1 and s pin-3/2 Ising fer-\nrimagnet on a square lattice are investigated by standard an d histogram Monte\nCarlo simulations. The critical temperature is studied as a function of a single-\nion anisotropy strength. The second order of the phase trans ition is established by\nfinite-size scaling for the entire boundary. Some previousl y obtained results, such as\na tricritical point, predicted by the mean field theory (MFT) and effective field the-\nory (EFT), or a first-order transition line separating two di fferent ordered phases,\nobtained bytheclustervariational theory (CVT),aredeeme dartifacts oftherespec-\ntive approximations. Sois a reentrant phenomenonproduced by CVT. Nevertheless,\nthe multicompensation behavior predicted by MFT and EFT was confirmed.\nKey words: Ferrimagnet, Mixed spin system, Phase diagram, Ising model , Monte\nCarlo simulation, Compensation temperature\nPACS:05.10.Ln, 64.60.De, 75.10.Hk, 75.30.Kz, 75.50.Gg\n1 Introduction\nThere have been a number of theoretical studies of mixed-spin Isin g fer-\nrimagnets as simple models of certain types of molecular-based magn etic ma-\nterials [1–5]. Besides other interesting properties, such as appear ance of a\nmulticritical behavior, they attract attention due to so-called com pensation\nbehavior with possible technological applications. Such a phenomeno n occurs\nat a compensation temperature, i.e., the temperature below the cr itical point\n∗Corresponding author.\nEmail addresses: milan.zukovic@upjs.sk (M.ˇZukoviˇ c),\nandrej.bobak@upjs.sk (A. Bob´ ak).\nPreprint submitted to Physica A 3 August 2018at which the sublattice magnetizations completely cancel out and th e total\nmagnetization changes sign. Generally speaking, from the previous studies it\ncan be concluded that the higher values of spins and the more comple x lattice\ntopologies used the richer behavior can be expected. However, th e increas-\ning complexity usually requires increasing simplifications in the approac hes\nfor the problem to be manageable. Thus nonperturbative approac hes, such as\nexact [6–10] or Monte Carlo (MC) [11–14], are so far limited to the simp lest\ncases of either the smallest spin values (i.e., mixed spin-1/2 and spin-1 ) or the\nlattice topology (e.g., honeycomb or Bethe lattices). As a conseque nce, the\nbehavior of the mixed spin-1/2 and spin-1 Ising system is rather well under-\nstood, while there are still disagreements among different theoret ical investi-\ngations of the mixed-spin systems with higher spin values, including th e spin-1\nand spin-3/2 case. The disagreements arise from the fact that du e to higher\ncomplexity mostly different approximative schemes with various degr ees of\napproximation have been employed, such as mean field theory (MFT) [15],\neffective field theory (EFT) [16–18] and cluster variational theory (CVT) [19].\nThese approximative approaches have been previously shown to pr oduce some\nartifacts, such as a tricritical point, predicted by MFT [20] and EFT [21] for\nthe mixed spin-1/2 and spin-1 system on a square lattice, that were not repro-\nduced either in numerical transfer matrix [12] or MC studies [11,12, 14]. To\nour knowledge, the results of these approximative studies for the mixed spin-1\nand spin-3/2 system have so far been verified by MC simulations only f or the\ncase of a simple cubic lattice [22]. However, for this particular case, t here were\nbasically no qualitative disagreements among the conclusions drawn f rom the\nrespective studies and the MC results only confirmed the previously obtained\nresults. On the other hand, for the cases of honeycomb and squa re lattices,\nthere are qualitative differences in the results, which have not been resolved\nyet.\nThe objective of this study is to focus on the case of the mixed spin- 1 and\nspin-3/2 Ising system with a uniform single-ion anisotropy on a squar e lattice,\nfor which the differences between the MFT, EFT and CVT results are the\nmost prominent. In particular, we aim to answer the following questio ns: (1)\nAre the phase transitions of second order for the entire range of the anisotropy\nstrength, as predicted by CVT, or is there a tricritical point separ ating lines of\nthe second- and first-order transitions, as predicted by both MF T and EFT?\n(2) Is there a reentrant phenomenon in the second-order phase boundary, as\nsuggested by the CVT results but not by MFT nor EFT? (3) Does the line\nof first-order transitions situated within the ordered ferrimagne tic region, ob-\ntained by CVT but not by MFT nor EFT, really exist? (4) Can the syste m\ndisplay up to two compensation points, as predicted by both MFT and EFT\n(not investigated by CVT)?\n22 Model and Monte Carlo simulations\nThe model of the mixed spin-1 and spin-3/2 Ising system on the squa re\nlattice is described by the Hamiltonian\nH=−J/summationdisplay\n(i,j)SA\niSB\nj−DA/summationdisplay\ni(SA\ni)2−DB/summationdisplay\nj(SB\nj)2, (1)\nwhereSA\ni=±3\n2,±1\n2forAions,SB\nj=±1,0 forBions,J <0 is the nearest-\nneighbor coupling parameter between the ions on A and B sublattices , and\nDA,DBare the single-ion anisotropies acting on the spin-3/2 and spin-1 ions ,\nrespectively. Inthisstudywewillconsider theanisotropyofaunifo rmstrength\ni.e.,D≡DA=DB.\nA simulated L×Lsquare lattice consists of two interpenetrating sublat-\ntices, each one comprising L2/2 sites. We consider linear lattice sizes rang-\ning from L= 20 up to L= 200 with the periodic boundary conditions im-\nposed. Initial spin states are randomly assigned and the updating f ollows the\nMetropolis dynamics. The lattice structure and the short range of the interac-\ntions enable vectorization of the algorithm. Since the spins on one su blattice\ninteract only with the spins on the other, each sublattice can be upd ated si-\nmultaneously. Thus one sweep through the entire lattice involves ju st two sub-\nlattice updating steps. For thermal averaging, we typically conside rN= 105\nMC sweeps in the standard and up to N= 107MC sweeps in the histogram\nMC simulations [23,24], after discarding another 20% of these numbe rs for\nthermalization. To assess uncertainity, we perform 10 runs, using different\nrandom initial configurations. Then the errors of the calculated qu antities are\ndetermined from the values obtained for those runs as twice of the standard\ndeviations.\nWe calculate the internal energy per site e=E/L2=∝angbracketleftH∝angbracketright/L2and the\nsublattice magnetizations per site\nmA= 2∝angbracketleftMA∝angbracketright/L2= 2/angbracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nASA\ni/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketrightBigg\n/L2, (2)\nmB= 2∝angbracketleftMB∝angbracketright/L2= 2/angbracketleftBigg\n−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nBSB\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketrightBigg\n/L2, (3)\nwhere∝angbracketleft···∝angbracketrightdenotes thermal averages. The total magnetization per site is\ndefined as\nm=∝angbracketleftM∝angbracketright/L2=∝angbracketleftMA+MB∝angbracketright/L2. (4)\nSince for ferrimagnets mcan vanish within the ordered phase at the compen-\nsation temperature, as an order parameter it is useful to define t he staggered\n3magnetization per site as\nms=∝angbracketleftMs∝angbracketright/L2=∝angbracketleftMA−MB∝angbracketright/L2. (5)\nFurther, the following quantities which are functions of the parame tersE\nor/andO(=M, M s) are defined: the specific heat per site c\nc=∝angbracketleftE2∝angbracketright−∝angbracketleftE∝angbracketright2\nL2kBT2, (6)\nthe direct ( O=M) and staggered ( O=Ms) susceptibilities per site χO\nχO=∝angbracketleftO2∝angbracketright−∝angbracketleftO∝angbracketright2\nL2kBT, (7)\nthe logarithmic derivatives of ∝angbracketleftO∝angbracketrightand∝angbracketleftO2∝angbracketrightwith respect to β= 1/kBT\nD1O=∂\n∂βln∝angbracketleftO∝angbracketright=∝angbracketleftOE∝angbracketright\n∝angbracketleftO∝angbracketright−∝angbracketleftE∝angbracketright, (8)\nD2O=∂\n∂βln∝angbracketleftO2∝angbracketright=∝angbracketleftO2E∝angbracketright\n∝angbracketleftO2∝angbracketright−∝angbracketleftE∝angbracketright, (9)\nthe Binder parameter U\nU= 1−∝angbracketleftM4∝angbracketright\n3∝angbracketleftM2∝angbracketright2, (10)\nand the fourth-order energy cumulant V\nV= 1−∝angbracketleftE4∝angbracketright\n3∝angbracketleftE2∝angbracketright2. (11)\nThe above quantities are useful for localization of a phase transitio n aswell\nas for determination of its nature. For example, the transition tem peratures\ncan be estimated from the locations of the peaks of the response f unctions c\nandχOor, with a higher precision, from the intersection of the Binder pa-\nrameterUcurves for different lattice sizes L. The first-order character of the\ntransition can be judged from the presence of discontinuities in the magneti-\nzations and energy, as well as from hysteresis loops. Furthermor e, the order of\nthe transition can be reliably established by a finite-size scaling (FSS) anal-\nysis. For instance, the energy cumulant Vexhibits a minimum near critical\ntemperature Tc, which achieves the value V∗=2\n3in the limit L→ ∞for\na second-order transition, while V∗<2\n3is expected for a first-order transi-\ntion [23,24]. Temperature-dependences of a variety of thermody namic quanti-\nties display extrema at the L-dependent transition temperatures, which at a\nsecond-order transition are known to scale with a lattice size as, fo r example:\nχO,max(L)∝LγO/νO, (12)\nD1O,max(L)∝L1/νO, (13)\n4D2O,max(L)∝L1/νO, (14)\nwhereνOandγOrepresent the critical exponents of the correlation length\nand susceptibility, respectively. More precise locations of the extr ema used in\nFSS can be obtained from histogram MC simulations [23,24], performe d at\ncriticality for each lattice size. In the case of a first-order transit ion, typically\na bimodality appears in the P(E) andP(O) histograms as Lincreases and\nthe quantities (6) −(11) display a volume-dependent scaling, ∝L2.\n3 Results and discussion\n[Fig. 1 about here.]\nThe critical temperature as a function of the anisotropy strengt hD, shown\ninFig.1,wasdeterminedfromthepositionsofthestaggeredsusce ptibilityχMs\npeaks obtained by standard MC simulations for a fixed lattice size of L= 40.\nWe checked that the peaks positions (the temperatures at which t hey occur)\ndonotsignificantly changewhenlarger Lisused. Forthecaseof DA=DB= 0\nwe estimated the critical temperature with relatively high precision f rom the\nBinder parameter analysis, using the lattice sizes from L= 40 up to 200\nandN= 5×106MC sweeps, as kBTc/|J|= 2.362±0.003. In line with\nthe MFT [15], EFT [16–18] and CVT [19] results, the critical temperat ure\ndecreases as the anisotropy is decreased from positive to negativ e values and\neventually vanishes at the exact value of D/|J|=−2. However, in contrast\nto the CVT results, no signs of the reentrant phenomenon was obs erved near\nD/|J|=−2. Namely, as temperature is increased from low values there is a\nsingle order-disorder phase transition for each value of D/|J|>−2 and no\nphase transition below D/|J|=−2.\n[Fig. 2 about here.]\nFurthermore, no indications of the additional first-order transit ion line\nwithin the ordered phase at low temperatures, found in the CVT stu dy, were\nobserved. In Fig. 2 we show anisotropy dependences of the stagg ered magneti-\nzationmsand energy e, along with the respective response functions, χMsand\nc, shown in insets, at kBT/|J|= 0.1. It is well known that discontinuities in\nphysical quantities associated with a first-order transition tend t o smear out\nin MC simulations for smaller lattice sizes. Therefore, in order to dete ct first-\norder transition features, we chose a relatively large size of L= 120 and also\nchecked for possible loops in the curves obtained by increasing and d ecreas-\ningD. According to [19], we should observe two anomalies associated with\ntwo phase transitions: the first-order one between two ordered phases and the\nsecond-order one between the ferrimagnetic and paramagnetic p hases. How-\never, as evidenced in Fig. 2, only one anomaly can be observed and, t herefore,\n5we conclude that there is only one phase transition which is from the o rdered\nferrimagnetic to the paramagnetic phase. This scenario agrees wit h the MFT\nand EFT predictions. However, the latter insist that in this low-temp erature\nregion, i.e., below the tricritical point ( Dt/|J|,kBTt/|J|) = (−1.9730,1.1606)\nfrom MFT and ( Dt/|J|,kBTt/|J|) = (−1.9981,0.6175) from EFT, the tran-\nsition is first-order, which does not appear so in our simulations, sinc e no\ndiscontinuities nor loops in the staggered magnetization and interna l energy\ncurves can be seen.\nNevertheless, the respective spike-like response functions, whic h are some-\nwhat reminiscent of afirst-order transition, promptedus tofurt her explore the\norder of the transition by running more extensive histogram MC simu lations\nfor even larger sizes Land performing FSS analysis. We considered the lattice\nsizes up to L= 200 and the number of MC sweeps N= 107, however, we did\nnot detect any signs of a bimodal distribution in the energy or stagg ered mag-\nnetization histograms at criticality that would signal a first-order t ransition.\nOn the contrary, FSS of the quantities (7) −(9), shown in Fig. 3, indicates that\nthe calculated susceptibility and correlationlength critical exponen ts,γMsand\nνMs, respectively, agree well with the standard 2D Ising ones, i.e., γI= 7/4\nandνI= 1. In the susceptibility exponent dependence we can see that a line ar\nregime in the log-log behavior is established only at L≈80 and, therefore,\nsufficiently large lattice sizes should be used. The second-order nat ure of the\ntransition is also confirmed by the scaling of the minima of the fourth- order\nenergy cumulant V(Fig. 4), which tend to the value V∗= 2/3 forL→ ∞\nand apparently do not scale with volume, as expected for the secon d-order\ntransition. Such thorough investigations were not done below kBT/|J|= 0.1\nand we cannot rule out the possibility of the tricritical point existenc e at still\nlower temperatures but we deem it unlikely and presume that the tra nsition\nremains second-order at all temperatures.\n[Fig. 3 about here.]\n[Fig. 4 about here.]\nFinally, we verified the MFT [15] and EFT [18] predictions about the\nmulticompensation behavior of the system. Namely, these theories predict\nthe existence of two compensation points at the anisotropy values close to\nD/|J|=−1.96. Such a behavior was indeed confirmed in our MC simulations.\nIn Fig. 5 we plot the total magnetization vs temperature curves fo r several\nvalues of the anisotropy strength d≡D/|J|. While for d1=−1.95 there\nis no compensation point, for the values of d2=−1.958 and d3=−1.97,\nthere are two compensation points below the respective critical te mperatures\ntc2(≡kBTc2/|J|)andtc3. Ford4=−1.98, thereis onlyonecompensation point\nbelowtc4. The entire curve of the compensation temperature kBTk/|J|as a\nfunction of the anisotropy strength D/|J|is shown in the inset of Fig. 1.\n6[Fig. 5 about here.]\n4 Conclusions\nWe have studied the critical and compensation properties of the mix ed\nspin-1 and spin-3 /2 Ising ferrimagnet with a uniform single-ion anisotropy on\na square lattice by Monte Carlo simulations. We focused on several c ontra-\ndictory results previously obtained by different approximative appr oaches. In\nparticular, we checked if the system for sufficiently large anisotrop y strength\ncan display a reentrant phenomenon and additional first-order ph ase transi-\ntion within the ordered ferrimagnetic phase, as predicted by CVT. N either\nof these predictions was confirmed. Instead, we found just one o rder-disorder\nphase boundary as a single-valued function of the anisotropy within the entire\nrange of values, in agreement with the MFT and EFT results. Thus, w hile\nfor the system on a cubic lattice the qualitative features predicted by CVT\nwere confirmed by the MC simulations [22], they were not confirmed on a\nsquare lattice. On the other hand, our findings neither support th e MFT and\nEFT predictions about the existence of a tricritical point at which th e transi-\ntion would change to the first order one. Nevertheless, the multico mpensation\nbehavior with two compensation points observed within the MFT and E FT\nstudies was reliably verified in the current MC simulations.\nAcknowledgments\nThis work was supported by the Scientific Grant Agency of Ministry o f Edu-\ncation of Slovak Republic (Grant VEGA No. 1/0128/08).\nReferences\n[1] T. Iwashita, N. Uryu, J. Phys. Soc. Japan 53 (1984) 721.\n[2] H.F. Verona de Resende, F.C. S´ aBarreto, J.A. Plascak, P hysica A 149 (1988)\n606.\n[3] O. Kahn, Molecular Magnetism, VCH, New York, 1993.\n[4] T. Kaneyoshi, Y. Nakamura, J. Phys.: Condens. Matter 10 ( 1998) 3003.\n[5] T. Kaneyoshi, Y. Nakamura, S. Shin,J. Phys.: Condens. Ma tter 10 (1998) 7025.\n[6] L.L. Goncalves, Phys. Scripta 32 (1985) 248.\n7[7] A. Lipowski, T. Horiguchi, J. Phys. A: Math. Gen. 28 (1995 ) L261.\n[8] M. Jaˇ sˇ cur, Physica A 252 (1998) 217.\n[9] A. Dakhama, Physica A 252 (1998) 225.\n[10] M. Jaˇ sˇ cur, J. Streˇ cka, Condens. Matter Phys. 8 (2005 ) 869.\n[11] G.M. Zhang, C.Z. Yang, Phys. Rev. B 48 (1993) 9452.\n[12] G.M. Buendia, M. Novotny, J. Phys.: Condens. Matter 9 (1 997) 5951.\n[13] M. Godoy, W. Figueiredo, Physica A 339 (2004) 392.\n[14] W. Selke, J. Oitmaa, J. Phys.: Condens. Matter 22 (2010) 076004.\n[15] O.F. Abubrig, D. Horv´ ath, A. Bob´ ak, M. Jaˇ sˇ cur, Phys ica A 296 (2001) 437.\n[16] A. Bob´ ak, Physica A 258 (1998) 140.\n[17] A. Bob´ ak, Physica A 286 (2000) 531.\n[18] A. Bob´ ak, O.F. Abubrig, D. Horv´ ath, J. Magn. Magn. Mat er. 246 (2002) 177.\n[19] J.W. Tucker, J. Magn. Magn. Mater. 237 (2001) 215.\n[20] T. Kaneyoshi, J.C. Chen, J. Magn. Magn. Mater. 98 (1991) 201.\n[21] A. Bob´ ak, M. Jurˇ ciˇ sin, Physica A 240 (1997) 647.\n[22] Y. Nakamura, J.W. Tucker, IEEE Trans. Magn. 38 (2002) 24 06.\n[23] A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61 (19 88) 2635.\n[24] A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63 (19 89) 1195.\n8List of Figures\n1 The critical temperature kBTc/|J|as a function of the\nsingle-ion anisotropy strength D/|J|. The inset shows the\nenlarged picture in a low-temperature region with both the\ncritical (kBTc/|J|) and compensation ( kBTk/|J|) temperature\ncurves. 10\n2 (a) The internal energy eand specific heat c(inset) per\nsite and (b) the staggered magnetization msand staggered\nsusceptibility χMs(inset) per site, as functions of the\nanisotropy strength D/|J|, atkBT/|J|= 0.1. 11\n3 Scaling behavior of the maxima of the quantities (7) −(9)\nfor the parameter O=Msi.e.,χMs,max,D1Ms,max\nandD2Ms,max, in a log-log plot at the critical point\n(Dc/|J|,kBTc/|J|) = (−1.9975,0.1). The slopes represent the\nvalues of the standard 2D Ising exponents ratios γI/νI= 7/4\nand 1/νI= 1. 12\n4 The minimum of the fourth-order energy cumulant Vas a\nfunction of L−2. The solid horizontal line denotes V∗= 2/3. 13\n5 The total magnetization vs temperature curves for several\nvalues of the anisotropy strength d≡D/|J|. There is no\ncompensation point for d1=−1.95, two compensation points\nford2=−1.958 andd3=−1.97 and one compensation point\nford4=−1.98.tci≡kBTci/|J|,i= 1,2,3,4, denote the critical\ntemperatures corresponding to the respective values of di. 14\n9−2 −1 0 100.511.522.53\nD/|J|kBTc/|J|\n00.20.40.60.8\n−2−1.98−1.96\nD/|J|kBTc/|J|\nkBTk/|J|\nFig. 1. The critical temperature kBTc/|J|as a function of the single-ion anisotropy\nstrength D/|J|. The inset shows the enlarged picture in a low-temperature r egion\nwith both the critical ( kBTc/|J|) and compensation ( kBTk/|J|) temperature curves.\n10−2−1.995 −1.99−1.985 −1.98−1.5−1−0.50\nD/|J|e\n−2−1.995−1.9901000200030004000500060007000\nD/|J|c\n(a) \n−2−1.995 −1.99−1.985 −1.9800.20.40.60.81\nD/|J|ms\n−2−1.995−1.9905001000150020002500\nD/|J|χM\ns(b) \nFig. 2. (a) The internal energy eand specific heat c(inset) per site and (b) the\nstaggered magnetization msand staggered susceptibility χMs(inset) per site, as\nfunctions of the anisotropy strength D/|J|, atkBT/|J|= 0.1.\n112.533.544.555.523456789\nln(L)• ln(χM\ns,max), slope = 7/4 \n× ln(D1M\ns,max), slope = 1      \n♦ ln(D2M\ns,max), slope = 1\nFig. 3. Scaling behavior of the maxima of the quantities (7) −(9) for the parameter\nO=Msi.e.,χMs,max,D1Ms,maxandD2Ms,max, in a log-log plot at the critical point\n(Dc/|J|,kBTc/|J|) = (−1.9975,0.1). The slopes represent the values of the standard\n2D Ising exponents ratios γI/νI= 7/4 and 1/νI= 1.\n1200.511.522.5\nx 10−30.6580.660.6620.6640.6660.668\nL−2Vmin\nFig. 4. The minimum of the fourth-order energy cumulant Vas a function of L−2.\nThe solid horizontal line denotes V∗= 2/3.\n130.20.30.40.50.6−0.01−0.0050  0.005\nkBT/|J|m\ntc3 tc2 d1=−1.95 \nd2=−1.958 \nd3=−1.97 \ntc4 \nd4=−1.98 \nFig. 5. The total magnetization vs temperature curves for se veral values of the\nanisotropy strength d≡D/|J|. There is no compensation point for d1=−1.95,\ntwo compensation points for d2=−1.958 and d3=−1.97 and one compensation\npoint for d4=−1.98.tci≡kBTci/|J|,i= 1,2,3,4, denote the critical temperatures\ncorresponding to the respective values of di.\n14"    },    {        "title": "2109.12377v2.Phase_transitions_in_rare_earth_ferrimagnets_with_surface_anisotropy_near_the_magnetization_compensation_point.pdf",        "content": "Phase transitions in rare-erth ferrimagnets with surface anisotropy near the\nmagnetization compensation point\nV.V. Yurlov,1, 2K.A. Zvezdin,2, 3,\u0003and A.K. Zvezdin2, 3\n1Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia\n2New Spintronic Technologies, Russian Quantum Center,\nBolshoy Bulvar 30, bld. 1, 121205 Moscow, Russia\n3Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia\n(Dated: December 13, 2022)\nWe report of a theoretical model for calculating the H-T phase diagrams of a rare-earth ferri-\nmagnet, taking into account anisotropies originated by both magnetization sublattices' and by the\nsurface. The possibility of an exchange spring formation due to surface anisotropy is considered.\nThis situation is realized in heterostructures containing a ferrimagnet and a heavy metal. We derive\nthe stability lose lines of the collinear phase from the free energy of the two sublattice ferrimagnet.\nWe numerical calculate the magnetic phase diagrams for the cases when the magnetic \feld applied\nalong and perpendecular to the easy axis. We demonstrate that tricritical point down at the low \feld\nrange due to surface anisotropy e\u000bect. Moreover, the line of the \frst order phase transition between\nangular and collinear phases reduces due to surface anisotropy. In the case when magnetic \feld is\napplied perpendicular to the easy axis we show the possibility of the \frst order phase transition\nbetween two collinear phases in contrast to the phase diagram without surface anisotropy.\nI. INTRODUCTION\nRare-earth-transition metal (RE-TM) compounds is\na class of magnetic materials that attracts particu-\nlar attention in a wide range of di\u000berent areas such\nas spintronics[1, 2], optospintronics[3] and ultrafast\nmagnetism[4]. This rare-earth ferrimagnetic (FiM) thin\n\flms can be applied for technological uses such as ul-\ntrafast memory devices[5] or high density recording[6{\n8]. Depending on the composition, FiM \flms may have\nthe magnetization compensation point TMwhere the an-\ntiferromagneticaly coupled RE and TM magnetizations\ncompensate each other[9]. This point plays an impor-\ntant role for studying the magnetic phase transitions or\nmagnetization dynamics of the FiM \flms[10{12].\nStudying of the magnetic phase diagrams[13] is partic-\nular of a interest for a better understanding the magne-\ntization dynamics in ferrimagnets. Recent experiments\nwith ferrimagnets such as GdFeCo, GdCo and TbFe\ndemonstrate anomalous hysteresis loops near the mag-\nnetization compensation point[14{17]. In particular, in\nthe GdFeCo ferrimagnet, triple hysteresis loops are ob-\nserved above the magnetization compensation tempera-\nture [14]. At the same time, experiments with TbFeCo\nwith Ta capping layer[18] show that triple loops can ap-\npear to the left of the compensation point. To explain\nthis anomalous hysteresis loop, theoretical models[19]\nwere constructed, in which the interplay of the surface\nanisotropy, and anisotropies of both sublattices led to a\nmodi\fcation of the phase diagrams. The thickness and\n\fnite size of a ferrimagnetic \flm can also signi\fcantly\na\u000bect the spin-reorientation transitions and change the\nphase diagram. Theoretical [20, 21] and experimental[22]\n\u0003zvezdin.ka@phystech.edustudies on the e\u000bect of the surface on the spin dynamics\nwere carried out for example for nanowires[23] and for\nvarious ferrimagnetic materials[24, 25]. However, given\nthe new experimental and theoretical results, this area\nrequires further study, and the in\ruence of the surface\ne\u000bects on the FiM phase diagram deserves particular at-\ntention.\nIn this work we study the magnetic phase diagram for\nthe FiM layer taking into account anisotropies of the both\nmagnetization sublattices and the surface anisotropy. We\nderive the stability lose lines of the collinear phase from\nthe free energy of the two sublattice ferrimagnet. In-\n\ruence of the surface exchange anisotropy can be taken\ninto account by introducing the dimensionless parameter\nwhich modi\fes the e\u000bective anisotropy of the ferrimag-\nnet. We numerical calculate the magnetic phase diagram\nfor two di\u000berent directions of the external magnetic \feld\nand show the lines of the second and the \frst order phase\ntransitions. We demonstrate that surface anisotropy can\nshift down at lower \feld range the tricritical point and\nreduce the line of the \frst order phase transition between\nangular and collinear phases. In the case when magnetic\n\feld is applied perpendicular to the easy axis we show\nthat the surface anisotropy enables the \frst order phase\ntransition between two collinear phases.\nII. MODEL AND BASIC EQUATIONS\nTo obtain the magnetic phase diagram for ferrimagnets\nwith the surface anisotropy we use the e\u000bective thermo-\ndynamic potential[13]. We assume that transition metal\n(d-sublattice) is saturated due to large d-d interactions\n(of the order of 106\u0000107Oe) and rare-earth (f-sublattice)\nis considered as a paramagnetic in the e\u000bective mag-\nnetic \feld. The applicability of this model is substanti-\nated by the hierarchy of exchange interactions[26]. Thus,arXiv:2109.12377v2  [physics.app-ph]  12 Dec 20222\nthe e\u000bective thermodynamic potential without surface\nanisotropy can be written as:\n\b =\u0000MdH\u0000ZHeff\n0Mf(x)dx\u0000Ka+ \bex;(1)\nwhere Mdis magnetization of the transition metal sub-\nlattice, Mfis magnetization of the rare-earth ions which\nare saturated in the e\u000bective \feld Heff=H\u0000\u0015Md,\n\u0015is the f-d exchange constant, His external magnetic\n\feld,Kais magnetic anisotropy energy and \b exis non-\nuniform exchange energy. The magnetization of the f-\nsublattice can be described by the Brillouin function\nMf(x) =\u0016BgJBJ\u0010gJ\u0016Bx\nkT\u0011\n, wheregis Lande g-factor,\nJis total angular momentum of the rare-earth ions, \u0016B\nis Bohr magneton. The anisotropy energy of the ferri-\nmagnet is[14, 19]:\nKa=\u0000Kdsin2 \u0000Kf\u0010\u0015Mdsin \nHeff( )\u00112\n; (2)\nwhereKdandKfare unaxial anisotropy constants of\nd- and f- sublattices, respectively,  is an angle be-\ntween magnetization of the d-sublattice and the easy\nmagnetization axis. Non-uniform exchange energy can\nbe written as \b ex=A(r )2whereAis the ex-\nchange sti\u000bness constant. Now we should take into ac-\ncount the surface e\u000bect as induced exchange magnetic\nanisotropy. This energy takes the form Fs=kdsin2 s+\nkf(\u0015Mdsin s)2=H2\neff( s), where sis magnitude of the\nangle on the surface of the \flm, kdandkfconstants\nof the d- and f- sublattice surface magnetic anisotropy.\nFurther expressions is written in the assumption that\nmagnetic \flm is homogeneous in the plane of the \flm.\nWe consider the \flm with thickness jzj< d. Exchangesurface anisotropy is equal at the edges of the \flm\nkd(\u0000d) =kd(d) andkf(\u0000d) =kf(d) which means that\nthe symmetrical arrangement of magnetization is most\nbene\fcial. As a result we can say that d =dzj0= 0.\nFinaly, we obtain the free energy of the ferrimagnet by\nintegrating (1) over the \flms volume\nF=Zd\n0\bdz+Fs=Zd\n0n\nA(r )2\u0000MdH\u0000\nZHeff\n0Mf(x)dx+Kdsin2 +Kf\u0010\u0015Mdsin \nHeff( )\u00112o\ndz\n+kdsin2 s+kf\u0010\u0015Mdsin s\nHeff( s)\u00112\n:\n(3)\nEquation (3) will be used below to construct the magnetic\nphase diagram.\nIII. MAGNETIC PHASE DIAGRAM\nWe obtain here the magnetic phase diagram of ferri-\nmagnetic material with surface anisotropy using the free\nenergy (3). We consider two cases with di\u000berent direc-\ntions of the external magnetic \feld H: (a) when the mag-\nnetic \feld is applied along the easy axis and (b) magnetic\n\feld is perpendicular to the easy axis. For the case (a) we\nsuppose that  =\u0012and for the case (b) {  =\u0019=2\u0000\u0012,\nwhere\u0012is the angle between magnetization of the d-\nsublattice and external magnetic \flm. Without loss of\ngenerality, we consider the case (a) when the magnetic\n\feld is aligned with the easy axis. For the case (b), the\nconclusions given below are also valid.\nWe vary the functional (3) and obtain the Euler-\nLagrange equations and boundary conditions to study\nthe free energy:\n\u0001\u0012=MdH\n2Asin\u0012n\n1\u0000\u0015\u001f(\u0012) +Kf\u0010\u0015Md\nHeff(\u0012)\u001121\nMdH\u0010\n2 cos\u0012\u0000\u0015MdHsin2\u0012\nH2\neff(\u0012)\u0011\n+2Kd\nMdHcos\u0012o\n;\nd\u0012\ndz\f\f\f\f\ns=\u0000kf\n2A\u0010\u0015Md\nHeff(\u0012s)\u00112\nsin\u0012s\u0010\n2 cos\u0012s\u0000\u0015MdHsin2\u0012s\nH2\neff(\u0012s)\u0011\n\u00002kd\n2Asin\u0012scos\u0012s\nd\u0012\ndz\f\f\f\f\n0= 0;(4)\nwhere\u001f=Mf(\u0012)=Heff(\u0012), indexsin the second equa-\ntion de\fnes the \flm boundary. Note that the analytical\nsolution of these equations has some di\u000eculties associ-\nated with the de\fnition of the \frst integral of the di\u000ber-\nential equation. However, the e\u000bect of surface anisotropy\ncan be taken into account by considering the lines of sta-\nbility loss of the collinear phases. Having this is mind\nwe linearize the (4) near the lines of stability loss \u0012= 0\nand\u0012=\u0019. Let us give analytical expressions only forthe case\u0012= 0. We look for a solution of the linearized\nequations in the following form:\n\u0012=\u0012(z) expif{xx+{yyg; (5)\nwhere {{{is the vector lying in the plane of the magnetic\n\flm. We obtain second order di\u000berential equation for\neigenvalues of the Sturm{Liouville type. After some cal-\nculations we obtain a transcendental expression for the3\nH\nMf Md\nTM\nRP\nP'CB'A'\nC' E'A\nBH\nMf Md\nTMR'E\nC'H\nMf MdAB\nB'\nE'EA'\nCRP\nTMH\nMf Md(a)\n(b) (c)\nFIG. 1. (a) H\u0000Tphase diagram of the ferrimagnet in the\nmagnetic \feld applied along the easy axis in the high \feld\nrange; solid lines show the second order phase transition be-\ntween collinear and angular phases, the dotted lines show\nthe second order phase transition between collinear phases;\n\u0012is an angle between external magnetic \feld and magneti-\nzation of the d-sublattice. b) The zoomed-in phase diagram\nnear the tricritical points PandP0; lineTMR0Rshows the\n\frst order phase transition between collinear phases, lines RP\nandR0P0show the \frst order phase transition between an-\ngular and collinear phases. c) The zoomed-out phase dia-\ngram near the magnetization compensation temperature TM.\nLinesACandBEis the stability lose lines when the surface\nanisotropy is zero; lines A0C0andB0E0is the stability in the\npresence of the surface anisotropy. All diagrams constructed\nfor 0<heff<1,Keff>0,keff<0.\nvector {{{:\np\n{2d2+\u00142d2tanhf{2d2+\u00142d2g=\n\u0000d\nAn\nkf\u0010\u0015Md\nHeff(0)\u00112\n+kdo\n;(6)\nwhere\u00142=MdH\n2Af1\u0000\u0015\u001f(0) +2Kf\nMdH(\u0015Md\nHeff(0))2+2Kd\nMdHg.\nThe stability condition for collinear phases (when mag-\nnetizations of both sublattice are parallel and \u0012= 0 or\n\u0012=\u0019) is that the equation has no real solutions. Car-\nrying out similar reasoning for the \u0012=\u0019, we obtain the\nlines of stability loss of the collinear phases:\n(B0E0) : 1\u0000\u0015\u001f(0) +Keff(0)\nMdH(1\u0000heff) = 0;\n(A0C0) : 1\u0000\u0015\u001f(\u0019)\u0000Keff(\u0019)\nMdH(1\u0000heff) = 0;(7)\nPHMf Md\nTMHMf MdTM\nA\nCA'\nC'B\nEE'B'A\nCE'B' A'\nC'B\nE(a)\n(b)FIG. 2. a) H\u0000Tphase diagram of the ferrimagnet in the\nmagnetic \feld directed perpendecular to the easy axis near\nthe magnetization compensation point; solid lines show the\nsecond order phase transition between collinear and angular\nphases, the dotted lines show the second order phase transi-\ntion between collinear phases; lines ACandBEis the sta-\nbility lose lines when the surface anisotropy is zero; lines\nA0C0andB0E0is the stability in the presence of the sur-\nface anisotropy; this diagram is constructed for heff<\u00001,\nKeff>0,keff<0;\u0012is an angle between external magnetic\n\feld and magnetization of the d-sublattice. b) The magnetic\nphase diagram near the compensation temperature TM; dia-\ngram is constructed for heff>1,Keff>0,keff>0.\nwhereKeff=Kf(\u0015Md\nHeff)2+Kd,heffis the solution of the\nequationj\u000esj= (\u001b\nheff)1=2tanh\u00001(\u001b\nheff)(1=2)andkeff=\nkf(\u0015Md\nHeff)2+kd<0,\u001bis material surface parameter with4\npositive value, \u000es= (kf(\u0015Md\nHeff)2+kd)d=A. Similarly, it is\npossible to obtain lines of stability loss when the magnetic\n\feld is perpendicular to the easy axis of the ferrimagnet.\nFor this case, the stability loss lines will be written in the\nform:\n(B0E0) : 1\u0000\u0015\u001f(0)\u0000Keff(0)\nMdH(1 +heff) = 0;\n(A0C0) : 1\u0000\u0015\u001f(\u0019) +Keff(\u0019)\nMdH(1 +heff) = 0;(8)\nwhereheff is the solution of the equation \u000es=\n(\u001b\nheff)1=2tanh\u00001(\u001b\nheff)(1=2). Here we should note that\nthe present theory applicable in the microscopic range\nford\u001810\u00007\u000410\u00006m.\nRecent researches show that anisotropy of the RE\nsublattice can be larger then the one of the TM\nsublattice[19]. Therefore, let us investigate how lines of\nthe \frst and second phase transitions are changed due\nto the exchange surface anisotropy. By using the equa-\ntions (7), Euler-Lagrange equations (4), free energy (3)\nand methods which are described in [13] we calculate nu-\nmerically the magnetic phase diagram of ferrimagnetic\n\flm with the exchange surface anisotropy (see in Fig. 1\nand Fig. 2). For the calculations we use the GdFeCo\nparameters: Md(0) = 4:5\u0016B=f:u: ,Mf(0) = 7\u0016B=f:u: ,\nKd= 0:1\u0001105erg=cc ,Kf= 0:9\u0001105erg=cc ,Hex=\n\u0015Md\u0018106Oe,TM\u0019263K. Magnetic phase diagrams\nin Fig. 1 and Fig. 2 represent the areas of the collinear\nphase where \u0012= 0 (purple area in Fig. 1 and Fig. 2),\n\u0012=\u0019(green area in Fig. 1 and Fig. 2) and noncollinear\nphase\u0012=\u0012(T;H) (yellow area in Fig. 1 and Fig. 2).\nBlue are in Fig. 1 and Fig. 2 shows the di\u000berent between\ncollinear and noncollinear phases for the ferrimagnet with\nand without e\u000bect of the surface anisotropy.\nIV. RESULTS AND DISCUSSION\nMagnetic phase diagram in Fig. 1 and Fig. 2 demon-\nstrate three di\u000berent phases of the ferrimagnetic \flm.\nThis phases are: the collinear phase at the high tem-\nperature range ( \u0012= 0 purple area), the collinear phase\nat the low temperature range ( \u0012=\u0019green area) and\nthe noncollinear phase \u0012=\u0012(T;H) which is indicated as\nyellow area.\nLet us discuss now the case (a) when external magnetic\n\feld is co-directed with easy axis of ferrimagnet. Fig.\n1(a) show the phase diagram at the high \feld range. Dot-\nted lines show the second order phase transition between\ntwo collinear phases. The zoomed area of the diagram\nin Fig. 1(a) is shown in the Fig. 1(c). Lines ACand\nBEshow the lines of a second-order phase transition be-\ntween collinear and non-collinear phases. Note that the\nsolid lines denote the second-order phase transition be-\ntween the collinear and the angular phases. The dotted\nlines indicate the phase transition between two collinear\nphases. The grey doted line in Fig. 1(a) demonstratethe situation when Heff(T) = 0. These lines ( ACand\nBE) demonstrate the case when the surface anisotropy is\nzero (heff= 0). If the surface anisotropy a\u000bects the mag-\nnetic system, than the e\u000bective anisotropy Keffchanges,\nas follows from the (7) and the ACandBElines turn\ninto theA0C0andB0E0. In Fig. 1(b) and Fig. 1(c), the\nblue color indicates the di\u000berence between two cases de-\nscribed above. Fig. 1(b) and Fig. 1(c) show that surface\nanisotropy plays a signi\fcant role near low values of the\nmagnetic \feld H\u0003\u0018(2Keff\u0015)1=2. With an increase of\nthe magnitude of the magnetic \feld, the lines AC,BE\nandA0C0,B0E0quickly approach to each other. Fig. 1 is\nplotted for the heff\u00180:5. TheTMR0Rline in Fig. 1(b)\nshows a \frst-order phase transition line between collinear\nphases whereF(0) =F(\u0019). TheRPline is the \frst-order\nphase transition line between the angular and collinear\n\u0012= 0 phases. Pis the tricritical point. Note that this\nline is located to the right of the magnetization compen-\nsation point TMdue to the in\ruence of the anisotropy\nof the rare-earth sublattice. Note, that tricritical point\nPmay located to the left from the compensation due to\nmodifying the surface of the ferrimagnetic by the heavy\nmetal \flm such as Ta[18]. The line RPtransforms into\nanR0P0due to the exchange surface anisotropy. The\n\frst-order phase transition between collinear and non-\ncollinear phases is reduced under the in\ruence of surface\nanisotropy. Thus, the regions of phase transitions near\nthe compensation point can change signi\fcantly due to\nsurface anisotropy.\nThe similar situation realises for the case (b)when the\nmagnetic \feld is perpendicular to the easy axis of the\nferrimagnet. Fig. 2(b) shows that the angular phase\nexpands due to surface anisotropy when the heff>0.\nHowever, the most interesting e\u000bect can be seen if the\nheff<0. In this case, the spins are pinned on the surface\nof the ferrimagnet. As a result, the transition between\ncollinear phases in the low-\feld region can occur through\nthe \frst order phase transition. This e\u000bect is demon-\nstrated in the Fig. 2(a). In the case described above,\nthe linesACandBEturns intoA0C0andB0E0. It also\nshould be noted, that if the anisotropy of the d-subluttice\nis higher than the one of the f-sublattice than the \frst\norder phase transition between the collinear phase \u0012=\u0019\nand the angular phase is possible because tricritical point\nin this case is lower than magnetization compensation\ntemperature.\nV. CONCLUSION\nThe magnetic phase diagram are studied for the\nGdFeCo ferrimagnet in presence of the surface magnetic\nanisotropy for the two di\u000berent cases: magnetic \feld ap-\nplied along and perpendecular to the easy axis. The sta-\nbility lose lines are derived from the free energy of the fer-\nrimagnet in the assumption that f-sublattice anisotropy\nis larger than d-sublattice one. In this particular case\nthe tricritical point lies above the compensation temper-5\nature. We numerical calculate the phase diagram and\nshow the lines of the second and \frst order phase tran-\nsition. We show that in the case when magnetic \feld is\nalong the easy axis the stability lose lines and tricriti-\ncal point are falling down in the low \feld range due to\nsurface anisotropy. Moreover, the area of the \frst order\nphase transition between angular and colliniar phase nar-\nrows due to surface e\u000bects. In other case, when the mag-\nnetic \feld is perpendecular to the easy axis we show thepossibility of the realization the \frst order phase transi-\ntion between collinear phases due to surface anisotropy.\nFor the both cases which are described above the sur-\nface anisotropy change phase diagram signi\fcantly only\nin the low magnetic \feld. In the high \feld range the sta-\nbility lose lines for the cases with surface anisotropy and\nwithout one fast approach to each to each other. This\n\fndings may be useful for theoretical and experimental\nstudy of the spin reorientation transitions. This research\nhas been supported by RSF grant No. 22-12-00367.\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] S. Bader and S. Parkin, Annual Review\nof Condensed Matter Physics 1, 71 (2010),\nhttps://doi.org/10.1146/annurev-conmatphys-070909-\n104123.\n[3] T. J. Huisman, C. Ciccarelli, A. Tsukamoto,\nR. V. Mikhaylovskiy, T. Rasing, and A. V. Kimel,\nApplied Physics Letters 110, 072402 (2017),\nhttps://doi.org/10.1063/1.4976202.\n[4] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod.\nPhys. 82, 2731 (2010).\n[5] R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert,\nS. Salahuddin, and J. Bokor, Phys. Rev. B 95, 180409\n(2017).\n[6] A. Fert and L. Piraux, Journal of Magnetism and Mag-\nnetic Materials 200, 338 (1999).\n[7] G. Srinivasan, B. U. M. Rao, J. Zhao, and M. S.\nSeehra, Applied Physics Letters 59, 372 (1991),\nhttps://doi.org/10.1063/1.105462.\n[8] H. Tabata and T. Kawai, Applied Physics Letters 70, 321\n(1997), https://doi.org/10.1063/1.118202.\n[9] R. Moreno, T. A. Ostler, R. W. Chantrell, and\nO. Chubykalo-Fesenko, Phys. Rev. B 96, 014409 (2017).\n[10] V. Kimel, A and M. Li, Nat Rev Mater 4, 189{200 (2019),\nhttps://doi.org/10.1038/s41578-019-0086-3.\n[11] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen,\nA. Kirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99,\n217204 (2007).\n[12] V. V. Yurlov, K. A. Zvezdin, G. A. Kichin, M. D. Davy-\ndova, A. E. Tseplina, N. T. Hai, J.-C. Wu, S.-Z. Ciou,\nY.-R. Chiou, L.-X. Ye, T.-H. Wu, R. C. Bhatt, and A. K.\nZvezdin, Applied Physics Letters 116, 222401 (2020),\nhttps://doi.org/10.1063/5.0010687.[13] A. Zvezdin (Elsevier, 1995) pp. 405{543.\n[14] J. Becker, A. Tsukamoto, A. Kirilyuk, J. C. Maan,\nT. Rasing, P. C. M. Christianen, and A. V. Kimel, Phys.\nRev. Lett. 118, 117203 (2017).\n[15] K. Okamoto and N. Miura, Physica B: Condensed Matter\n155, 259 (1989).\n[16] T. Chen and R. Malmh all, Journal of Magnetism and\nMagnetic Materials 35, 269 (1983).\n[17] Scienti\fc Reports 5, 18377 (2015).\n[18] M. Davydova, P. Skirdkov, K. Zvezdin, J.-C. Wu, S.-\nZ. Ciou, Y.-R. Chiou, L.-X. Ye, T.-H. Wu, R. C. Bhatt,\nA. Kimel, and A. Zvezdin, Phys. Rev. Applied 13, 034053\n(2020).\n[19] M. D. Davydova, K. A. Zvezdin, J. Becker, A. V. Kimel,\nand A. K. Zvezdin, Phys. Rev. B 100, 064409 (2019).\n[20] G. Sayko, S. Utochkin, and A. Zvezdin, Journal of Mag-\nnetism and Magnetic Materials 113, 194 (1992).\n[21] W. Jiang, J. nan Chen, B. Ma, and Z. Wang, Physica\nE: Low-dimensional Systems and Nanostructures 61, 101\n(2014).\n[22] S.-h. Noh, W. Na, J.-t. Jang, J.-H. Lee, E. J. Lee,\nS. H. Moon, Y. Lim, J.-S. Shin, and J. Cheon,\nNano Letters 12, 3716 (2012), pMID: 22720795,\nhttps://doi.org/10.1021/nl301499u.\n[23] W. Jiang, F. Zhang, X.-X. Li, H.-Y. Guan, A.-B. Guo,\nand Z. Wang, Physica E: Low-dimensional Systems and\nNanostructures 47, 95 (2013).\n[24] J. Slonczewski, Journal of Magnetism and Magnetic Ma-\nterials 117, 368 (1992).\n[25] K. Zhang and D. R. Fredkin, Jour-\nnal of Applied Physics 79, 5762 (1996),\nhttps://aip.scitation.org/doi/pdf/10.1063/1.362180.\n[26] A. Zvezdin, V. M. Matveev, A. A. Mukhin, and A. I.\nPopov (Nauka, Moscow, 1985)."    },    {        "title": "2312.01630v2.Ground_State_Phase_Diagram_of__1_2_1_2_1__Mixed_Diamond_Chains.pdf",        "content": "arXiv:2312.01630v2  [cond-mat.str-el]  2 Mar 2024Journal of the Physical Society of Japan FULL PAPERS\nGround-State Phase Diagram of (1/2,1/2,1) Mixed Diamond Ch ains\nKazuo Hida∗\nProfessor Emeritus, Division of Material Science, Graduat e School of Science and Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n(Received )\nThe ground-state phases of mixed diamond chains with ( S,τ(1),τ(2)) = (1/2,1/2,1), where Sis the\nmagnitude of vertex spins, and τ(1)andτ(2)are those of apical spins, are investigated. The two apical\nspins in each unit cell are coupled by an exchange coupling λ. The vertex spins are coupled with the top\nand bottom apical spins by exchange couplings 1 + δand 1−δ, respectively. Although this model has\nan infinite number of local conservation laws for δ= 0, they are lost for finite δ. The ground-state phase\ndiagram is determined using the numerical exact diagonaliz ation and DMRG method in addition to the\nanalytical approximations in various limiting cases. The p hase diagram consists of a nonmagnetic phase\nand several kinds of ferrimagnetic phases. We find two differe nt ferrimagnetic phases without spontaneous\ntranslational symmetry breakdown. It is also found that the quantized ferrimagnetic phases with large\nspatial periodicities present for δ= 0 are easily destroyed by small δand replaced by a partial ferrimagnetic\nphase. The nonmagnetic phase is considered to be a gapless To monaga-Luttinger liquid phase based on the\nrecently extended Lieb-Schultz-Mattis theorem to the site -reflection invariant spin chains and numerical\ndiagonalization results.\n1. Introduction\nIn low-dimensional frustrated quantum magnets, the\ninterplay of quantum fluctuation and frustration leads\nto the emergence of various exotic quantum phases.1,2)\nThe conventional diamond chain3,4)is known as one of\nthe simplest examples in which an interplay of quan-\ntum fluctuation and frustration leads to a wide variety of\nground-state phases. Remarkably, this model has an in-\nfinite number of local conservation laws, and the ground\nstates can be classified by the corresponding quantum\nnumbers. If the two apical spins have equal magnitudes,\nthe pair of apical spins in each unit cell can form a non-\nmagnetic singlet dimer and the ground state is a di-\nrect product of the cluster ground states separated by\nsinglet dimers.3,4)Nevertheless, in addition to the spin\ncluster ground states, various ferrimagnetic states and\nstrongly correlated nonmagnetic states such as the Hal-\ndane state are also found when the apical spins form\nmagnetic dimers. In these cases, all the spins collectively\nform a correlated ground state over the whole chain.\nIn the presence of various types of distortion, the spin\ncluster ground states also turn into highly correlated\nground states. Extensive experimental studies have been\nalso carried out on the magnetic properties of the nat-\nural mineral azurite which is regarded as an example of\ndistorted spin-1/2 diamond chains.5,6)\nOn the other hand, the cases with unequal apical spins\narelessstudied. In this case,the apicalspins cannotform\na singlet dimer. Hence, all spins in the chain inevitably\nform a many-body correlated state. As a simple example\n∗E-mail address: hida@mail.saitama-u.ac.jpof such cases, we investigated the mixed diamond chain\nwith apical spins of magnitude 1 and 1/2, and vertex\nspins, 1/2 in Ref. 7 assuming that the exchange inter-\nactions between the vertex spins and two apical spins\nare equal to each other. In the absence of coupling λbe-\ntween two apical spins, we found a quantized ferrimag-\nnetic (QF) phase with the spontaneous magnetization\nper unit cell mspquantized to unity as expected from\nthe Lieb-Mattis (LM) theorem.8)With the increase of λ,\nwe found an infinite series of QF phases with msp= 1/p,\nwherepis a positive integer (1 ≤p <∞) that increases\nwithλ. Finally, the nonmagnetic Tomonaga-Luttinger\nliquid (TLL) phase sets in at a critical value of λ=λc.\nThe width and spontaneous magnetization of each QF\nphase tend to infinitesimal as λapproaches λc.\nIf the two apical spins have different magnitudes, how-\never, it is natural to assume that the exchange interac-\ntions between these two kinds ofapicalspins and the ver-\ntex spins are also different. We examine this case in the\npresent work.Two QF phaseswithout translationalsym-\nmetry breakdown are found. The QF phases with large\npare replaced by a partial ferrimagnetic (PF) phase in\nwhich the magnetization varies continuouslywith the ex-\nchange parameter. With the help of numerical diagonal-\nization results, the nonmagnetic phase is considered to\nbe a TLL phase consistent with the Lieb-Schultz-Mattis\n(LSM) theorem9–13)that is recently extended to site-\nreflection invariant spin chains.11–13)\nThispaperisorganizedasfollows.InSect.2,themodel\nHamiltonian is presented. In Sect. 3, various limiting\ncases are examined analytically. In Sect. 4, the classical\nground state is analytically determined. In Sect. 5, the\n1J. Phys. Soc. Jpn. FULL PAPERS\nground-statephasediagramdeterminedbythenumerical\ncalculation is presented and the properties of each phase\nare discussed. The last section is devoted to a summary\nand discussion.\n2. Model\nWe investigate the ground-state phases of mixed dia-\nmond chains described by the following Hamiltonian:\nH=L/summationdisplay\nl=1/bracketleftBig\n(1+δ)Sl(τ(1)\nl+τ(1)\nl−1)\n+(1−δ)Sl(τ(2)\nl+τ(2)\nl−1)+λτ(1)\nlτ(2)\nl/bracketrightBig\n,(1)\nwhereSl,τ(1)\nlandτ(2)\nlare spin operators with magni-\ntudesSl=τ(1)\nl= 1/2 andτ(2)\nl= 1, respectively. The\nnumber of unit cells is denoted by L, and the total num-\nber of sites is 3 L. The lattice structure is depicted in Fig.\n1. We consider the region λ≥0 and 1 ≥δ≥ −1. For\nSl1+δ\n1−δ1+δ\n1−δτl(1)\nSl+1\nτl(2)S=τ(1)=1/2\nτ(2)=1λ\nFig. 1. Structure of the diamond chain investigated in this work.\nδ= 0, (τ(1)\nl+τ(2)\nl)2commutes with the Hamiltonian (1)\nfor alll. In Ref. 7, we made use of this property to de-\ntermine the ground-state phase diagram. In the present\nwork, we examine the general case of δ/negationslash= 0.\n3. Analytical Results\n3.1 Lieb-Mattis theorem\nWe start with several limiting cases where we can ex-\namine the ground state analytically. Before going into\nthe discussion of specific cases, we briefly introduce the\nLieb-Mattis theorem which is useful to determine the\nspontaneous magnetization in the absence of frustration.\nLet us consider a Heisenberg model\nHH=/summationdisplay\ni,jJijSiSj (2)\ndefined on a lattice consisting of two sublattices A and\nB. Here, Siis the spin operator on the i-th site with\narbitrary magnitude. It is assumed that the magnetic\ninteractions Jijsatisfy the following condition:\n(1) Iftwospinsareondifferentsublattices,then Jij≥0.\n(2) If twospins areon the same sublattice, then Jij≤0.\n(3) All spins are connected by magnetic interaction.\nThese assumptions imply the absence of frustration.\nThen, the spontaneous magnetization Mspof the ground(c) TLL(a) LM1\n(b) LM2\n(d) p=2(b’) LM21+δ\nλ\n1−δ\n1+δλ\n1−δ1/2⇑⇑ ⇑1+δ\nλ\n1−δ\n1+δ\nλ\n1−δ⇑ ⇑1+δλ\n1−δ1/2⇑⇑ ⇑\nFig. 2. Schematic spin configurations in the ground-state\nphases. The ovals in (b’), (c), and (d) mean that the two spins\nin an oval form a spin-doublet state. The open arrows express the\ntotal spins Tlin ovals. In (c), the antiferromagnetic long-range or-\nder depicted in the figure melts due to the quantum fluctuation\nresulting in the nonmagnetic TLL.\nstate is given by Msp=|SA−SB|, whereSAandSB\nare the sums of the magnitudes of the spins on the A-\nsublattice and B-sublattice, respectively.8)\n3.2λ= 0\nIf−1< δ <1, the system is unfrustrated and the\nground state is the QF phase with msp= 1 according\nto the LM theorem.8)In this case, the sublattice A con-\nsists of the sites occupied by Sl, and the sublattice B,\nthose occupied by τ(1)\nlandτ(2)\nl. Hence, SA=L/2 and\nSB= 3L/2 which gives Msp=Landmsp=Msp/L= 1.\nThe numerical analysis in Sect. 5 shows that this phase\nsurvives even in the weakly frustrated regime of small λ.\nHereafter, this phase is called the LM1 phase. Schematic\nspin configuration is presented in Fig. 2(a).\n3.3δ= 1\nIfλ >0, the ground state is the QF phase with\nmsp= 1 according to the LM theorem. In this case, the\nsublattice A consists of the sites occupied by τ(1)\nl, and\nthe sublattice B, those occupied by Slandτ(2)\nl. Hence,\nSA=L/2 andSB= 3L/2 which gives Msp=Land\nmsp=Msp/L= 1. The numerical analysis in Sect. 5\nshows that this phase survives even in the weakly frus-\ntrated regime of small 1 −δ. Hereafter, this phase is\ncalled the LM2 phase. The schematic spin configuration\npresented in Fig. 2(b) demonstrates that this phase is\ndistinct from the LM1 phase. This is also numerically\nconfirmed in Sect. 5.\n2J. Phys. Soc. Jpn. FULL PAPERS\n3.4λ≫1+δ,1−δ\nIn the limit λ→ ∞, all pairs of τ(1)\nlandτ(2)\nlform\ndoublet states ||Tl,Tz\nl/angbracketrightwithTl= 1/2 andTz\nl=±1/2\nwhereTl≡τ(1)\nl+τ(2)\nl. They are expressed using the\nbasis/vextendsingle/vextendsingle/vextendsingleτ(1)z\nl,τ(2)z\nl/angbracketrightBig\nas\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,1\n2/angbracketrightbigg\n=/radicalbigg\n1\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,1\n2/angbracketrightbigg\n−/radicalbigg\n2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1,−1\n2/angbracketrightbigg\n,(3)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,−1\n2/angbracketrightbigg\n=/radicalbigg\n1\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,−1\n2/angbracketrightbigg\n−/radicalbigg\n2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1,1\n2/angbracketrightbigg\n.(4)\nFor large but finite λ,TlandSl′(l′=lorl+1) are cou-\npled by an effective Heisenberg interaction Jeff. IfJeff\nis antiferromagnetic, the ground state is the nonmag-\nnetic TLL state as schematically shown in Fig. 2(c). If it\nis ferromagnetic, the ground state is the QF state with\nmsp= 1 as shown in Fig. 2(b’). This ground state con-\nfiguration is continuously deformed to that of the LM2\nphase depicted in Fig. 2(b) by reducing the coefficient\nof|0,1/2/angbracketrightin Eq. (3) continuously. Hence, this QF state\nalso belongs to the LM2 phase. We estimate Jeffwithin\nthe first order perturbation calculation with respect to\n(1±δ)/λas\nJeff=1\n3(3−5δ). (5)\nHence, we find the phase boundary between the TLL\nphase and the LM2 phase is given by δ= 0.6 for large\nenoughλ.\n3.5δ=−1\nThe system is unfrustrated and the ground state is the\nnonmagnetic phase with msp= 0 according to the LM\ntheorem. In this case, the sublattice A consists of the\nsites occupied by τ(2)\nl, and the sublattice B, those occu-\npied bySlandτ(1)\nl. Hence, SA=LandSB=Lwhich\ngivesMsp= 0 and msp=Msp/L= 0. According to the\nnumerical calculation of Sect. 5, this phase continues to\nthe TLL phase discussed in Sect. 3.4.\n3.61+δ≃0andλ≃0\nFor 1 +δ=λ= 0,Slandτ(2)\nlform a ferrimagnetic\nchain with msp= 1/2, andτ(1)\nlare free spins with mag-\nnitude 1/2. We take the zeroth order Hamiltonian as\nH0=L/summationdisplay\nl=1/bracketleftBig\n(1−δ)Sl(τ(2)\nl+τ(2)\nl−1)/bracketrightBig\n,(6)\nand the total spin of H0is defined by\nStot\n0=L/summationdisplay\nl=1(Sl+τ(2)\nl). (7)The perturbation part of the Hamiltonian is rewritten\nas,\nH1=L/summationdisplay\nl=1/bracketleftBig\n(1+δ)(Sl+Sl+1)+λτ(2)\nl/bracketrightBig\nτ(1)\nl.(8)\nWithin the subspace of the ferrimagnetic ground states\nofH0withStot\n0=L/2, each ground state is specified by\nStot\n0z=−L/2,...,L/2. From rotational symmetry, the\ntwo-spin effective Hamiltonian Heff\nlforStot\n0andτ(1)\nlis\nwritten down by their inner product as\nHeff\nl=J′\neff\n(L/2)Stot\n0τ(1)\nl. (9)\nIt should be noted that the terms of higher powers of\nStot\n0τ(1)\nlareunnecessarysince Stot\n0τ(1)\nlcantakeonlytwo\nvaluesStot\n0/2 and−(Stot\n0+1)/2. Then, the total effective\nHamiltonian Heffis given by\nHeff=L/summationdisplay\nl=1Heff\nl. (10)\nTo determine J′\neff, we estimate the expectation values of\nH1andHeffin the state |F/angbracketright=|Stot\n0=L/2;Stotz\n0=L/2/angbracketright\nto find\n/angbracketleftF|Heff|F/angbracketright=L/summationdisplay\nl=12J′\neff\nL/angbracketleftF|Stotz\n0|F/angbracketrightτ(1)z\nl\n=J′\neffL/summationdisplay\nl=1τ(1)z\nl, (11)\n/angbracketleftF|H1|F/angbracketright=L/summationdisplay\nl=1/bracketleftBig\n2(1+δ)/angbracketleftF|Sz\nl|F/angbracketright\n+λ/angbracketleftF|τ(2)z\nl|F/angbracketright/bracketrightBig\nτ(1)z\nl.(12)\nComparing (11) and (12), we find\nJ′\neff= 2(1+δ)/angbracketleftF|Sz\nl|F/angbracketright+λ/angbracketleftF|τ(2)z\nl|F/angbracketright.(13)\nThe expectation values /angbracketleftF|Sz\nl|F/angbracketrightand/angbracketleftF|τ(2)z\nl|F/angbracketright\nare calculated by the numerical diagonalization for L=\n2,4,6,12 and 14. After two steps of Shanks transforma-\ntion,14)we find\n/angbracketleftF|Sz\nl|F/angbracketright ≃0.7924871, (14)\n/angbracketleftF|τ(2)z\nl|F/angbracketright ≃ −0.2924871. (15)\nThus, we can determine the phase boundary between\nthe LM1phaseandthe nonmagneticphaseas λ/(1+δ)≃\n0.73815 where the sign of J′\neffchanges.\n4. Classical Phase Diagram\nBefore the description ofthe numericalphase diagram,\nwe examine the classical limit. We regard all spins as\nclassical vectors with fixed magnitudes. The magnitudes\nofSlandτ(1)\nlare denoted by s, and that of τ(2)\nl, byαs.\n3J. Phys. Soc. Jpn. FULL PAPERS\nWe assume a uniform ground-state spin configuration in\nthe form,\nSl= (0,ssinϕ,scosϕ),\nτ(1)\nl= (0,ssinθ,scosθ),\nτ(2)\nl= (0,0,αs),(16)\nas depicted in Fig. 3. We take the direction of τ(2)\nlasz-\ndirection. The nonuniform configurationssuch as period-\n1+δ\nλ\n1−δ\nτ(2)\nlτ(1)\nl\nSl+1 Slθ\nϕ ϕ\nFig. 3. Definition (16) of spin angles θandϕ.\ndoubledstatesandspiralstatesarealsoconsidered.How-\never, they turned out to have higher energies.\nThe ground-state energy per unit cell is given by\nE= 2s2(1+δ)cos(θ−ϕ)\n+2αs2(1−δ)cosϕ+αs2λcosθ.(17)\nMinimizing Ewith respect to θandϕ, we have\n∂E\n∂θ=−2s2(1+δ)sin(θ−ϕ)−αs2λsinθ= 0,(18)\n∂E\n∂ϕ= 2s2(1+δ)sin(θ−ϕ)\n−2αs2(1−δ)sinϕ= 0. (19)\nLet us start with trivial solutions.\n(1)θ=ϕ= 0\nSl= (0,0,s),\nτ(1)\nl= (0,0,s),\nτ(2)\nl= (0,0,αs).(20)\nThis is a ferromagnetic phase with spontaneous\nmagnetization msp= (2 +α)sper unit cell. This\nphase is not realized in the parameter regime con-\nsidered ( λ >0 and−1≤δ≤1).(2)θ=ϕ=π:\nSl= (0,0,−s),\nτ(1)\nl= (0,0,−s),\nτ(2)\nl= (0,0,αs).(21)\nForα= 2, this is a N´ eel-type ground state with\nlong-range antiferromagnetic order. However, once\nthe quantum fluctuation is switched on, it is ex-\npected that this state turns into the nonmagnetic\nphasewith msp= 0owingtotheone-dimensionality.\nHence, this corresponds to the classical counterpart\nof the nonmagnetic phase.\n(3)θ= 0,ϕ=π:\nSl= (0,0,−s),\nτ(1)\nl= (0,0,s),\nτ(2)\nl= (0,0,αs).(22)\nThis is the classical counterpart of the LM1 phase\nwith spontaneous magnetization msp=αsper unit\ncell.\n(4)θ=π,ϕ= 0 :\nSl= (0,0,s),\nτ(1)\nl= (0,0,−s),\nτ(2)\nl= (0,0,αs).(23)\nThis is the classical counterpart of the LM2 phase\nwith spontaneous magnetization msp=αsper unit\ncell.\n(5) Nontrivial solution :\nAssuming sin θ/negationslash= 0 and sin ϕ/negationslash= 0, we find\ncosϕ=λ(1+δ)\n4α(1−δ)2−αλ\n4(1+δ)−(1+δ)\nαλ,(24)\ncosθ=2(1−δ2)\nαλ2−(1+δ)\n2α(1−δ)−α(1−δ)\n2(1+δ),(25)\nafter some elementary manipulations from (18) and\n(19). The spontaneous magnetization mspper unit\ncell is given by\nm2\nsp= (αs)2+2αs2(cosθ+cosϕ)\n+2s2/parenleftbigg\n1+cosϕcosθ−λ\n2(1−δ)(1−cos2θ)/parenrightbigg\n.\n(26)\nThis state corresponds to the PF phase.\nThe phase boundary between the PF phase and other\nphases can be obtained in the following way,\n(1) PF-LM1 ( θ= 0,ϕ=π) phase boundary :\n4J. Phys. Soc. Jpn. FULL PAPERS\nSetting cos θ= 1 in (25), we have λ=λc1where\nλc1=2(1−δ2)\nα(1−δ)+(1+δ). (27)\nThe value of cos ϕatλ=λc1is obtained by substi-\ntuting (27) into (24) as\ncosϕ=−1. (28)\nThis implies that λc1corresponds to the PF-LM1\nphase boundary.\n(2) PF-LM2 ( θ=π,ϕ= 0) and PF-nonmagnetic ( θ=\nϕ=π) phase boundary :\nSetting cos θ=−1 in (25), we have λ=±λc2where\nλc2=2(1−δ2)\n(1+δ)−α(1−δ). (29)\nThe value of cos ϕatλ=±λc2is obtained by sub-\nsstituting (29) into (24) as\ncosϕ=±1. (30)\nThis implies that λc2and−λc2correspond to the\nPF-LM2 and PF-nonmagnetic phase boundary, re-\nspectively.\nThe classical phase diagram obtained in this section is\nshown in Fig. 4 for αs= 1 and s= 1/2.\n−1 0 1012\nPFnonmagneticLM2\nLM1λ\nδ\nFig. 4. Ground-state phase diagram in the classical limit with\nαs= 1 and s= 1/2.\n5. Numerical Results\nWe have carried out the numerical exact diagonaliza-\ntion forL= 4,6 and 8 with the periodic boundary con-\ndition to determine the phase boundary from the values\nof the spontaneous magnetization. The extrapolation of\nthe transition point to the thermodynamic limit is car-\nried out using the Shanks transform.14)If the data for\nlarger systems are necessary, the DMRG calculation for−1 0 1012\nLM2 TLLλ\nδLM1PFp=2PFPF\nFig. 5. Ground-state phase diagram. The open circles are the\nphase boundaries estimated from the numerical exact diagon aliza-\ntion data extrapolated to the thermodynamic limit from L= 4,6\nand 8 by the Shanks transform.14)The double circles are the phase\nboundaries estimated from the DMRG data for L= 48. The trian-\ngles atδ= 0 are the phase boundaries between the infinite series\nof QF phases determined in Ref. 7. The QF phases with p >2\nfor finite δare not shown, since they survive only for invisibly\nsmallδin the present scale. The deviation from the scaling rela-\ntion ∆E∼1/LforL= 18 and 24 is significant in the shaded area.\nThe curves are guides for the eye.\nL= 48 is carried out with the open boundary condition.\nThe obtained phase diagram is shown in Fig. 5.\n5.1 Ferrimagnetic phases with msp= 1\nAs in the classical case, the two QF phases (LM1,\nLM2) with msp= 1 do not form a single phase but are\nseparatedbythePFphase,theTLLphase,andthe p= 2\nQF phase. The δ-dependence of the spontaneous magne-\ntizationmspforL= 48calculatedbythe DMRG method\nis shown in Fig. 6 for λ= 0.6. This behavior shows that\na PF phase with msp<1 intervenes between the two QF\nphases with msp= 1. The corresponding behavior in the\nclassical limit is also shown in Fig. 7. The angles θand\nϕvary byπacross the PF phase. This behavior explic-\nitly shows that the LM1 and LM2 phases are different\nphases.\n5.2 The fate of the infinite series of QF phases\nFigure 8 shows the λ-dependence of the spontaneous\nmagnetization for (a) δ= 0.1 and (b) 0.02. The corre-\nsponding figure for δ= 0 taken from Ref. 7 that shows\nthe presence of an infinite series of QF phases is also\nshown as Fig. 8(c) for comparison. For δ= 0.02, the\nQF phases with msp= 1 (p= 1) and msp= 1/2 (p= 2)\nremain finite and the structures survive around the mag-\nnetizations corresponding to p= 3 and 4. For δ= 0.1,\nonly the QF phases with msp= 1 (p= 1) and msp= 1/2\n(p= 2) remain finite, while those with p≥3 are smeared\nout and replaced by a PF phase. These results suggest\n5J. Phys. Soc. Jpn. FULL PAPERS\n0.4 0.6 0.8 100.51msp\nλ=0.6\nδp=1\nL=48\nFig. 6. δ-dependence of mspin the ground state for λ= 0.6\ncalculated by the DMRG method for open chains with L= 48.\n0.4 0.6 0.8 100.51\nmsp\nθ/π ϕ/πλ=0.6\nδ\nFig. 7. δ-dependence of mspin the classical ground state for\nλ= 0.6. The angles θandϕare also plotted.\nthat the QF phases become more fragile with increas-\ningp. The corresponding curves in the classical limit are\nshown in Fig. 9. The QF phases vanish in the classical\ncase showing that these are essentially quantum effects.\nThe fragility of the QF phase with large pcan be un-\nderstood in the following way: If the spontaneous break-\ndown of the translational invariance were absent, the QF\ngroundstates with p≥2 areabsent and the groundstate\nis a PF state that is regarded as a magnetized TLL.15)\nThe low energy effective Hamiltonian in this phase is\ngiven by the U(1) compactified boson field theory with\nthe TLL parameter Kas follows:\nH(0)\nB=1\n2π/integraldisplay\ndx/bracketleftbigg\nK(πΠ)2+/parenleftbigg1\nK/parenrightbigg\n(∂xφ)2/bracketrightbigg\n,(31)\nwhereφis a boson field defined on a circle φ∈[0,√\n2π),\nand Πis the momentum density field conjugate to φ. The\nspin wave velocity is set equal to unity. Extending the0.6 0.7 0.8 0.900.51msp\nδ=0.1\nλp=1\np=2\np=3\np=4L=48(a)\n0.6 0.7 0.8 0.900.51msp\nδ=0.02\nλp=1\np=2\np=3\np=4L=48(b)\n0.6 0.7 0.800.51\nλmsp\np≤38λc(∞)−~0.807p=2p=1\np=3\np=4(c)\nFig. 8. λ-dependence of the spontaneous magnetization in the\nground state for (a) δ= 0.1 and (b) δ= 0.02 calculated by the\nDMRGmethod foropen chains with L= 48. (c)The corresponding\nfigure for δ= 0 is taken from Ref. 7.\nbosonization procedure of Ref. 16 to the case of mixed\nspin chains, a translation by one unit cell results in the\nshift of the boson field as\nφ→φ+√\n2π(Suc−msp), (32)\nwhereSucis the sum of the spin magnitudes in a unit\ncell. In the present case, Suc= 2. We consider the case\n6J. Phys. Soc. Jpn. FULL PAPERS\n0 1 2 301msp\nλδ=0\nδ=0.02\nδ=0.1\nFig. 9. λ-dependence of the spontaneous magnetization in the\nground state for δ= 0, 0.02 and 0.1 in the classical limit.\nmsp= 1/pthatcorrespondstothevalueof mspintheQF\nphase with period p. Then, taking the compactification\ncondition into account, the shift (32) is rewritten as\nφ→φ−√\n2π/p. (33)\nThe leading perturbation invariant under the shift (33)\nis given by\nH(1)\nB=c/integraldisplay∞\n−∞dxcos(√\n2pφ), (34)\nwherecis a constant. Although this operator is transla-\ntionally invariant, if it is relevant, the phase φis pinned\ntooneofthe minimaof ccos(√\n2pφ)andthe translational\ninvariance is spontaneously broken. Since the scaling di-\nmension of the operator(34) is xp=p2K/2,it is relevant\nifxp<2. Although the p-dependence of Kis unknown,\nassuming that it is moderate, the main pdependence of\nxpcomes from the factor of p2. This explains why the\nQF phases with large pare more fragile than those with\nsmallp.\n5.3 Nonmagnetic phase\nFor larger λ, the nonmagnetic phase appears and it\ncontinues to the TLL phase for large λdiscussed in Sect.\n3.4. Since the sum of the spin magnitudes in a unit cell is\nan integer, the conventional LSM theorem9,10)does not\nexclude the unique gapped phase. However, our model\n(1) is invariant under the site-reflection about the vertex\nspinSlwhose magnitude is 1/2. Hence, our model sat-\nisfies the condition to exclude the unique gapped phase\nin the recent extension of the LSM theorem to the site-\nreflection invariant spin chains.11–13)Taking the continu-\nity to the TLL phase in the limit λ→ ∞into account,\nthewholenonmagneticphaseisconsideredtobetheTLL\nphase. This is confirmed by the numerical diagonaliza-\ntion calculation of the singlet-triplet energy gap ∆ E. It\nis checkedthat ∆ Eapproximatelyscaleswith the system0 1 2024\nλL∆Eδ=−0.3\nL=4\nL=6\nL=8(a)\n0 1 200.51\nλL∆Eδ=0.3\nL=4\nL=6\nL=8(b)\n0 1 2 3 400.10.20.3\nλL∆Eδ=0.5\nL=4\nL=6\nL=8(c)\nStot>1 Stot=1\nFig. 10. λ-dependence of the scaled gap L∆Eof the lowest ex-\ncitations for (a) δ=−0.3, (b) 0.3 and (c) 0.5 with L= 4,6 and\n8. The open and filled symbols are excitations with Stot= 1 and\nStot>1, respectively.\nsizeLas ∆E∼1/Las shown in Fig.10(a) for δ=−0.3\nand (b) for 0.3. Similar analyses are also carried out for\nseveral other values of δ. In the vicinity of the PF phase\nindicated by the shaded areaofFig. 5, however,the devi-\nationfromthescalingrelation∆ E∼1/Lissignificantas\nshown in Fig. 10(c). Nevertheless, this area shrinks with\nthesystemsize.Hence,itislikelythatthewholenonmag-\n7J. Phys. Soc. Jpn. FULL PAPERS\nnetic phase is a TLL phase. It should be also remarked\nthat the nonmagnetic ground state for δ= 0 is also a\nTLL phase7)since it is exactly mapped onto the ground\nstate of the spin-1/2antiferromagneticHeisenbergchain.\nForλ/greaterorsimilarλc, whereλcis the nonmagnetic-ferrimagnetic\ntransition point, the ferrimagnetic state with total spin\nStot>1 comes down resulting in the level-crossing with\nthe nonmagnetic state at λc. Their energies measured\nfrom the ground state are plotted by the filled symbols\nin Fig. 10(c). These ferrimagnetic states are macroscop-\nically different from the nonmagnetic state in the ther-\nmodynamic limit and cannot be regarded as elementary\nexcitations in the nonmagnetic state, even though they\nhave the next-lowest energy for finite-size systems. Un-\nfortunately, in the region where this type of state has the\nnext-lowest energy, it is difficult to identify the singlet-\ntriplet gap, since we can calculate only several lowest\neigenvalues by the Lanczos method we employ in this\nwork.\n6. Summary and Discussion\nThe ground-state phases of mixed diamond chains (1)\nare investigated numerically and analytically. For com-\nparison, the ground-state phase diagram of the corre-\nsponding classical model is calculated analytically. In\nthe quantum case, the ground-state phase diagram is de-\ntermined using the numerical exact diagonalization and\nDMRG method in addition to the perturbation analysis\nforvariouslimitingcases.Thegroundstateofthepresent\nmodel has a rich variety of phases such as the two kinds\nof QF phases with msp= 1, the QF phase with sponta-\nneous translational symmetry breakdown, the PF phase,\nand the nonmagnetic TLL phase.\nThefate oftheinfinite seriesofQFphasesobservedfor\nδ= 0 is also investigated numerically. It turned out that\nthe QF phases with largespatial periodicities pare easily\ndestroyed by small δand replaced by the PF phase. The\ninterpretation of this behavior is also discussed using the\nbosonization argument.\nIn the nonmagnetic phase, the unique gapped ground\nstate is excluded based on the recently extended LSM\ntheorem.11–13)Combined with the numerical calculation\nofthe energygap, this regionis consideredto be the TLL\nphase.Sofar,theexperimentalmaterialscorrespondingtothe\npresent mixed diamond chain arenot available.However,\nconsidering the rich variety of ground-state phases, the\nexperimental realizationofthe present model is expected\nto produce a fruitful field of quantum magnetism. With\nthe recent progress in the synthesis of mixed spin com-\npounds,17)we expect the realization of related materials\nin the near future.\nThenumericaldiagonalizationprogramisbasedonthe\nTITPACK ver.2 coded by H. Nishimori. Part of the nu-\nmerical computation in this work has been carried out\nusing the facilities of the Supercomputer Center, Insti-\ntute for Solid State Physics, University of Tokyo, and\nYukawaInstitute Computer Facility at KyotoUniversity.\n1)Introduction to Frustrated Magnetism: Materials, Experi-\nments, Theory , ed. C. Lacroix, P. Mendels, and F. Mila\n(Springer Series in Solid-State Sciences, Springer, Heide lberg,\n2011).\n2)Frustrated Spin Systems , ed. H. T. Diep, (World Scientific,\nSingapore, 2013) 2nd ed.\n3) K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Condens.\nMatter8, 6405 (1996).\n4) K. Hida and K. Takano, J. Phys. Soc. Jpn. 86, 033707 (2017).\n5) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.\nTonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta,\nPhys. Rev. Lett. 94, 227201 (2005).\n6) H.Kikuchi,Y.Fujii,M.Chiba,S.Mitsudo,T.Idehara,T.T one-\ngawa, K. Okamoto, T. Sakai, T. Kuwai T, K. Kindo, A. Mat-\nsuo, W.Higemoto, K.Nishiyama, M.Horovi´ c, and C.Bertheir ,\nProg. Theor. Phys. Suppl. 159, 1 (2005).\n7) K. Hida, J. Phys. Soc. Jpn. 90, 054701 (2021).\n8) E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).\n9) E.Lieb, T.Schultz, and D.Mattis, Ann.Phys. 16, 407 (1961).\n10) H. Tasaki, J. Stat. Phys. 170653 (2018).\n11) Y. Fuji, Phys. Rev. B 93104425 (2016).\n12) H. C. Po, H. Watanabe, C.-M. Jian, and M. P. Zaletel, Phys.\nRev. Lett. 119, 127202 (2017).\n13) Y.Ogata, Y.Tachikawa, and H.Tasaki, Commun.Math.Phys .\n385, 79 (2021).\n14) D. Shanks, J. Math. Phys. 34, 1 (1955).\n15) S.C.Furuyaand T.Giamarchi,Phys.Rev.B 89,205131 (2014).\n16) M. Oshikawa, M. Yamanaka and I. Affleck, Phys. Rev. Lett.\n78 1984 (1997)\n17) H.Yamaguchi, Y.Iwasaki, Y.Kono, T.Okita, A.Mat-\nsuo, M. Akaki, M. Hagiwara, and Y. Hosokoshi, Phys.\nRev. B102, 060408(R) (2020) and references therein.\n8"    },    {        "title": "1910.05918v1.Ultrafast_domain_wall_motion_in_ferrimagnets_induced_by_magnetic_anisotropy_gradient.pdf",        "content": " \n \nUltrafast domain wall motion in ferrimagnet s induced by magnetic anisotropy gradient  \n \nW. H. Li1, Z. Jin1, D. L. Wen1, X. M. Zhang1, M. H. Qin1,*, and J. –M. Liu2 \n1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nand Institute for Advanced Materials, South China Academy of Advanced Optoelectronics , \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstructures and Innovative Center for Advanced \nMicrostructures, Nanjing University, Nanjing 210093, China  \n \n[Abstract]  The ultrafast magnetic dynamics in compensated ferrimagnets not only provides \ninformation similar to antiferromagnetic dynamics, but more importantly opens new \nopportunities for future spintronic devices  [Kim et al., Nat. Mater. 16, 1187 (2017) ]. One of \nthe most essential issues  for device design is s earching for low-power -consuming and \nhigh-efficient methods of controlling domain wall. In this work, we propose to use the \nvoltage -controlled magnetic anisotropy gradient as an excit ation source to drive the domain \nwall motion in ferrimagnet s. The ultrafast wall motion under the anisotropy gradient is \npredicted  theoretically based on the collective coordinate theory , which is also confirmed by \nthe atomistic micromagnetic simulations. The antiferromagnetic spin dynamics is realized at  \nthe angular momentum comp ensation point , and t he wall shift ing has a constant speed under \nsmall gradient and can be slightly accelerated under large gradient due to the broadened wall \nwidth during the motion. For nonzero net angular momentum, the Walker breakdown occurs \nat a critical anisotropy gradient  significantly depend ing on the second  anisotropy  and \ninterfacial Dzyaloshinkii -Moriya interaction , which is  highly appreciated for further \nexperiments  including the materials selection  and device geometry design.  More importantly , \nthis work unveils a low -power -consuming and high-efficient method of controlling the \ndomain wall in ferrimagnets, benefiting to future spintronic applications.  \n \nKeywords: magnetic  dynamic s, domain wall , magnetic  anisotropic gradient , ferrimagnets  \n \n                                                             \nEmail: qinmh@scnu.edu.cn  I. Introduction  \nAntiferromagnetic materials show fast magnetic dynamics and produce non -perturbin g \nstray fields , attributing to their zero magnetization and ultralow susceptibility . These \nadvantages make them promising candidates for next generation of high-density and  \nhigh-speed spintronic devices .1-5 However, the magnetic field immunity of antiferromagnetic \nmaterials  also hinders the detection and manipulation of magnetic states .6-8 Thus, it is still \nchallenging to  experimental ly study the  antiferromagnetic  spin dynamics , although several \nstimuli have been predicted to drive the fast domain wall motion in the earlier theoretical \nworks .9-18 Therefore, a reliable and direct detection of the magnetic states  remains to be a \ncommon issue for antiferromagnetic spintronic researches.  \nTo overcome this deficiency, an immediate alternative strategy is to consider \nferrimagnetic (FiM) systems where the fast magnetic dynamics in the vicinity of  angular \nmomentum compensation temperature  TA can be achieved,19 at which the net momentum \nvanishes  while the net magnetic moment is nonzero . It has been theoretically predicted and \nexperimentally confirmed that the FiM dynamics at TA is similar to the antiferromagnetic \ndynamics . More importantly, the magnetic states of a FiM system at TA can be effectively \ndetected and addressed through the magnetoelectric20-22 and magneto -optical23 responses, \nbenefiting from their nonzero magnetic moment , and thus highly appreciated .  \nIn fact , the magnetic field - and electrical current -driven fast domain wall motion s in \nangular momentum compensated ferrimagnets have been experimentally reported, \nrespectively.19,24-26 Also , the Walker breakdown field, under which the domain wall begins to \nprecess and reaches to a threshold speed, is significantly increased and the domain wall \nmobility is extensively enhanced when the net angular momentum approaches to zero . At TA, \nthe field diverg es, and the domain wall speed keeps increasing linearly with field due to the \nexcluded Walker breakdown, exactly the same as in antiferromagnets. For example, the \ndomain wall speed as high as ~ 20 km s-1T-1 was reported at TA in rare earth  3d transition \nmetal ferrimagnets.19 Thus, the magnetic dynamics in ferrimagnets at TA not only provides \nequivalent information for antiferromagnetic spin dynamics, but more importantly opens new \nopportunities for future spintronic devices.  \nOn the other hand , searching for well-controlled and low -power -consum ed method s to \nmodulat e FiM domain w all is one of the most important issues for spintronic device operation , \nnoting that  the shortcomings of these proposed schemes  may be detrimental for future  \napplications . For instance, the dispersion characteristic of magnetic field generally limits the \ndensity of ferrimagnetic  elements and hinders the further optimization of device dimension. Moreover, some of the  electrical current related schemes normally generate Joul e heating  and \nunnecessary energy loss , significantly affecting the data transportation process where a stable  \noperating temperature is benefiting . Along this line, electric field control could be highly \npreferred,27 to be explained in detail below.   \nFirst, numerous experiments have revealed the voltage control of magnetism. For \nexample, the voltage induced magnetic anisotropy gradient has been experimentally reported \nin magnetic heterostructures through elaborate structure design.28-30 Under such a gr adient, \nthe magnetic domain wall tends to move towards the low anisotropy side in order to save free \nenergy. As a matter of fact, the anisotropy gradient has been proven to efficiently drive the \nskyrmions motion and antiferromagnetic domain wall motion,31-33 and this scheme could  be \nalso utilized to control the FiM domain wall motion. More importantly, this alternative \nscheme is promising for future spintronic applications considering  the low-energy cost and \nthe high operating efficiency. However, as far as  we know, few works on this subject have \nbeen reported , while the dynamics of FiM domain wall under anisotropy gradient  is certainly \nan urgent topic to be understood , in order to provide instruction for future experiments and \npromot e the application proces s for spintronics.  \nIn this work, we study the domain wall dynamics of ferrimagnets under an anisotropy \ngradient, using the collective coordinate theory and atomistic Landau -Lifshitz -Gilbert (LLG) \nsimulations. It is demonstrated  that the wall speed and precession direction depend closely on \nthe net angular momentum. At the angular momentum compensation point, the Walker \nbreakdown vanishes and the wall moves at a maximal speed, similar to the case of \nantiferromagnetic dynamics. It will be shown that the wall remain s to shift at a const ant speed \nunder small gradient, while the motion can be slightly accelerated under large gradient due to \nthe broadened wall width during the motion. Furthermore, for a nonzero angular momentum, \nthe Wa lker breakdown gradient could be modulated by utilizing a  second  anisotropy and the \ninterfacial Dzyaloshinkii -Moriya (DM) interaction . These results provid e useful information \nfor future material design and spintronic applications.   \n \nII. Analytical analysis  and numerical simulation  \nWe investigate theoretically the  domain wall motion for ferrimagnet s such as rare earth \nand transition metal compounds , whose magnetic structure is depicted in Fig. 1 (a) where  the \nspins of two inequivalent sublattices ar e coupled antiferromagnetically .34 We set n1,2(r, t) (n1 \n= -n2), M1,2 (M1,2 = M1,2·n1,2), 1,2, g1,2, and 1,2 to be the local unit vector at time t and position r, magnetization  moment , gyromagnetic ration, Landé -g factor, and Gilbert damping \nconstant of the two sublattices. Thus, the spin density of the sublattice i is given by si = Mi/i \nwith i = giB/ћ, where µB is the Bohr magneton. It is noted that the net magnetization M = \nM1 + M2 is nonzero at TA where the net angular momentum s = s1  s2 = 0, because of the \ndifferent Landé -g factors between the two sublattices.  \n \n2.1. Analytical treatment  \nFollowing  the collective coordinate approach , the low -temperature magnetic dynamics of \nFiM model  is described by the Lagrangian density L = LB  U with the spin Berry phase LB \nand the potential -energy density U.19,35 In detail, the Berry phase is associated with the \nstaggered spin density s = (s1 + s2)/2 and the net spin density s, which  can be described \nby:18,19,35 \n( ) ( ) ,BsLs      n n m a n n\n             (1) \nwhere n  (n1 – n2)/2, and m  (n1 + n2)/2, ṅ represents  the derivative with respect to time, \na(n) is the vector potential generated by a magnetic monopole of unit charge satisfying n  a \n= n. The potential -energy density is given by  \n2\n2 2 2 ()( ) ( ).2 2 2 2 2ex\nz x y zA K z k DU n n       mn e n n\n          (2) \nHere, the first and second terms are  the inhomogeneous and homogeneous exchange energ ies \nwhere Aex > 0 is the exchange stiffness  and  is the magnetic susceptibility . The third term is \nthe easy -axis anisotropy along the z axis (nanowire axis)  with positive K which changes \nlinearly with the z-coordinate  K(z) = K0  z·dK/dz. The fourth term is the so-called second \nanisotropy or intermediate  anisotropy defined along the x axis with k > 0, and this anisotropy \nshould be weaker than the easy -axis anisotropy  along the z-axis. The last term is the \ninterfacial DM interaction with D > 0 and ey is the unit vector in the y direction . To obtain an \nmore explicit expression of the  Lagrangian density, w e replace m with m = s ṅ  n,36,37 and \nobtain  \n2 2 2 2( ) ( ) ( ),2 2 2 2 2ex\ns z x y zA K k DL n n         n a n n n e n n\n     (3) \nwhere   s2 parametrizes the inertia of dynamics. The dissipative dynamics can be described by introducing the Rayleigh function density R = sṅ2/2 with s  1s1  2s2 \naccounting for the energy and spin loss due to the magnetic dynamics.38 \nNow we discuss the low -energy dynamics of FiM domain wall. Following the earlier \nwork, we introduce  two collective coordinates, the position q(t) and azimuthal angle (t) in Eq. \n(3) to characterize the FiM domain wall  under an anisotropy gradient . We consider the Walker \nansatz39 for the domain wall  profile: n(z, t) = (sech(( z-q)/)cos, sech(( z-q)/)sin, \ntanh(( z-q)/)) where   is the domain wall  width.  After a pplying the Euler -Lagrange equation, \nwe obtain t he equations of motion for the two coordinates : \n/, Mq G Mq F  \n              (4) \n00 / sin cos sin , I Gq I k D        \n          (5) \nwhere M = 2/ is the mass  with  the cross -sectional area of the domain wall , I = 2 is \nthe moment of inertia, G = 2s is the gyrotropic coefficient,  = s is the relaxation time, F \n= 4·dK/dz is the force exerted by an anisotropy gradient , k0 = 2k, and D0 = D/2.  \nA specific solution to Eq. ( 4) and Eq. (5) for k = D = 0 gives the domain wall velocity v \nand domain wall plane precession  speed : \n2\n22,\nsdKvs s dz\n\n               (6) \n.svs\n                 (7) \nEq. (6) shows that velocity v increases linearly with dK/dz and reaches the maximum at \nthe angular momentum compensation point TA where  s vanishes  (s ~ 0). To illustrate that \nthis velocity can be high in real materials, one gives a crude estimation of v by taking  the \nwell-known FiM compound GdFeCo as an example .19,24,26 Setting the internal parameters \nexchange stiffness Aex = 50 pJ/m, anisotropy constant at high anisotropy end  K0 = 0.5 MJ/m3, \nM1 = 440 kA/m, M2 = 400 kA/m, 1 = 2 = 0.01, g1 = 2.2, and g2 = 2.0 , one obtains a wall \nmotion velocity v ~ 1.2 km/s  at the compensation point  under an  anisotropy gradient dK/dz = \n300 GJ/m4, comparable to the current - and the field -driven motions  for antiferromagnetic \ndomain wall motions . Furthermore, as shown in Eq. (7), the domain wall plane rotates with \nthe domain wall propagation  without any favored orientation due to k = 0, which is closely  dependent  of s.  \n \n2.2. Numerical calculation  \nIn order to check the validity of the above analytical treatment , we also perform the \nnumerical simulations based on the atomistic LLG  equation. Here, the corresponding \none-dimensional discrete Hamiltonian is given by :40 \n22\n11 ( ) ( ) ( ),zx\ni i i i x i i i i\ni i i iH=J K S K S        S S D S S\n     (8) \nwhere t he first term is the exchange interaction with  J = 1, Si is the normalized spin moment \nvector at lattice site i. The second term is the anisotropy energy with the easy axis along the \nz-direction, and the anisotropy constant at site i is described by Ki = K 0 – ia·K where K \ndescribes the anisotropy gradient magnitude, a is the lattice constant . The third term is the \nsecond  anisotropy Kx along the x-axis, and the last term is the DM interaction with Di = (0,  Dy, \n0). \nThen, t he dynamics is investigated by solving the stochastic LLG equation,41-43  \n 2( ) ,(1 )ii\ni i i i i\nii tM   SS H S H\n          (9) \nwhere Hi = − H/Si is the effective field. Without loss of generality, we set the damping \nconstants 1 = 2 = 0.01, the gyromagnetic ratio s 1 = 1.1  and 2 = 1.0  corresponding  to the \nLandé  g-factors g1 = 2.2 and g2 = 2.0  for the two sublattices .44  \nTo investigate the dynamics in the vicinity of the momentum compensation point,  several \nsets of (M1, M2) are employed, as listed in Table I. Unless stated elsewhere, the LLG \nsimulations are performed on a 1  1  400 lattices  with open boundary conditions using the \nfourth -order Runge -Kutta method with a time step t = 1.0  10−4 s/Jeff where s is the \nsaturation  moment and eff = (1 + 2)/2. After a sufficient relaxation of the domain structure , \nthe anisotropy gradient is applied to drive the domain wall  motion , as schematically depicted \nin Fig. 1 (a). \nAs a matter of fact, a comparison between the analytical  treatment and the atomic model \ncan be useful in qualitative sense . It is seen from the atomistic model that various torques act \non the wall spins .16 The two spins neighboring the central wall spin deviate differently from  the ea sy-axis with 1 > 2, resulting in the net damping torque d from the exchange \ninteraction on the central spin, as depicted in Fig. 1(b). The damping torques  d ~ − S  (S  \nH) point in an opposite direction on the two sublattices and drive the wall motion. Moreover , \nthe precession torques p ~ − S  H pointing into the same direction on the two sublattices are \nunequal in magnitude in the case of s  0, resulting in the precess ion of the wall plane with \nthe wall propagation , in agreement with Eq. (7). For s = 0, torques p on the two sublattices \nare equal and the domain wall plane is fixed.  \nThus, the fast domain wall motion and the precession of the wall plane in ferrimagnets \nare theoretically revealed and qualitatively confirmed by the atomic model simulation s. \nSubsequently, we present the analytically derived and numerically calculated results to \ndemonstrate the  quantitative consistence between the analytical derivation and atomistic \nsimulation  on one hand, and more importantly to unveil the FiM dynamics in details.   \n \nIII. Results and discussion  \n3.1. Domain wall dynamics  \nWe first present the domain wall dynamics by discussing the wall velocity and precession \nspeed as a function of the anisotropy gradient respectively. Fig. 2(a) shows the numerically \nsimulated (empty points) and Eq. (6) -based calculated (solid lines) wall v elocit y v as a \nfunction of K for various δs and K0 = 0.01 J, Kx = 0 and Dy = 0. It is seen that the simulated \ndata fit the calculations  perfectly , confirming the validity of the analytical treatment . Here, \ntwo issues deserve highlighting. First, the driving torque increases with the increasing  K, \nwhich significantly enhances the wall motion speed. Specifically, v increases linearly with K, \nnoting that here only low anisotropy gradient  is considered and the domain wall width is \nhardly changed during the motion. Second, for a fixed K, v increases with decreasing \nmagnitude of s, and reaches to the maximum at the angular momentum compensation point \ns = 0, as clearly shown in Fig. 2(b) w here v(s) curves for various K are presented.  \nWe then discuss the domain wall plane precession which appears for a nonzero s in \naccompanying with the wall motion, as shown in Fig. 2(c) where the angular velocity (d/dt) \nof the plane  as a function of K is plotted . Also two issues are highlighted . First, t he angular \nvelocity increases linearly with K or the wall speed  v. In comparison with the dynamics for s = 0 where the domain wall plane is fixed, the wall plane precession leads to additional \nenergy dissipation, resulting in the low wall mobility for nonzero s under the same K. \nSecond and more  interestingly, the precession direction of the wall plane depends on the sign \nof s. Specificall y, the wall plane precesses clockwise around the easy -axis for s > 0, while \ndoes counter clockwise for s < 0, as clearly shown in Fig. 2(d) where the x and y components \nof local quantity n, nx and ny, are presented at various times for s > 0 (top half) and s < 0 \n(bottom half) . With the wall motion, opposite precession directions are clearly observed in the \ntwo cases , in consistent with the theoretical predict ion in  Eq. (7).   \nSo far, the anisotropy gradient driven domain wall motion in the vicinity of the angular \nmomentum compens ation point of ferrimagnets has been clearly uncovered in our theoretical \nanalysis and LLG simulations. In experiments, anisotropy gradient could be induced by \ntuning electric field on particular heterostructures , and efficiently drives the domain wall \nmotion generating Joule heat much less than those electrical current related methods . Thus, \nthe proposed method in the work is expected to be both low power -consuming and \nhigh-efficient, which is essential for future spintronic applications.  \n \n3.2. Roles of internal parameters  \nBased on the good consistency between the analytical analysis and numerical calculations, \none is able to discuss the roles of various internal parameters. An unveiling of these roles \nwould be highly appreciated for practical applicatio ns including the materials selection, \ndevice geometry design, and performance optimization.   \nFirst, the anisotropy constant K0 determines the wall width  which can be estimated by \na(J/2K0)1/2, and in turn affects the wall speed which increases with  as demonstrated in Eq. \n(6). Thus, contra ry to K, a large  K0 results in a small  and makes the wall motion slow, as \nconfirmed in our simulations. In Fig. 3(a), the simulated and calculated speeds as functions of \nK0 for various K at the angular momentum compensation point  are plotted, not only showing \nthe excellent consistence between the simulation and analytical derivation but also clearly \nrevealing that the anisotropy magnitude enables an decelerated wall motion. In addition, an \nenhanced damping term a lways reduces the wall mobility,45,46 and v decreases with the \nincrease of the damping constant α. As a matter of fact, v linearly  increases with 1/ α, as shown in Fig. 3(b) where presents the simulated and calculated  v as functions of 1/ α at δs = 0 for \nfixed K0 and K. \nIn the above analysis, the wall width  is simply set  to be unchanged during the wall \nmotion, which well describes the case of very low anisotropy gradient. However, when the \nwall shifts under a high gradient, the wall is considerably enlarged,  resulting in an accelerated \ndomain wall motion. This phenomenon has been also observed  in our simulations (dashed \nlines) and calculations (solid lines)  in Fig. 4 which give the evolution of the wall position (Fig. \n4(a)) and the local  wall velocity (Fig. 4(b)) for various K at δs = 0. In this case, the wall \nwidth could be  updated to  = a(J/2Kc)1/2 with Kc the anisotropy on the wall central spin. A \nconstant velocity is obtained  under a low gradient K ~ 0.5  10-5J/a, while an acceleration  of \nthe wall motion  under a high gradient K ~ 2  10-5J/a is clearly observed.     \nSecond, the intermediate anisotropy Kx could be non -negligible  in some FiM materials , \nand affects the wall motion.  Subsequently, we check the effect of Kx on the dynamics of \ndomain wall in ferrimagnets. The time evolutions of the wall position for various Kx for δs = \n0.022 are presented in Fig. 5(a) which exhibits three types of wall motion .16 As discussed \nabove, the wall has no favored orientation for Kx = 0 and rotates continuously and moves \nconstantly with a reduced speed. The consideration of the intermediate anisotropy suppresses \nthe rotation of the wall plane, and in turn  significantly affects the wall motion . In the case of \nsmall anisotropy of Kx = 2.5  10-5 J, the Wal ker breakdown occurs under the anisotropy \ngradient K larger than the threshold value KWB. Here , the Walker breakdown gradient \nKWB can be estimated by Kxs/4s.47,48 In the case of high anisotropy Kx = 10  10-5 J, the \nprecession of  the domain wall is completely suppressed  for the considered K, resulting in the \nwall motion with a maximal velocity. On the other hand, the wall motion at the angular \nmomentum compensation point  δs = 0 where no precession of the wall is available is \nindependent of the anisotropy  Kx, as clearly shown in Fig. 5(b) where presents the mean \nvelocity of the domain wall as a function of Kx for δs = 0 and δs = 0.0 22. Moreover, f or a fixed \ngradient K below the Walker breakdown  KWB, δs = 0.022 is with a magnetization  smaller \nthan δs = 0, and the domain wall motion for δs = 0.0 22 is slightly faster than δs = 0.  \nThird, a DM interaction  could be  induced at interface between heavy metal and \nferrimagnet  and modulated efficiently  through elaborate heterostructure design.  Similarly, the interfacial DM interaction Dy(0, 1, 0) also suppresses the precession of the wall plane and \nspeed s up the wall motion  below KWB. In Fig. 5(c), the simulated velocit ies as a function of \nDy for δs = 0 and δs = 0.0 22 for K = 0.5  10-5J/a is plotted, revealing the critical DM \ninteractions Dc which can be given by  |Dc| = 8δs2·KWB / πs for nonzero δs.26,47,48 Under a \nfixed K, the Walker b reakdown occurs for | Dy| < |Dc|, while vanishes for | Dy| > |Dc|. The \nsimulated | Dc| (empty points) as a function  of δs for various K is presented in Fig. 5(d), well \nin consistent with the theoretical derivation  (solid lines) . Thus, this prediction could be used \nto improve the Walker breakdown field and to enhance the domain wall mobility, which is \nvery meaningful for spintronic applications.   \n Thus, the domain wall motion depending on the internal parameters has been clearly \nunveiled, which definitely provides usefu l information for future material selection and device \ndesign. For example, high domain wall mobility could be available in ferrimagnet with not \ntoo large K0 and considerable second anisotropy, as suggested in our calculations. Moreover, a \nlarge DM interac tion generated in interface between heavy metal and ferrimagnet \nsignificantly improves the Walker breakdown field and ensures the fast domain wall motion. \nOf cause, these  predictions given here deserve  to be checked in further experiments.    \n \nIV. Conclusion  \nTo summarize , we have studied analytical ly and numerically the dynamics  of the domain \nwall in  ferrimagnet s driven by the magnetic anisotropy gradient.  The wall moves  towards the \nlow anisotropy  side to release the free energy  and reaches to a maxim al velocity  at the angular \nmomentum compensation  point  where exhibits the antiferromagnetic dynamics . Moreover,  \nthe net spin angular momentum determines  not only the wall velocity but also the precession \ndirection of the domain wall  plane . Furthermore, for nonzero net angular momentum, Walker \nbreakdown occurs under a critical anisotropy gradient which significantly depends on the \nintermediate anisotropy and interfacial DM interaction. This work unveils a low \npower -consuming and also high -efficient method  of controlling the domain wall in \nferrimagnets, benefiting to future experiments design and spintronic applications . \n \n Acknowledgment  \nThe work is supported by the National Key Projects for Basic Research of China (Grant \nNo. 2015CB921202), and  the Natural Science Foundation of China (No. 51971096 ), and the \nScience and Technology Planning Project of Guangzhou in China (Grant No. 201904010019),  \nand the Natural Science Foundation of Guangdong Province (Grant No. 2016A030308019).  \n  References:  \n \n1. T. Jungwirth, X. Marti, P. Wadle y and J. Wunderlich, Nat. Nano technol. 11, 231 (2016).  \n2. P. Wadley et al., Science 351, 587 (2016).  \n3. J. Železný, P. Wadley, K. Olejník, A. Hoffmann and H. Ohno, Nat. Phys. 14, 220 (2018).  \n4. J. Torrejon et al., Nature 547, 428 (2017).  \n5. P. Park et al., npj Quantum Mater . 3, 63 (2018) . \n6. O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. 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Liu , Phys. Rev. B 100, 020402 (R) (2019) . \n48. A. Mougin, M. Cormier, J. Adam, P. Metaxas, and J. Ferr,  Europhys. Lett. 78, 57007 \n(2007).  \n  \n  Table I. Parameters chosen for the simulation.  \n \nParameter  1 2 3 4 5 6 7 \nM1 1.13 1.12 1.11 1.1 1.09 1.08 1.07 \nM2 1.0 1.04 1.02 1.0 0.98 0.96 0.94 \ns -0.03273  -0.0218  -0.0109  0 0.0109  0.0218  0.03273  \n \n \n \n \n \n   \n \nFIG.1. (color online) (a) Illustration of a domain wall in ferrimagnetic  nanowire under an \nanisotropy gradient. Here the asymmetry of the domain wall center is exaggerated. (b) A \nschematic depiction of torques acting on the central spins of the domain wall.  \n  \n \n \nFIG.2. (color online) The simulated (empty points) and calculated (solid lines) velocities as \nfunctions of (a) K for various s, and (b) s for various K for K0 = 0.01. (c) The simulated \n(empty points) and calculated (solid lines) angular velocit ies of the wall plane as function s of \nK for various s, and (d) the evolutions of the local nx and ny for s > 0 (top half) and s < 0 \n(bottom half) . The rotations of the wall plane are shown in the insert of (d).  \n  \n \n \nFIG.3. (color online) The simulated (empty points) and calculated (solid lines) velocities at \nthe momentum compensation point as functions of (a) K0 for various value of K for  = 0.01, \nand (b) 1/ for K0 = 0.01 J and K = 0.5  10-5 J/a. \n  \n \n \nFIG.4. (color online) The simulated (dashed lines) and calculated (solid lines) (a) evolutions \nof the wall positions for various K, and (b) instantaneous speed for various K.  \n \n  \n \n \nFIG.5. (color online) For K = 0.5 ×  10-5 J/a, the simulated (a) evolutions of the wall position \nfor various Kx, and  mean velocities as functions of (b) Kx and (c) Dy for δs = 0 and δs = 0.022. \n(d) The simulated (empty points) and calculated (solid lines) | Dc| as a function of δs for \nvarious K. \n \n"    },    {        "title": "1707.04854v1.Competing_magnetic_and_spin_gap_less_semiconducting_behaviour_in_fully_compensated_ferrimagnet_CrVTiAl__Theory_and_Experiment.pdf",        "content": "Competing magnetic and spin gap-less semiconducting behaviour in fully\ncompensated ferrimagnet CrVTiAl: Theory and Experiment\nY. Venkateswara,1,\u0003Sachin Gupta,1, 2,\u0003S. Shanmukharao Samatham,1\nManoj Raama Varma,3Enamullah,4K. G. Suresh,1,yand Aftab Alam4,z\n1Magnetic Materials Laboratory, Department of Physics,\nIndian Institute of Technology Bombay, Mumbai 400076, India\n2WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan\n3National Institute for Interdisciplinary Sciences and Technology (CSIR), Thiruvananthapuram, India;\n4Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India\n(Dated: July 18, 2017)\nWe report the structural, magnetic and transport properties of polycrystalline CrVTiAl alloy\nalong with \frst principles calculations. It crystallizes in the LiMgPdSn type structure with lattice\nparameter 6.14 \u0017A at room temperature. Absence of (111) peak along with the presence of a weak\n(200) peak indicates the antisite disorder of Al with Cr and V atoms. The magnetization measure-\nments reveal a ferrimagnetic transition near 710 K and a coercive \feld of 100 Oe at 3 K. Very low\nmoment and coercive \feld indicate fully compensated ferrimagnetism in the alloy. Temperature co-\ne\u000ecient of resistivity is found to be negative, indicating a characteristic of semiconducting nature.\nAbsence of exponential dependence of resistivity on temperature indicates a gapless/spin-gapless\nsemiconducting behaviour. Electronic and magnetic properties of CrVTiAl for three possible crys-\ntallograpic con\fgurations are studied theoretically. All the three con\fgurations are found to be\ndi\u000berent forms of semiconductors. Ground state con\fguration is a fully compensated ferrimagnet\nwith band gaps 0.58 eV and 0.30 eV for up and down spin bands respectively. The next higher\nenergy con\fguration is also ferrimagnetic, but has spin-gapless semiconducting nature. The highest\nenergy con\fguration corresponds to a non-magnetic gapless semiconductor. The energy di\u000berences\namong these con\fgurations are quite small ( <1 mRy=atom) which hints that at \fnite temperatures,\nthe alloy exists in a disordered phase, which is a mixture of the three con\fgurations. By taking\ninto account the theoretical and the experimental \fndings, we conclude that CrVTiAl is a fully\ncompensated ferrimagnet with predominantly spin-gapless semiconductor nature.\nPACS numbers: 75.50.Gg, 75.50.Pp, 75.50.Ee, 78.40.Fy, 61.43.-j, 85.75.-d, 31.15.A\nI. INTRODUCTION\nSpintronics is an emerging branch of electronics in\nwhich the spin degree of freedom is added to the charge\ndegree of electron to realize many advantages such as\nnon-volatility, high processing speed, low power con-\nsumption, high storage density etc. over the conventional\nelectronics.1{7The utilization of the spin degree of free-\ndom i.e., in spintronic devices, can be found in spin diodes\nused in magnetic hard disks, read heads, magnetoresis-\ntive random access memory (MRAM), spin transistors,\ntunnel diodes, vortex oscillators etc.8{12For realization\nof spintronic devices, special materials are required, for\nexample, their electrical conduction should be restricted\nto one type of spin carriers. Such a phenomenon is seen\nin half metallic ferromagnets (HMF), spin gap-less semi-\nconductors (SGS), semiconducting spin \flters etc.3,13{16\nAmong the discovered materials, fully compensated fer-\nrimagnetic (FCF) materials have gained a lot of inter-\nest recently.17{19Leuken and Groot showed theoretically\nthat this new class of materials can show 100 % spin po-\nlarization without having a net magnetic moment, and\nhence given the name half metallic antiferromagnetic ma-\nterials or fully compensated ferrimagnets.20However, for\nthe antiferromagnets, symmetry demands the same den-\nsity of states (DOS) for spin up and spin down bands.19,21Due to symmetric bands and DOS, both the spin chan-\nnels equally contribute to electrical conductivity, which\nresults in zero net spin-polarized current. Such a scenario\nis not always true for FCF materials, which usually con-\ntain three or more magnetic ions with moments aligned\nin such a way that the net magnetization is nearly zero.\nSome of the unique properties and advantages of FCF\nmaterials are (i) nearly zero magnetic moment which cre-\nates no external stray \felds, resulting in low energy losses\n(ii) spin sensitivity without stray magnetic \felds, which\nallows them not to disturb the spin character and make\nthem ideal for spin polarized scanning tunnelling micro-\nscope tips and improved density of circuit integration in\na chip22(iii) low shape anisotropy, which helps in appli-\ncations in spin injection etc. These properties make them\nsuperior compared to HMF materials and are very much\nin demand today.\nThough Heusler alloys are known for many decades,\nthey gained renewed interest because of the develop-\nments in the \feld of spintronics.4,23{27Ozdogan et al.\nstudied the electronic and magnetic properties of qua-\nternary Heusler alloys (QHAs) theoretically by using\nthe full-potential non-orthogonal local-orbital minimum-\nbasis band structure scheme (FPLO).28Among the stud-\nied alloys, CrVTiAl (CVTA) has attracted a lot of inter-\nest and the preliminary band structure studies indicated\nit to be an antiferromagnetic semiconductor. Later, itarXiv:1707.04854v1  [cond-mat.mtrl-sci]  16 Jul 20172\nFIG. 1. Schematic density of states (DOS) of various types of\nsemiconductors. a) conventional semiconductor (CS) in which\nboth up ( \") and down ( #) spin bands have \fnite and equal\nband gaps. b) a gapless semiconductor (GS) where both the\nspin bands have vanishing band gap. c) a magnetic semicon-\nductor (MS) in which the band gaps of up and down spin\nbands are \fnite but unequal. d) a spin-gapless semiconduc-\ntor in which any one band (up or down) is gapless while the\nother is with \fnite gap.\nwas found that it is a fully compensated ferrimagnet with\ndistinct magnetic moments at Cr, V and Ti ions.29,30\nSchematic density of states of di\u000berent classes of semi-\nconductors are displayed in Fig. 1. Figure 1 (a) is a\nconventional semiconductor (CS) in which both spin up\nand spin down bands have equal band gap (\u0001 Eg). In\nthermal equilibrium, the intrinsic charge carrier concen-\ntration is given by31\nni= 2\u0012kBT\n2\u0019~2\u00133=2\n(memh)3=4e\u0000(\u0001Eg=2kBT):(1)\nHereme(mh) is the e\u000bective mass of electron(hole).\nThe conductivity in CS is dominated by exponential\nterm. Figure 1 (b) is the special case in which gap closes\n(\u0001Eg\u00190) (for example HgTe) in which nivaries as\nT3=2.32Figure 1 (c) is a typical magnetic semiconduc-\ntor in which band gap for each spin band is \fnite but\nnot equal, resulting in spin polarized intrinsic carriers\nand hence used in spin \flters. Figure 1(d) is the special\nclass of magnetic semiconductors in which one of the spin\nbands encounters zero gap (\u0001 Eg\"\u00190), while the other\nhas a \fnite gap.14,33,34In this case, the concentration of\nintrinsic spin up carriers ( n\"), which varies as T3=2dom-\ninates in comparison to that of spin down carries ( n#),\nwhich varies in an exponential manner and as a result,\nthe temperature dependence of the total concentration of\nintrinsic carriers slightly deviates from pure T3=2.\nAn experimental investigation carried out by\nStephen et al., has shown CVTA to be a magnetic\nsemiconductor,35but they attributed the resistivity be-\nhaviour to a combination of metallic and semiconducting\ncontributions. According to them, the magnetization\ndepends linearly on the \feld, indicating the antifer-\nromagnetic behaviour. However, a close inspection of\ntheir XRD pattern reveals small peaks near (220), which\nis indicative of secondary phase(s). In addition, theirsample shows a (111) peak with considerable intensity,\nunlike ours. The nearly equal electronegativities of Al\nand Cr/V causes the antisite disorder between these\nsites resulting the absence of superlattice (111) peak\nin the XRD.36In view of these di\u000berences and with\nthe aim of shedding more light into the anomalous\nproperties exhibited by this alloy, we have carried out\na combined theoretical and experimental study, which\npredicts entirely di\u000berent set of properties than what is\nreported earlier.\nII. EXPERIMENTAL AND THEORETICAL\nDETAILS\nPolycrystalline CrVTiAl alloy was prepared by arc\nmelting the stoichiometric proportions of constituent el-\nements with purity at least 99.99 %. Room temperature\nX-ray di\u000braction (XRD) patterns were collected using\nX'Pert Pro di\u000bractometer using Cu K \u000bradiation. The\ncrystal structure was analyzed by Rietveld re\fnement us-\ning FullProf suite.37\nThe crystal structure of QHAs of type XX0YZ (where\nX, X0, Y are transition elements and Z is the main group\nelement), can be described by three distinct (symme-\ntry inequivalent) possible arrangements of atoms.4,36The\nstructure consists of 4 wycko\u000b sites 4a, 4b, 4c and 4d. By\n\fxing Z at 4a site, the distinct con\fgurations are\n(I) X at 4b, X0at 4c and Y at 4d sites ,\n(II) X at 4c , X0at 4b and Y at 4d sites,\n(III) X at 4d , X0at 4c and Y at 4b sites\nrespectively. The structure factor for the \frst con\fgura-\ntion is given by\nFhkl= 4(fz+fye\u0019i(h+k+l)+fxe\u0019i\n2(h+k+l)+fx0e\u0000\u0019i\n2(h+k+l))\n(2)\nwith unmixed (hkl) values. Here fz,fy,fxandfx0are the\natomic scattering factors for the atoms Z, Y, X and X0\nrespectively. Therefore, the magnitudes of\nF111= 4[(fz\u0000fy)\u0000i(fx\u0000fx0)] (3)\nF200= 4[(fz+fy)\u0000(fx+fx0)] (4)\nF220= 4[fz+fy+fx+fx0] (5)\nare used to classify the order of the crystal structure.\nMagnetization measurements (from 2-400 K) were per-\nformed implementing zero \feld cooled warming (ZFCW)\nand \feld cooled warming (FCW) protocols in 500 Oe\nusing vibrating sample magnetometer (VSM, Quantum\nDesign). High temperature magnetization measurement\nwas carried out using (MPMS) in \feld warming (FW)\nmode at 1 kOe. Resistivity ( \u001a) measurements were car-\nried out using Physical Property Measurement System\n(PPMS) by four probe method applying 5 mA current.3\nA. Theoretical Details\nSpin-resolved Density Functional Theory (SDFT) as\nimplemented in Quantum Espresso (QE) package38was\nused to calculate the band structure and magnetic prop-\nerties of CVTA. The exchange-correlation functional\nwas taken within the generalized gradient approxima-\ntion (GGA) in the parametrization of Perdew-Burke-\nErnzerhof (PBE).39The pseudo potentials with Pro-\njector Augmented-Wave method40were generated using\nPSlibrary and QE. Self consistent calculations were car-\nried out using 24\u000224\u000224 k-point mesh with Methfessel-\nPaxton smearing of width 0.005 Ry, resulting in 413 k-\npoints in the irreducible wedge of the Brillouin zone. The\nenergy convergence criterion was set to 10\u00009Ry. The ki-\nnetic energy of the plane wave expansion (energy cuto\u000b\nEcut) was restricted to 60 Ry and charge density expan-\nsion to 700 Ry. Non-self consistent \feld calculations were\ncarried out using 48 \u000248\u000248 k-point grid. Projected den-\nsity of states (DOS) were extracted with an energy width\nof 0.0025 Ry.\nThermal charge carrier concentration was calculated\nusing theoretical DOS, D(E). The electron density\nabove the Fermi energy ( EF) at \fnite temperature T is\nD(E)f(E). Hence the total number of thermally created\nelectrons is\nne(T) =Z1\nE=0D(E)f(E)dE: (6)\nHereEFis taken as the reference level, f(E) = 1=(1 +\nexp(\u0000E=k BT)) is the Fermi function. In a similar man-\nner, the total number of thermally created holes can be\nfound by the expression\nnh(T) =Z0\nE=\u00001D(E)[1\u0000f(E)]dE: (7)\nIn an intrinsic semiconductor, at thermal equilibrium,\nthe number of thermal electrons is equal to the number\nof created holes i.e., nei=nhi=pnenh. In addition to\nthe thermally created charge carriers, there exists \fnite\nnumber of charge carriers ne0even atT= 0. So the total\nnumber of intrinsic carriers at a given Tisn= 2pnenh+\nne0. To obtain spin resolved total carriers one has to\nreplaceD(E) with spin resolved density of states. The\nintrinsic spin polarization is obtained by the following\nexpression\nP(T) =n\"(T)\u0000n#(T)\nn\"(T) +n#(T)\u0002100: (8)\nIII. EXPERIMENTAL RESULTS\nA. Crystal Structure\nCVTA is found to crystallize in the LiMgPdSn (space\ngroup F \u001643m, # 216) prototype structure (or Y struc-\nture) with a lattice parameter of aexp= 6.14 \u0017A. Figure\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s50/s52 /s50/s53 /s50/s54 /s50/s55 /s50/s56 /s50/s57 /s51/s48 /s51/s49\n/s40 /s41\n/s32/s32\n/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108\n/s32/s73\n/s111/s98/s115/s45/s73\n/s99/s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s40/s49/s49/s49/s41\n/s40/s50/s48/s48/s41\n/s40/s50/s50/s48/s41\n/s40/s51/s49/s49/s41\n/s40/s50/s50/s50/s41\n/s40/s52/s48/s48/s41\n/s40/s51/s51/s49/s41\n/s40/s52/s50/s48/s41\n/s40/s52/s50/s50/s41\n/s40/s53/s49/s49/s41/s40/s51/s51/s51/s41\n/s40/s52/s52/s48/s41\n/s40/s53/s51/s49/s41\n/s40/s52/s52/s50/s41 /s40/s54/s48/s48/s41\n/s40/s54/s50/s48/s41\n/s40/s53/s51/s51/s41/s67/s114/s86/s84/s105/s65/s108\n/s40 /s41\n/s40/s50/s32/s48/s32/s48/s41\n/s32/s32FIG. 2. Rietveld re\fned room temperature XRD of CVTA.\nThe super-lattice (111) re\rection is absent whereas a weak\n(200) peak is present (see inset).\n2 shows the XRD pattern for CVTA. Inset shows the\nzoomed region near (200) peaks. The absence of (111)\nand the presence of (200) superlattice peaks usually in-\ndicate the existence of B2 type disorder. For this type of\nantisite disorder to occur, there should be simultaneous\ndisorder between two pairs of atoms occupying octahe-\ndral sites and tetrahedral sites i.e., disorder between one\npair of X and X0and another pair of Y and Z. Because\nof this, the resulting structure resembles the CsCl type\nstructure. Rietveld re\fnement with B2 disorder did not\n\ft well for (200) peak. Observed peak intensity was much\nless than the calculated value for this peak. Subsequently\nit was \ftted to DO 3type anti-site disorder which yielded\ngood agreement between the experimental data and the\ntheoretical pattern. As Cr, V and Al have nearly same\nelectronegativity values, it is more probable to have anti-\nsite disorder among these atoms. Due to the large ionic\nradius and least electronegativity, Ti ions are less prone\nto have antisite disorder with other atoms. However, it is\nto be noted that XRD analysis alone cannot completely\nresolve the structural disorder in this alloy.\nB. Magnetic and Transport Properties\nFor QHAs composed of atleast two elements having less\nthan half-\flled d electrons, the saturation magnetization\nobeys the Slater Pauling (SP) rule,28,41,42\nM=N\u000018\u0016B=f:u: (9)\nwhereNis the total number of valence electrons in the\nalloy.\nFigure 3 (a) and (b) show the M-T and M-H data\nwhich clearly indicate a very small moment of CVTA\n(\u001810\u00003\u0016B=f:u:) and the magnetic ordering tempera-\nture is high (\u0018710 K, see the inset of Fig. 3(a)). M-H\ncurve, as shown in Fig. 3(b), has a low, nonzero hystere-\nsis (see also the inset). The behaviour remains almost4\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s50/s54/s48/s50/s55/s48/s50/s56/s48/s50/s57/s48/s51/s48/s48/s32/s40 /s45/s99/s109/s41/s67/s114/s86/s84/s105/s65/s108/s32/s32/s32/s48/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101\n/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s72/s32/s61/s32/s53/s48/s48/s32/s79/s101/s90/s70/s67/s87/s32 /s32/s77/s32/s40/s49/s48/s45/s52\n/s32\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s70/s67/s87\n/s67/s114/s86/s84/s105/s65/s108\n/s72/s32/s61/s32/s49/s32/s107/s79/s101\n/s32/s32\n/s32/s32/s70/s87/s77/s32/s40/s49/s48/s45/s52\n/s32\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s77/s32/s40/s49/s48/s45/s51\n/s32\n/s66/s47/s102/s46/s117/s41\n/s72/s32/s40/s107/s79/s101/s41/s64/s51/s32/s75\n/s67/s114/s86/s84/s105/s65/s108/s72\n/s99/s111/s101/s114/s126/s32/s49/s48/s48/s32/s79/s101/s32/s32\n/s40/s98/s41/s40/s99/s41\n/s40/s97/s41\nFIG. 3. (a) Temperature dependence of magnetization in\nZFCW and FCW modes. The inset shows the temperature\ndependence of magnetization in high temperature regime. (b)\nThe \feld dependence of magnetization. The inset shows the\nmagnetization in zoomed scale. (c) The temperature depen-\ndence of electrical resistivity in di\u000berent \felds.\nthe same at high T. Observation of extremely low mo-\nment with \fnite hysteresis indicate the strong possibility\nof fully compensated ferrimagnetic nature, as also found\nin our simulation. The outcome of nearly zero moment\nis consistent with the SP rule, which is a prerequisite\nfor spintronics materials. Stephen et. al. ,35on the other\nhand, reported a linear M-H curve, which may be due to\nthe presence of small impurities present in their sample.\nFigure 3(c) shows temperature dependence of resis-\ntivity in di\u000berent \felds. The resistivity shows negative\ntemperature coe\u000ecient, suggesting semiconducting be-\nhaviour. In intrinsic semiconductors, the variation of \u001a\nwith T is dominated by exponential term, as shown in Eq.\n1. Hence the absence of such a term in CVTA indicates\neither gapless or spin-gapless semiconducting nature.\nIV. THEORETICAL RESULTS\nWe have fully relaxed the experimentally formed crys-\ntal structure (space group # 216) in three distinct con\fg-\nurations, I, II and III, as described in Sec. II. The whole\nidea of performing these simulations was to get a better\nunderstanding of the XRD results (e.g. absence of (111)\npeak) which is not enough to clarify a few of the struc-\ntural aspects. Table I shows the relaxed lattice param-\neter (a0), total and atom projected magnetic moments\nand the total energy ( E0) for the three con\fgurations.TABLE I. Relaxed lattice parameter ( a0), atom-projected\nmagnetic moments, total moments ( \u0016B) and total energy ( E0)\nfor the three con\fgurations I, II and III of CVTA.\nTypea0(\u0017A)mCrmVmTimTotalE0(Ry/atom)\nI 6.08 0.00 0.00 0.00 0.00 -171.370934\nII 6.15 2.25 -1.26 -0.98 0.00 -171.370997\nIII 6.19 2.80 -2.29 -0.49 0.00 -171.371658\nAmong these, the con\fguration III was found to be\nenergetically the most stable one with lattice parameter\na0= 6:19\u0017A. The total energy di\u000berence among the three\ncon\fgurations is less than 1 mRy =atom which hints that\nat \fnite temperature CVTA could be a mixture of these\nthree con\fgurations, responsible for the observed disor-\nder. In order to understand the e\u000bect of this disorder,\nwe studied all the three con\fgurations in detail.\n/s45/s50/s45/s49/s48/s49\n/s87 /s32/s32/s76/s32 /s32/s32/s32/s88/s32/s85/s44/s75/s32/s32 /s32/s32/s32/s87 \n/s32/s45/s50/s45/s49/s48/s49\n/s87 /s32/s32/s76/s32 /s32/s32/s32/s88/s32/s85/s44/s75/s32/s32 /s32/s32/s32/s87 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49\n/s56 /s54 /s52 /s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49\n/s48 /s50 /s52/s45/s50/s45/s49/s48/s49/s69/s110/s101/s114/s103/s121/s32/s69/s40/s107/s41/s45/s69\n/s70/s32/s40/s101/s86/s41/s73\n/s73/s73\n/s73/s73/s73\n/s77/s111/s109/s101/s110/s116/s117/s109/s32/s40/s107/s41/s32/s32/s32/s32/s32/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s45/s102/s46/s117/s46/s41/s32/s32/s32/s32/s32/s32/s77/s111/s109/s101/s110/s116/s117/s109 /s32/s40/s107/s41\nFIG. 4. Spin polarized band structure and DOS for the con\fg-\nurations, I, II and III of CVTA at relaxed lattice parameters\n(a0). Left-side bands correspond to spin up while right-hand\nside is for spin down.\nCalculated spin polarized band structure and DOS for\nall the three con\fgurations are shown in Fig. 4. Cal-\nculations for all the three con\fgurations were initiated\nwith a ferrimagnetic arrangement of spins with moments\nat Cr-atoms aligned antiparallel to those of V and Ti.5\nFIG. 5. (a) Fermi surface (FS) is same for both up and\ndown spin bands of con\fguration I. It has spherical shape\nand crosses at X point. (b) Fermi surface for the spin down\nband of con\fguration II. It has oblate shape and crosses at X\npoint. Con\fguration III does not have any FS.\nThis was done keeping in mind the vanishingly small ex-\nperimental net moment (see previous section) and other\ntheoretical reports where ferrimagnetic arrangement was\nproposed to be the stable phase. In our case, con\fgura-\ntion I converges to a non-magnetic phase with identical\nspin up and down bands, and consequently has the lowest\nmagnetic ordering temperature. Both spin up and down\nbands are gapless with nearly zero DOS at EF, indicating\nthe gapless nature. It acquires an indirect band gap with\nconduction band minima touching at X-point and valence\nband maxima at other k-point. Figure 5(a) shows the\nFermi surface plot for con\fguration I. As expected, both\nspin up and down Fermi surfaces are identical, with tiny\nspherical shape and are shared by neighbouring Brillouin\nzone. The essential features of DOS and band structure\nremain unchanged at aexpand hence its physical proper-\nties (transport and magnetic) are robust.\nIn the case of con\fguration II, irrespective of the ini-\ntial magnetic moments at each site, the calculations con-\nverged in a ferrimagnetic arrangement with Cr moments\naligned antiparallel to V and Ti. For this con\fguration,\nDOS and band structure for spin down channel mostly\nresemble that of con\fguration I, except for the shape\nand size of Fermi surface at X-point, indicating gapless\nnature for spin down channel. The shape of the Fermi\nsurface, as shown in Fig. 5(b) is oblate, centred at X-\npoint and equally shared by neighbouring Brillouin zone.\nThe size of the surface is more than double that of con-\n\fguration I. There is almost no change in the DOS and\nband structure of spin down channel with a0andaexp,\nindicating that its spin down gapless nature is robust\nagainst small changes in a. On the other hand, DOS and\nband structure for spin up channel shows a clear indirect\nband gap of \u0001 E\"\ng= 0:36 eV, revealing semiconducting\nnature. There is no observable change in \u0001 E\"\ngwith lat-\ntice parameter indicating that its semiconducting nature\nis also robust against small changes in a. Due to the ab-\nsence of symmetric DOS and band structure along with\nrelatively high absolute magnetization (4.64 \u0016B=f:u:), itsmagnetic ordering temperature is expected to be very\nhigh. Such a phase with zero gap in one spin band and\n\fnite gap in the other gives rise to a fully compensated\nferrimagnetic, spin-gapless semiconductor.\nSimilar to con\fguration II, con\fguration III is also\nfound to be ferrimagnetic. However the later has \u0001 E\"\ng=\n0:58 eV and \u0001 E#\ng= 0:30 eV ata0, indicating that it is\na magnetic semiconductor. Spin up and down gaps are\nreduced to 0.55 eV and 0.25 eV respectively at aexp. Ab-\nsence of Fermi surfaces (no states at EF) for both spin\nchannels also con\frms the magnetic semiconducting na-\nture. Presence of large exchange splitting gaps and a\nlarge absolute magnetization (5.92 \u0016B=f:u:) indicates a\nlarge magnetic ordering temperature.\nAll the properties of con\fguration III (such as mag-\nnetic state, \u0001 E\";#\ngetc.), which corresponds to ground\nstate one, are in good agreement with the earlier reports\nby Ozdogan et. al with exception of sign of moments on\nall ions. As a result, DOS and band structure are in-\nterchanged for spin up and down electrons. In addition,\nthey reported a small positive moment on Al ions which\nalso has an opposite sign in the current study. Notably,\nCr and V moments are aligned in opposite directions and\nthe resulting moment is compensated by Ti ions (see Ta-\nble I).\nIt is important to note that all the con\fgurations show\ndi\u000berent forms of semiconductors, with two of them con-\nverged into the ferrimagnetic state. In order to under-\nstand the true nature of the semiconductor, it is quite\nrelevant to study the temperature dependence of the\ntransport properties i.e., intrinsic carrier concentration\n(n) and spin polarization ( P). Figure 6 shows the tem-\nperature dependence of these quantities for the three con-\n\fgurations. Con\fguration I, being almost non magnetic,\nhas negligible spin polarization ( P), although they indeed\nhave a \fnite carrier concentration due to small states at\nthe Fermi level ( EF), For con\fguration II and III, the spin\nup carrier concentration is negligibly small due to vanish-\ning states at EF. The spin down carrier concentration,\non the other hand, is large but has very di\u000berent nature\nof T-dependence for the two con\fgurations (II and III).\nInterestingly, in the case of con\fguration III, nshows an\nalmost straight line behaviour, which is neither exponen-\ntial nor T3=2. The magnitude, however, is much smaller\n(n\u00182) compared to the other two con\fgurations. Spin\npolarization values for these two con\fgurations are high.\nV. DISCUSSION AND CONCLUSION\nExperimental results reveal that CrVTiAl is a fully\ncompensated ferrimagnet with DO 3disorder among Al,\nCr and V atoms. First principle calculations within GGA\napproximation predict con\fguration III as the ground\nstate, which is a fully compensated ferrimagnet with un-\nequal band gaps for spin up and down channels. How-\never, the energy di\u000berences among all the three con\fg-\nurations (I, II and III) are small, which indicates the6\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s53/s49/s48/s49/s53\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s32/s110/s110/s32/s32/s32/s110/s32 /s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73/s73\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s57/s57/s46/s55/s57/s57/s46/s56/s57/s57/s46/s57/s49/s48/s48/s46/s48\n/s80\n/s32/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s110 /s32/s40/s49/s48/s49/s55\n/s47/s99/s99/s41\n/s32/s32\n/s110\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s32/s32\n/s32/s32/s110 /s32/s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s32/s83/s112/s105/s110/s45/s117/s112/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110 /s41\n/s32/s83/s112/s105/s110/s45/s100/s110/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110 /s41\n/s32/s84/s111/s116/s97/s108/s32/s67/s97/s114/s114/s105/s101/s114/s115/s32/s40/s110/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s50/s52/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s110/s110/s32\n/s32/s110 /s32/s40/s49/s48/s49/s57\n/s47/s99/s99/s41/s67/s114/s86/s84/s105/s65/s108/s45/s73/s73/s73\n/s57/s57/s46/s48/s57/s57/s46/s50/s57/s57/s46/s52/s57/s57/s46/s54/s57/s57/s46/s56/s49/s48/s48/s46/s48\n/s80\n/s32/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41/s32/s32\n/s110/s110 /s32/s49/s48/s49/s54\n/s47/s99/s99\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s40/s97/s41/s40/s98/s41/s40/s99/s41\nFIG. 6. Spin resolved intrinsic carrier concentrations (left axis) and spin polarization (right axis) vs. temperature (between\n3-400 K) for con\fguration I (left), II (middle) and III (right panel) for CVTA.\npossibility of a mixed phase or disordered phase at \f-\nnite T. One can consider the disordered state as a lin-\near combination of all the three independent con\fgura-\ntions (with probabilities proportional to their Boltzmann\nfactors) yielding either spin-gapless, gapless semiconduc-\ntor or magnetic semiconductor with reduced band gaps.\nThis concept originates from the fact that the disordered\nKohn Sham (KS) orbitals (\b dis) can be written as linear\ncombination of KS orbitals (\b) of each con\fguration i.e.,\n\bdis=cI\bI+cII\bII+cIII\bIIIwhereci/exp(\u0000Ei\n0=kBT).\nSuch a linear combination is possible due to the fact that\nall the three pure con\fgurations have vanishing states\nat the Fermi level, unlike most of the reported cases\nin which certain con\fgurations alone have \fnite states\natEF. At the observation level, the above linear com-\nbination presents itself as having a predominantly SGS\nproperty, because the \frst term is energetically least at-\ntainable and non magnetic while the third term is not\ne\u000bective as the DOS ( D\";#(E)) is zero at the Fermi level\n(E\";#\ng\u001dkBT). The last scenario can change if there are\nimpurities, which will alter the SGS nature seen in our\nsample.35In conclusion, we identify the true crystallographic and\nmagnetic ground states of CrVTiAl with the help of a\njoint theoretical and experimental investigation. While\nthe magnetic ground state is uniquely identi\fed as a fully\ncompensated ferrimagnet, the balance between spin gap-\nless nature and the magnetic semiconducting nature ap-\npears to be quite delicate according to the theoretical\ncalculations. 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Radu1\n1Helmholtz-Zentrum Berlin f ur Materialien und Energie, Albert-Einstein-Str.15, 12489 Berlin, Germany\n2Institute for Materials Research, Helmholtz-Zentrum Geesthacht, 21502 Geesthacht, Germany\n3Deutsches Elektronen-Synchrotron DESY, Notkestrae 85, 22607 Hamburg, Germany\n4Max-Planck-Institut fr Intelligente Systeme, 70569 Stuttgart, Germany\nOwing to the experimental discovery of magnetic skyrmions stabilized by the Dzyaloshinskii-\nMoriya and/or dipolar interactions in thin \flms, there is a recent upsurge of interest in magnetic\nskyrmions with antiferromagnetic spins in order to overcome the fundamental limitations inherent\nwith skyrmions in ferromagnetic materials. Here, we report on the observation of compact ferrimag-\nnetic skyrmions for the class of amorphous alloys consisting of 4f rare-earth and 3d transition-metal\nelements with perpendicular magnetic anisotropy, using a DyCo 3\flm, that are identi\fed by combin-\ning x-ray magnetic scattering, scanning transmission x-ray microscopy, and Hall transport technique.\nThese skyrmions, with antiparallel aligned Dy and Co magnetic moments and a characteristic core\nradius of about 40 nm, are formed during the nucleation and annihilation of the magnetic maze-like\ndomain pattern exhibiting a topological Hall e\u000bect contribution. Our \fndings provide a promising\nroute for fundamental research in the \feld of ferrimagnetic/antiferromagnetic spintronics towards\npractical applications.\nStabilized by the bulk Dzyaloshinskii-Moriya interac-\ntion (DMI), skyrmion lattices with swirling magnetic tex-\ntures have been originally predicted by Bogdanov et al.\n[1, 2] and later observed in non-centrosymmetric sin-\ngle crystal helimagnets [3{13]. The equilibrium condi-\ntions [14, 15], which satisfy the stability criteria for the\nformation of a robust Bloch-skyrmions lattice, \fll only\na narrow pocket of the magnetic phase diagram, usu-\nally of few tens of milli-Tesla and few degrees Celsius\naround the onset of the ordering temperature which is\nwell below room temperature. Di\u000berent approaches have\nbeen pursued to either stretch the skyrmion pocket to-\nwards room temperature, e.g., by reducing the dimen-\nsionality of the crystal using epitaxial thin \flms [16, 17]\nor by making use of interfacial DMI present in ultrathin\n\flms and multilayers leading to the formation of N\u0013 eel\nskyrmions [18{27]. For the latter case, recent observa-\ntions revealed that skyrmions can be created and ma-\nnipulated at room temperature, which makes such topo-\nlogical protected states particularly exciting for future\nspintronic applications [5, 7, 18, 28{30]. As exciting, it\nis suggested that skyrmions can occur also in systems\nwhich do not exhibit an intrinsic DMI, but a so called\ne\u000bective DMI that may result from mechanisms involv-\ning magnetic frustration like curvature-induced DMI [31{\n34], noncolinear type of magnetic interactions [35], and\nstray \felds for systems with relatively strong perpendic-\nular magnetic anisotropy [36].\nIn practice, however, the relative slow and pinning-\ndominated current-driven dynamic behaviors of the fer-\nromagnetic skyrmions [21, 23, 26] will limit their poten-\ntial for future spintronic applications. Moreover, the\ninteraction between spin-polarized currents and ferro-\nmagnetic skyrmions result in a non-collinear skyrmion\nmovement in respect to the current \row direction known\nas the skyrmion Hall e\u000bect (SkHE) [10, 25, 37]. Theskyrmion Hall e\u000bect, which is detrimental for applica-\ntions, is proportional to the net magnetization. Hence,\nantiferromagnetic (AF) skyrmions were proposed to sup-\npress the SkHE [38, 39], however, such skyrmions have\nbeen experimentally realized so far only in synthetic an-\ntiferromagnets [40{42]. Ferrimagnetic systems may open\nan alternative route towards the realization of skyrmion\nsystems suitable for applications. Particularly, ferrimag-\nnetic \flms of amorphous alloys consisting of 4f rare-earth\nand 3d transition-metal elements (RE-TM alloys) ex-\nhibiting perpendicular magnetic anisotropy are attract-\ning very much attention because of their versatile mag-\nnetic properties [43{50]. First realizations of skyrmions\nin ferrimagnetic RE-TM alloys were reported only re-\ncently, e.g., showing a promising reduction of the SkHE\n[51] or even a complete suppression of the SkHE at the\nangular momentum compensation temperature [52].\nFerrimagnetic RE-TM alloy \flms with perpendicular\nmagnetic anisotropy [50, 54, 55] are generally expected\nto host an eventual skyrmion state due to the bulk DMI\nresulting from the asymmetric distribution of the elemen-\ntal content [53], or the weak interfacial DMI induced by\nthe capping layer[56]. Notably, the absence of a center\nof inversion symmetry in amorphous materials inherently\ncauses a non-vanishing DMI [57]. For the particular case\nof DyCo xa few studies exist that indicate the existence\nof intrinsic nocollinear magnetic ground states for both\nsingle crystal and amorphous \flms [48, 62, 63]. For in-\nstance, for amorphous DyCo 3:4it is observed that the Dy\nmoments exhibit a sperimagnetic arrangement, whereas\nthe Co sublattice is ferromagnetically ordered [63].\nHere, we report on the observation of skyrmions in\nthis class of materials by investigating a ferrimagnetic\nDyCo 3(50 nm)/Ta(3 nm) \flm by means of x-ray mag-\nnetic circular dichroism (XMCD), small angle x-ray res-\nonant magnetic scattering (SAXRMS), high-resolutionarXiv:2008.00725v1  [cond-mat.mtrl-sci]  3 Aug 20202\n0.02 0.04 0.06 0.080.20.40.60.81.0\n Normalized intensity I\n II\n III\n IV\n V\n VI\n VII\nVIII\nQ (nm-1)Qφ\n(II)\n(III)\n(IV)\n(V)(VI)Qmax(b)\n-0.6 0.6 \nH (T) T=90K\n-101  M/Ms\n(I)(II)(III)(IV)(V)(VI)(VII)(VIII)\nBeamstop\nSampleX-ray\nCR/CL(a)\n(III)\n(I) (II)\n(III) (IV)\n(V) (VI)\n(VII) (VIII)(c) (d)Size of magnetic \nunit cell (nm)Hcr\n-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5200300400\n  \nμ0H (T)Maze-like domainsNucleation\nAnihilationSingle domain\nSingle domain\n(e)\nFIG. 1. X-ray magnetic scattering results for DyCo 3at 90 K.\n(a) Layout of the SAXRMS experiments: the maze-like do-\nmains produce donut-shaped scattering pattern recorded by\na charge-coupled device (CCD) camera. (b) Perpendicular\nmagnetic hysteresis loop of DyCo 3obtained by XMCD. (I)-\n(VII) mark the \felds for which the scattering patterns in\n(c) were required. (d) Intensity pro\fles extracted from the\ndi\u000braction pattern via azimuthal averaging of the scattering\npatterns. The solid lines are \fts to a split Pearson type VII\ndistribution to determine Q max. (e) Field dependent sizes of\nthe magnetic unit cell of DyCo 3.\nscanning transmission X-ray microscopy (STXM) with\ncircular polarized x-rays, and Hall transport technique.\nThe skyrmion phase is observed for magnetic \felds in\nthe transition region to a single domain state, providing\nan additional topological Hall resistance that is linked\nto the topology of the skyrmion's magnetization texture\n[58]. The magnetic microstructure of the skyrmions is\nfound to consist of a broad shell and a point-like core, re-\nferred to as compact skyrmions [13, 59{61], which can be\nexplained by the intrinsic magnetic material constants,\nthereby providing access to the strength of a DMI-like\ne\u000bect of the DyCo 3\flm.\nThe in\ruence of a perpendicular magnetic \feld on\nthe domain structure of DyCo 3\flm was investigated at\n90 K by means of SAXRMS. Using a beam diameter of\n120\u000250\u0016m2, this study con\frms that the statements\nmade in the following, in particular in the context ofskyrmion formation, are valid over the whole macroscopic\nsample. In order to avoid overexposure of the CCD de-\ntector, the direct beam was blocked by a beamstop with\n300\u0016m in diameter (Fig. 1a). The photon energy of\nthe circularly polarized X-rays was \fxed at 1294.3 eV\ncorresponding to the Dy M 5absorption edge, where the\nXMCD e\u000bect is maximized [54] (see Fig. 2a). Fig. 1b\nshows the magnetic hysteresis loop obtained by XMCD\nat Dy M 5-edge and Fig. 1c shows the corresponding\nSAXRMS patterns for di\u000berent magnetic \felds labeled\nas (I) to (VIII) (-0.18 to 0.43 T). The sheared regions\nof the magnetic hysteresis loop and the corresponding\ndonut-shaped scattering patterns as shown in Fig. 1c, re-\n\rect a maze-like domain con\fguration with alternating\nup and down domains. The radius of the ring with the\nmaximum scattering intensity at Qmax corresponds to\nthe periodicity of the domain pattern, which is twice of\nthe average domain size [64{66]: D=\u0019=Q max. Fig. 1d\nshows the azimuthally averaged intensity vs scattering\nvector Q of the patterns given in Fig. 1c.\nTo determine the scattering vector Q maxof the spectra\nS(Q) and hence the average domain size D as a function\nof magnetic \feld, they were \ftted using a split Pearson\ntype VII distribution [65]:\nS(Q) =S0[1 +(Q\u0000Qmax)2\nm\u000b2]\u0000m(1)\nThe D vs\feld behavior is given in Fig. 1e. The size of\nthe magnetic unit cell decreases with increasing the \feld\nuntil the maze domain phase with a similar size of up\nand down domains is reached (0.05-0.26 T, states (IV)-\n(VI)). The average domain size (half of the magnetic unit\ncell) evolves from a high value equal to \u0018200 nm right\nafter the nucleation \feld region down to about \u0018115 nm,\nclose to the anihilation region of these magnetic textures.\nNote that at the border to the nucleation and anihilation\nregions, the reciprocal map exhibits a broad scattering\npattern, which does not form a well de\fned donut-like\nshape. This is indicative for the formation of small mag-\nnetic domains.\nTo reveal the details of the nature of the magnetic do-\nmain structures, in particular the eventual formation of\nferrimagnetic skyrmions in this system, we performed\nelement-speci\fc STXM in the presence of an external\nperpendicular magnetic \feld at 90 K. The results are\nsummarized in Figure 2. The polarized x-ray absorption\nspectra (XAS) at the Co L 2;3and Dy M 4;5edges, which\nwere acquired at negative \feld saturation with circularly\nleft (CL, blue curve) and right (CR, red curve) polarized\nx-rays, are shown in Fig. 2a. The XMCD signal, negative\nfor Dy at the Dy-M 5edge and positive for Co at the Co-\nL3edge, revealing the ferrimagnetic alignment between\nthe Dy and Co moments, with the magnetization of Dy\noriented along the magnetic \feld [54]. The insets of Fig.\n2a show STXM images of the skyrmions for both the Co\nL3and Dy M 5edge. At the \feld of \u00160Hz=\u00000:09T,3\n-0.090TCo L3-edge\n-0.090\nTDy M5-edge\n1290 1300 1310 1320 1330 134002\n1760 780 800 820 840\n0.00.51.01.5\nCL\nCR\nCL\nCRXAS (arb. unit)Co L_edge\nDy M_edge\nPhoton energy (eV)Photon energy (eV)(a) (b)\n2μmCo L3-edge-STXM-0.075T -0.045T -0.0T\n+0.05T\n+0.15T +0.24T(c)\n1μm\n+0.05T\n+0.15T\n+0.24T(d)μ2 m Co L3-edge\n-0.110T\n1μm\n(I)\n(II)\n(III)\n(IV)(I) (II) (III)\n(IV)\nFIG. 2. STXM of domain structures at 90 K. (a) The X-ray\nabsorption spectra at Co L 2;3and Dy M 4;5edges acquired\nfor CL and CR polarization and negative \feld saturation,\nshowing the ferrimagnetic alignment of Dy and Co moments.\nThe insets show STXM images for CL polarization of ferri-\nmagnetic skyrmions at a \feld of -0.09 T with opposite con-\ntrast for Co and Dy sites. (b),(c) STXM images acquired at\nthe Co L 3edge and di\u000berent perpendicular magnetic \felds\nas labelled, displaying ferrimagentic skyrmions and maze-like\ndomain structures (d) and the bottom panel in (b) show cor-\nresponding high resolution images.\nferrimagnetic skyrmions are observed for both Co-L 3and\nDy-M 5edges. The inverted contrast particularly reveals\nthe ferrimagnetic nature of the skyrmions.\nStarting from magnetic saturation the nucleation of\nmultiple isolated ferrimagnetic skyrmions is observed at\na \feld of -0.110 T as shown in Fig. 2b. Some individ-\nual skyrmions with quasi-circular shape are shown with\nhigher magni\fcation at the bottom of panel (b), demon-\nstrating skyrmion dimensions in the range of 100 nm.\nWhen the external magnetic \feld is further increased,\nthe ferrimagnetic skyrmions begin to merge, transform-\ning into a maze-like domain structure up to +0.24 T, as\ndepicted in Fig. 2c, d. The latter images demonstrate\nthe growth of Co down domains at the expense of up\ndomains with increasing \feld strength.\nIn order to quantify the lateral skyrmion pro\fle in\ngreat detail, we performed high resolution STXM for CL\nand CR polarized x-rays (Fig. 3a) resonantly tuned to\nthe Co L 3-edge for an external \feld of \u00160Hz= -0.09\nT, where isolated skyrmions are present. A signi\fcant\ngradient in the transmitted intensity can be clearly seen\ndown to the center of the skyrmions, indicating the for-\nmZ,Co=+1\nmZ,Co=-1\n2R\nwCo magnetization Mz/MSCo magnetization Mz/MS\n2R\nwY-position (nm)\nX-position (nm)\nf\n0 2 4 6090180\nr/w  Y-axis. R=35 nm\n X-axis. R=42 nm\n κ=0.5\n κ=0.7\n κ=0.8\n κ=0.9\nκ=0.95θ(degree)FIG. 3. Magnetic pro\fle of ferrimagnetic skyrmions. High\nresolution STXM images showing the transmitted X-ray in-\ntensityI+andI\u0000for CL (a) and CR polarized x-rays\n(b), respectively, resonantly tuned to the Co L 3-edge at\n90K and\u00160Hz=-0.09T. (c) Map of the the magnetic con-\ntrast , i.e., perpendicular component of Co magnetization\nmz;Co =Mz;Co=MS;Co (MS;Co: Co saturation magnetiza-\ntion), obtained from the XMCD intensity given as ln(I+=I\u0000).\nThe rather broad skyrmion wall width wand the narrow\nskyrmion radius Rresult in a loss of the magnetic contrast\nin the skyrmion center. Vertical (d) and horizontal (e) mag-\nnetic pro\fle of the ferrimagnetic skyrmion picked from (c) as\nindicated by the green circle. The dots give the experimen-\ntal pro\fles while the green curves show the skyrmion pro\fle\naccording to Eq. (2) which is obtained by a deconvolution of\nthe experimental with the Gaussian beam pro\fle (FWHM of\n65 nm). The green curves are the calculated convolutions of\nthe beam pro\fle and the skyrmion pro\fle perfectly describ-\ning the experimental data. (f) Cross sections of the selected\nskyrmion (green and blue dots) shown together with calcu-\nlated pro\fles (solid lines) of isolated magnetic skyrmions for\nvarious\u0014taken from Ref [64].\nmation of compact skyrmions with the inner domain re-\nduced to a point-like core [13]. In contrast to this, bub-\nble skyrmions exhibit thin domain wall widths and broad\nskyrmion radii [36]. For compact skyrmions the skyrmion\npro\fle can be modeled by the Walker-like domain wall\nsolution of a standard 360\u000e-N\u0013 eel wall:\n\u0002z(r) = 2 arctanfsinh(R=w)\nsinh(r=w)g (2)4\nwhere \u0002,R, andwde\fnes the polar angle of the mag-\nnetization at position r, the skyrmion radius and mag-\nnetic wall width, respectively. The magnetic contrast is\nshown in Fig. 3c, which is a convolution of the skyrmion\npro\fle and a two dimensional Gaussian function with a\nFWHM of 65 nm, corresponding to the lateral resolu-\ntion of the STXM measurements. Hence, the reason for\nthe apparent vanishing magnetic contrast in the center of\nthe skyrmions stems from the comparable size of lateral\nresolution and skyrmion radius.\nA quasi-circular ferrimagnetic skyrmion was selected\nfrom Fig. 3c (green circle) to investigate its pro\fle along\ntwo orthogonal directions (Fig. 3d, e). The skyrmion\npro\fles are well reproduced by Eq. 2 with a domain wall\nwidth ofw= 50\u00065 nm and a skyrmion core or center\nradius ofR=35\u00065 and 42\u00065 nm for the pro\fles along\nthe vertical and horizontal direction, respectively.\nAccording to Eq. 2, the classi\fcation of skyrmions,\nwhether bubble-like or compact with a point-like core, is\nbased on the R/w ratio: for R \u001dw, bubble skyrmions ex-\nist, while for R/w <\u00181 compact skyrmions are formed. As\npointed out by B uttner et al., the shape of the skyrmions\nis determined by the parameter \u0014=\u0019Di=(4p\nAKeff),\nwith D i, A and K effrepresenting the DMI constant,\nexchange sti\u000bness, and e\u000bective anisotropy, respec-\ntively [36]. The e\u000bective anisotropy consists of magne-\ntocrystalline and shape anisotropy. The saturation mag-\nnetization per DyCo 3cluster is 2.5-3.5 \u0016Bbelow 150 K\nand drops to 1.0 \u0016Babove 200 K [54]. Thus, at tem-\nperatures below 90 K used in this study the magneto-\nstatic interaction is rather strong providing a signi\fcant\ncontribution to the stabilization of skyrmions. The mag-\nnetic pro\fle of the skyrmion shown in Fig. 3d and 3e can\nbe well described for \u0014in the range of 0.5 to 0.7 (see\nFig. 3f). Considering an exchange sti\u000bness of A=6 pJ/m\n[50] and an e\u000bective anisotropy of K eff=2:5\u0002104J=m3,\nthe DMI constant is estimated to be in the range of\n0.25-0.35 mJ/m2, which is in good agreement with the\nvalue of 0.18 mJ/m2determined for a 70 nm-thick DyCo 4\n\flm[48]. The domain wall width can be calculated to be\nw=\u0019p\nA=K eff= 49nm, which is also in agreement to\nthe value of 50 nm obtained from the skyrmion pro\fle\n(Fig. 3d-e). In the presence of e\u000bective DMI the do-\nmain wall energy \u001bw= 4p\nAKeff\u0000\u0019Dicorresponds to\n0.61 mJ/m2.\nTo investigate the electrical signature of the ferrimag-\nnetic skyrmions, magnetotransport measurements were\nperformed using a current density of j= 4\u0002106Am\u00002\ndriven along the x-axis and the magnetic \feld applied\nperpendicular to the plane of the \flm, i.e., H kz (Fig. 4a).\nThe Hall resistivity \u001axy(Fig. 4e) and longitudinal re-\nsistivity\u001axx(Fig. 4f) are measured at T=250, 175 and\n50 K. The STXM images taken at slightly di\u000berent but at\ncomparable temperatures of 90, 200, and 300K (Fig. 4b-\nd) show maze-like domain states in the demagnetized\nstate whose domain size strongly increases with tempera-\n2 μm\n300 K 90 K 200 K \n(b) \n(c) \n(d)\n-6 -4 -2 0 2 6-0.06-0.04-0.020.00[ρxx-ρxx(0)] /ρxx(0) \nμ0Hz (T)4250 K\n175 K\n50 K50 K175 K250 K\nρxx-1.0 -0.5 0.0 0.5 1.0-0.20.00.2\n  ρxy (μΩ cm)\nμ0Hz (T)ρxy\nρxyρxy(μΩ cm)\nHz\nHz-1.0 -0.5 0.0 0.5 1.0-0.20.00.2\n  \n ρxy\n ρxymodel\nμ0Hz (T)ρ xy(μΩ cm)T\nHz\n-1.0-0.50.0 0.5 1.0-0.020.000.02\n  \nμ0Hz (T)xyzjρxy\nHzρxx (a)\n \n(e)\n \n(f) \n(g)\n \n(h)T=50K\nFIG. 4. Topological Hall e\u000bect of DyCo 3. (a)Schematics for\nthe longitudinal ( \u001axx) and Hall resistivity ( \u001axy) measurements\nimpression using perpendicular magnetic \felds (H z). (e)\u001axy\nand (f)\u001axxvsHzbehavior for 50K, 175 K, and 250 K. STXM\nimages at T=90 (b), 200 (c), and 300K (d). (g) \u001axy(Hz) and\n\u001amodel\nxy (Hz) at 50 K shown for positive/negative \feld sweep di-\nrections. The latter is obtained from the \fts of the XMCD in-\ntensity. (h) Topological Hall signal \u001aT\nxy=\u001axy(Hz)-\u001amodel\nxy (Hz),\nfor both \feld sweep directions, which is non-zero in the \feld\nregion where individual skyrmions in a ferrimagnetic matrix\nare formed.\nture. Hence, the domain wall density as well the conven-\ntional positive domain wall resistance in \u001axx(Hz), i.e.,\nthe di\u000berence in resistivity at zero \feld and saturation\n\u0001\u001aDW, strongly increases with decreasing temperature\n(Fig. 4(f)).\nIn the absence of topological entities the Hall resistivity\n\u001axy(Hz) consists of contributions from conventional and\nanomalous Hall e\u000bects [67]:\n\u001amodel\nxy (Hz) = R 0Hz+ R SMz(Hz) (3)\nwith R 0the conventional Hall coe\u000ecient, and R Sthe cu-\nmulative anomalous Hall contribution from skew scatter-\ning, side-jump scattering, and the intrinsic (momentum\nspace) Berry curvature mechanisms [67].\nIn order to demonstrate for a topological Hall signal\ncaused by the skyrmions [27], the measured \u001axy(Hz) sig-\nnal at 50 K (see Fig. 4e) is compared with \u001amodel\nxy (Hz),\nwhich is obtained from the Mz(Hz) behavior measured\nby XMCD at the same temperature according to Eq. 3,\nalso accounting for the small normal Hall e\u000bect contribu-\ntion. Obviously, in the region where individual skyrmions\nare present, normal and anomalous Hall e\u000bects cannot\nfully describe the data, therefore indicating the pres-\nence of a topological Hall e\u000bect (THE) contribution:\n\u001aT\nxy(H) =\u001axy(Hz)\u0000\u001amodel\nxy (Hz), which is related to the\nchirality of the local domain structures. Such a addtional5\nsignal is clearly observed as shown in Fig. 4h. The max-\nimum value of the THE of \u0018\u00060.03\u0016\ncmis similar to\nthe one observed in Ir/Fe/Co/Pt multilayers [27].\nThe THE has been successfully used to indicate the\npresence of ferromagnetic skyrmions in bulk systems and\nmultilayers. Trivial N\u0013 eel or Bloch walls with chiral num-\nber of zero will not cause a THE. In ferrimagnets, how-\never, a contribution to the THE may arise from a canting\nbetween the Co and Dy moments within the chiral do-\nmain wall region. Future investigations focusing on de-\ntails of the spin structure of the domain wall will help\nto clarify if additional contributions to the THE in ferri-\nmagnets exist, as speculated above.\nIn conclusion, we have combined soft x-ray scatter-\ning and imaging techniques with transport measurements\nto investigate the formation of compact ferrimagnetic\nskyrmions in DyCo 3thin \flms with perpendicular mag-\nnetic anisotropy. Isolated skyrmions are imaged at 90 K\nby STXM for narrow perpendicular magnetic \feld re-\ngions. Starting at the magnetic saturation ( mz= -1),\nthe skyrmions nucleate at \u00160Hz= -0.11 T continuing\nto merge when positively sweeping the magnetic \feld.\nA second skyrmion regime is reached when applying\ncounter \felds ( Hz>0.25 T) that gradually annihilate the\nmaze-like domain pattern. The detection of a topological\nHall e\u000bect contribution indicates the existence of isolated\nskyrmions in broader \feld regimes at a lower temperature\nof 50 K. The magnetic microstructure of the skyrmions is\nfound to consist of a small core region of 40 nm and a sur-\nprisingly broad outer wall width of 50 nm. The skyrmion\npro\fle can be reproduced when considering the intrinsic\nmagnetic material parametes including a DMI constant\nof about 0.3 mJ/cm2, which is consistent with previous\nstudies on DyCo x\flms. The promising properties of fer-\nrimagnetic skyrmions in DyCo xalloy thin \flm systems in\ncombination with the possibility to easily tune their mag-\nnetic properties by varying their stoichiometry might be a\npromising route for skyrmions to be used in practical ap-\nplications in future spintronic devices. In particular, the\ndetrimental skyrmion Hall e\u000bect can be minimized when\nsetting the compensation temperature close to room tem-\nperature, which will possibly enable the realization of\nfunctional skyrmion devices that can be operated under\nambient conditions. Future systematic investigations of\nthe skyrmion pocket in the magnetic phase diagram for\nDyCo xand TM-RE alloys in general will reveal their full\npotential.\nThe authors acknowledge the \fnancial support for the\nPM2-VEKMAG beamline by the German Federal Min-\nistry for Education and Research (BMBF 05K10PC2,\n05K10WR1, 05K10KE1) and by HZB. A.P.-K. grate-\nfully acknowledges support from the DFG via Sonder-\nforschungsbereich (collaborative research center) SFB\n925 (subproject B3) and S. Rudor\u000b is acknowledged for\ntechnical support.SUPPLEMENTARY-METHODS\nThe samples were prepared by magnetron sputtering\n(MAGSSY chamber at BESSY) in an argon atmosphere\nof 1:5\u000210\u00003mbar with a base pressure of 5 \u000210\u00009mbar\nat a deposition temperature of 300 K. Si 3N4membranes\nwith a surface area of 5 \u00025 mm2and a thickness of 100 nm\nwere used as substrates for the soft x-ray transmission\nmeasurements including SAXRMS and STXM. A 3 nm\nthick Ta capping layer was grown on the DyCo 3layer to\nprevent surface oxidation.\nSAXRMS experiments have been performed at the\nVEKMAG end-station at the PM2 beamline, Helmholtz-\nZentrum Berlin (HZB). The di\u000bracted x-rays are col-\nlected on a Peltier-cooled square-shaped CCD detector\ncovering 2.1\u000eat the working distance of this study. The\nSAXRMS spectra (Fig. 1d) were retrieved by azimuthal\naveraging of the 2D patterns (Fig. 1c) after background\nsubtraction and masking of beamstop shadow and charge\nscattering streaks from the membrane edges. All intensi-\nties were normalized to the charge-scattering signal from\nthe membrane edges. The magnetic spectra were \ftted\nwith a split Pearson type VII distribution.\nElement-speci\fc STXM measurements were performed\nat MAXYMUS, beamline UE46 at HZB in the presence\nof an external magnetic \feld, H zparallel or antiparallel\nto the x-ray beam. 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Bagschik, et al., Employing soft x-ray resonant mag-\nnetic scattering to study domain sizes and anisotropy in\nCo/Pd multilayers, Phys. Rev. B 94, 134413 (2016).\n[67] N. Nagaosa, S. Onoda, A. H. MacDonald, & N. P. Ong,\nAnomalous Hall e\u000bect, Rev. Mod. Phys. 82, 1539-1592\n(2010)."    },    {        "title": "2304.14009v1.Effective_Tight_Binding_Model_of_Compensated_Ferrimagnetic_Weyl_Semimetal_with_Spontaneous_Orbital_Magnetization.pdf",        "content": "E\u000bective Tight-Binding Model of Compensated Ferrimagnetic Weyl Semimetal with\nSpontaneous Orbital Magnetization\nTomonari Meguro1,\u0003Akihiro Ozawa2,yKoji Kobayashi1,zand Kentaro Nomura1x\n1Department of Physics, Kyushu University, Fukuoka 819-0395, Japan and\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\nThe e\u000bective tight-binding model with compensated ferrimagnetic inverse-Heusler lattice\nTi2MnAl, candidate material of magnetic Weyl semimetal, is proposed. The energy spectrum near\nthe Fermi level, the con\fgurations of the Weyl points, and the anomalous Hall conductivity are\ncalculated. We found that the orbital magnetization is \fnite, while the total spin magnetization\nvanishes, at the energy of the Weyl points. The magnetic moments at each site are correlated with\nthe orbital magnetization, and can be controlled by the external magnetic \feld.\nI. INTRODUCTION\nTopological semimetals are new classes of materi-\nals characterized by the topologically-nontrivial gapless\nnodes. One of the representative systems is the Weyl\nsemimetal (WSM), a gapless semiconductor with non-\ndegenerate point nodes called Weyl points [1{3]. In mo-\nmentum space, these nodes behave as a source or sink\nof a \fctitious magnetic \feld (Berry curvature), which is\ndistinguished by the sign degrees of freedom, so-called\nchirality [4, 5]. The Weyl points with the opposite chi-\nrality must appear in pairs [6, 7]. Originating from the\nWeyl points, distinctive magnetoelectric e\u000bects, such as\nthe chiral magnetic e\u000bect [8{11], arise. Generally, to re-\nalize the WSM, one has to break either the inversion\nor time-reversal symmetry of Dirac semimetal [12{14],\nwhich has degenerate gapless linear dispersions. As early\ntheoretical works, the WSM phases in an antiferromag-\nnetic pyrochlore structure [2] and a topological insula-\ntor multilayer [3] were proposed. Especially, in Ref. 3,\nBurkov et al. discussed the anomalous Hall e\u000bect (AHE)\nin the WSMs with broken time-reversal symmetry, so-\ncalled magnetic Weyl semimetals (MWSMs). It was\nshown that the anomalous Hall conductivity (AHC) is\nproportional to the distance between the pair of Weyl\npoints with the opposite chirality.\nAfter these theoretical predictions, exploring for the\nWSM phases in speci\fc materials has been demonstrated.\nIn the early stage of the experimental studies, non-\nmagnetic WSMs with the broken inversion symmetry,\nsuch as TaAs, were examined [15, 16]. On the other\nhand, a great deal of e\u000bort has been devoted to realize\nthe MWSMs. After then, both theoretical and experi-\nmental studies have succeeded in discovering the MWSM\nphases in some systems, such as layered-kagome [17{40]\n\u0003meguro.tomonari@mbp.phys.phys.kyushu-u.ac.jp\nyPresent address: Institute for Solid State Physics, Univer-\nsity of Tokyo, Kashiwa 277-8581, Japan; akihiroozawa@issp.u-\ntokyo.ac.jp\nzPresent address: Physics Division, Sophia University, Chiyoda-\nku, Tokyo 102-8554, Japan; k-koji@sophia.ac.jp\nxnomura.kentaro@phys.kyushu-u.ac.jpand Heusler systems [41{47]. The layered-kagome sys-\ntems attract much attention from viewpoints of anoma-\nlous transport phenomena [18{26, 31, 32, 38] and various\nmagnetic orderings [17, 34, 36, 39, 40]. Antiferromag-\nnetic Mn3Sn shows the AHE even without net magneti-\nzation [18{24]. Ferromagnetic Co3Sn2S2shows the giant\nAHE and small longitudinal conductivity, resulting in the\nlarge anomalous Hall angle reaching 20% [26{30, 33, 35,\n37, 39]. As other promising candidates, recent studies\nreported ferromagnetic Heusler systems with relatively\nhigh Curie temperature TCcompared to those of the\nlayered-kagome systems. For example, Co2MnGa [41{\n43] withTC\u0019694 K [44] and Co2MnAl [45, 46] with\nTC\u0019724 K [47] are also studied. In addition to the\nlayered-kagome and Heusler systems, other systems such\nas EuCd2As2[48{50] and LnAlPn[51, 52] (Ln= lan-\nthanides,Pn= Ge, Si) are also reported as candidate\nmaterials of MWSMs. A wide variety of candidates for\nMWSMs has been explored and has in\ruenced on both\nthe \feld of topological materials and magnetism.\nThe magnetic Weyl semimetal phase has been re-\nported in compensated ferrimagnetic inverse Heusler al-\nloy Ti2MnAl [53] by \frst-principles calculations. In\nTi2MnAl, magnetic moments at two Ti sites are anti-\nparallel to that at the Mn site, showing zero net magne-\ntization. The transition temperature is determined to be\n650 K [54], which is comparable to those of other Heusler\nWeyl systems, such as Co2MnGa. Besides, Ti2MnAl\nexhibits a large AHE despite its vanishing total spin\nmagnetization, similar to Mn3Sn. However, in contrast\nto Mn3Sn, the density of states at the Weyl points in\nTi2MnAl is negligibly small, resulting in the manifesta-\ntion of a large anomalous Hall angle compared to other\nMWSMs. These properties indicate that the unique elec-\ntronic and magnetic structures of Ti2MnAl provide dis-\ntinctive magnetoelectric response and spintronic func-\ntionalities compared to conventional materials.\nIn order to study the magnetoelectric responses spe-\nci\fc to Ti2MnAl, a quantitative analysis is necessary.\nFirst-principles calculations are widely used to precisely\ncalculate the electronic structure, considering all the elec-\ntron orbitals. However, with the \frst-principles calcula-\ntions, it is generally di\u000ecult to study the magnetoelectric\nresponses related to the complicated spin texture, sucharXiv:2304.14009v1  [cond-mat.mes-hall]  27 Apr 20232\n(a) Lattice structure(b) Primitive cell\n(c) Inter-sublattice hopping(d) Intra-sublattice hoppingzyx\nTi1\nMn\nAl\nTi2\nFIG. 1. (a) Conventional unit cell of Ti 2MnAl. Ti and Mn\nare responsible for the ferrimagnetic ordering. (b) Primitive\nunit cell of Ti 2MnAl. In our model, only Ti1 (A), Ti2 (B),\nand Mn (C) sublattices are taken into account. (c) Inter-\nsublattice nearest-neighbor hoppings. (d) Intra-sublattice\nnearest-neighbor hoppings. Every sublattice has the same\nhopping vectors.\nas the magnetic domain wall. This is because the ma-\ntrix of the Hamiltonian becomes huge due to the lack\nof translational symmetry. Therefore, a simple tight-\nbinding model describing the Weyl points of Ti2MnAl is\nbene\fcial to calculate these magnetoelectric responses.\nIn this paper, we construct an e\u000bective tight-binding\nmodel of Ti2MnAl using a few orbitals. We consider the\nsingle orbitals of two Ti and Mn, spin-orbit coupling, and\nexchange interaction between compensated ferrimagnetic\nordering and itinerant electron spin. Using our model,\nwe study the electronic structure, AHE, spin and orbital\nmagnetizations, and magnetic anisotropy. We show that\nour model describes the con\fguration of the Weyl points\nthat is consistent with the results obtained by the \frst-\nprinciples calculations. Also, we discuss the control of\ncompensated ferrimagnetic ordering by orbital magneti-\nzation.\nII. MODEL\nIn this section, we introduce a simple tight-binding\nmodel of the compensated ferrimagnetic Weyl semimetal\nTi2MnAl. The crystal structure of Ti2MnAl is shown in\nFig. 1(a). Each sublattice (Ti1, Ti2, Mn, and Al) forms\na face-centered-cubic (FCC) lattice, and thus the prim-\nitive unit cell is the FCC type [Fig. 1(b)]. By focusing\non pairs of the sublattices, Ti1-Al (orange and gray) and\nTi2-Mn (red and blue) form the rocksalt structure. The\nrest of the combinations, e.g., Ti1-Ti2 or Ti1-Mn, form\ndiamond or zincblende structures.Our model consists of a single orbital from each of\nTi1 (A), Ti2 (B), and Mn (C) atoms. We neglect Al or-\nbitals, which are not responsible for the magnetism, for\nsimplicity. We explain our model Hamiltonian Hby di-\nviding into three components, H=Ht+Hexc+HSOC,\nwhereHtrepresents the hopping, Hexcthe exchange cou-\npling, andHSOCthe spin-orbit coupling (SOC).\nThe hopping component Htreads\nHt=\u0000X\nhijis\u0010\ntABay\nisbjs+tBCby\niscjs+tCAcy\nisajs+ h:c:\u0011\n\u0000X\nhijis\u0010\ntAAay\nisajs+tBBby\nisbjs+tCCcy\niscjs\u0011\n+X\nis\u0010\n\u000fAay\nisais+\u000fBby\nisbis+\u000fCcy\niscis\u0011\n; (1)\nwhereais,bis, andcisare the annihilation operators for\nelectrons at Ti1 (A), Ti2 (B), and Mn (C) sites, respec-\ntively. The \frst line corresponds to the inter-sublattice\nnearest-neighbor hopping [Fig. 1(c)]. The second line cor-\nresponds to the intra-sublattice nearest-neighbor hopping\n[Fig. 1(d)]. The third line is the on-site energy.\nThe exchange component is\nHexc=\u0000X\nim\u0001(JAsA;i+JBsB;i\u0000JCsC;i);(2)\nwheresA;i=ay\nis(\u001b)ss0ais0is the itinerant spin operator\nof A site, and the same applies to sBandsC.mis the\nunit vector that is parallel to the magnetic moment of Ti\n(A and B) and antiparallel to that of Mn (C). J\u000b(\u000b=\nA;B;C) are the coupling strength.\nThe SOCHSOCoriginates from the broken inversion\nsymmetry of the crystal structure. The dominant sym-\nmetry breaking comes from the imbalance between TI1\nand Al sublattices. We assume the amplitudes of the\nSOC terms for the Ti2-Ti2 and Mn-Mn hoppings are the\nsame, for simplicity. Since the atomic number of Ti and\nMn are close to each other, compared with Al, we neglect\nthe SOC for the Ti1-Ti1 hopping. The SOC term can be\ndescribed in a Fu-Kane-Mele-like form [55],\nHSOC=\u0000i8\u0015SOCp\n2a2X\nhijih\nby\nis(dBij\n1\u0002dBij\n2)\u0001(\u001b)ss0bjs0\n+cy\nis(dCij\n1\u0002dCij\n2)\u0001(\u001b)ss0cjs0i\n:(3)\nHere\u0015SOCis the strength of the SOC. d\u000bij\n1;2are the two\nnearest-neighbor hopping vectors from the site itojof\nthe sublattice \u000b. Note that dBij\n1;2=nlijanddCij\n1;2=\n\u0000nlij(lij= 1;2;3;4) withn1=a\n4(1;\u00001;\u00001),n2=\na\n4(\u00001;1;\u00001),n3=a\n4(\u00001;\u00001;1), andn4=a\n4(1;1;1).\nWe set the lattice constant a= 1 for simplicity. The\nhopping parameters are set to tAB= 1:1t0,tBC= 0:4t0,\ntCA= 1:2t0,tAA= 0:05t0,tBB= 0:85t0, andtCC=\n\u00000:05t0. On-site energies \u000fA=\u000fB=\u000fC=\u00002:15t0. The\nstrengths of the exchange coupling JA=JB= 0:7t0,3\nJC= 1:7t0. The strength of the SOC \u0015SOC =\u00000:2t0.\nWe set the energy unit t0= 0:33 eV. The parameters\nare set so that the energy bands, density of states, and\nAHCs become consistent with \frst-principles calculations\nas discussed later.III. ELECTRONIC STRUCTURE\nNow, we study the electronic structure of this model.\nThe momentum representation of the Hamiltonian is ex-\nplicitly expressed as,\nH(k) =0\n@\u0000tAAf(k)\u0000JAm\u0001\u001b\u0000tABg(k) \u0000tCAg\u0003(k)\n\u0000tABg\u0003(k)\u0000tBBf(k) +\u0015SOCR(k)\u0001\u001b\u0000JBm\u0001\u001b\u0000tBCP\ni=x;y;zcoski\n2\n\u0000tCAg(k)\u0000tBCP\ni=x;y;zcoski\n2\u0000tCCf(k)\u0000\u0015SOCR(k)\u0001\u001b+JCm\u0001\u001b1\nA:(4)\nHere,f(k) andg(k) correspond to the FCC and diamond\nhoppings, respectively, and are de\fned as,\nf(k)=4\u0012\ncoskx\n2cosky\n2+cosky\n2coskz\n2+coskz\n2coskx\n2\u0013\n;(5)\ng(k) = exp \n\u0000i4X\n\u0016=1n\u0016\u0001k!\n: (6)\nR(k) corresponds to the FCC hopping with SOC and is\nde\fned as,\nR(k) =0\nBB@sinkx\n2\u0010\ncosky\n2\u0000coskz\n2\u0011\nsinky\n2\u0000\ncoskz\n2\u0000coskx\n2\u0001\nsinkz\n2\u0010\ncoskx\n2\u0000cosky\n2\u00111\nCCA: (7)\nWe use the magnetic moment pointing in the out-of-\nplane direction as m= (0;0;1). By solving the eigen-\nvalue equationH(k)jn;ki=Enkjn;ki, the eigenvalues\nEnkand eigenstatesjunkiare obtained. Here, n=\n1;2;:::;6 is the band index labeled from the bottom. Fig-\nures 2(a) and 2(b) show the band structure along the\nhigh-symmetry lines and the density of states (DOS) as\na function of energy, respectively. The high-symmetry\nlines are shown in Fig. 2(d). We assume that the Fermi\nlevelEFis the energy which is satis\fed 4 =6 \flling, and\nis being set as E=t 0= 0. At the energy EF, we have\nthe Weyl points as discussed later. In Fig. 2(a), red and\nblue lines indicate up and down spin band, respectively,\nin the absence of SOC. Green lines indicate those in the\npresence of SOC. Here, we focus on the spin up bands\nnearE=t 0= 0:0. As shown in Fig. 2(b), the spin up\nbands show the local minimum of the DOS near the Fermi\nlevel (E=t 0= 0:0). This minimum is consistent with the\nresult obtained by the \frst-principles calculations and\nmay be signi\fcant to the large anomalous Hall angle [53].\nAs shown in the insets of Fig. 2(b), the energy spec-\ntrum of these majority spin bands show the gapless lin-\near dispersions, corresponding to the Weyl points, as dis-\ncussed later. On the other hand, down-spin state (blue)\nis gapped at the Fermi level ( E=t 0= 0:0).Next we study the Weyl points structure in Brillouin\nzone. We show that our model describes the Weyl points\nstructure similarly located to the result obtained by the\n\frst-principles calculations. From our model Hamilto-\nnianH(k) 24 gapless nodes (Weyl points) are obtained\nbetweenn= 4 band and n= 5 band. The kx-kzplane\nhas the Weyl points as shown in Fig. 2(e). Each Weyl\npoints with chirality + ( \u0000) is labeled by W+\n\u000b(W\u0000\n\u000b).\u000b\ndistinguish the pairs of the Weyl points. To characterize\nthese nodes by chirality, we compute the Berry curva-\nturebnk=rk\u0002ankof then= 4 band. Here, ankis\nthe Berry connection, de\fned as ank=\u0000ihn;kjrkjn;ki.\nFigure 2(f) shows the Berry curvature distribution on kx-\nkzplane. The strength of the shade of the arrows is the\namplitude of b(k). The red and blue circles indicate the\nsources (+) and sinks ( \u0000) of the Berry curvature. The\ncon\fguration of the Weyl points is consistent with those\nobtained by the \frst-principles calculations [53].\nIV. ANOMALOUS HALL EFFECT\nLet us study the AHE in this section. Using the Kubo\nformula [56], the anomalous Hall conductivities (AHCs)\n\u001bij(i6=j) can be expressed as follows,\n\u001bij=\u0000ie2\n~Z\nBZd3k\n(2\u0019)3X\nm6=nf(Enk)\u0000f(Emk)\nEnk\u0000Emk+i\u0011\n\u0002hn;kj~vijm;kihm;kj~vjjn;ki\nEnk\u0000Emk:(8)\nHere,f(E) = 1=(e\f(E\u0000EF)+ 1) is the Fermi-Dirac distri-\nbution function and ~v=@H(k)\n@kis the velocity operator.\nFigure 2(c) shows \u001bxy,\u001byz, and\u001bzxas a function of en-\nergy. Near the energy of the Weyl points, the \u001bxyis\nmaximized , while \u001byzand\u001bzxvanish. The value of \u001bxy\nat the peak is \u001bxy\u00190:80 [e2=ah] = 496 S=cm, which rea-\nsonably agrees with the result obtained by \frst-principles\ncalculations, 550 S =cm [53].\nThen we study the relation between the AHCs and\nthe con\fguration of the Weyl points. As discussed in\nthe introduction, when the Fermi level is located at the4\n-6.-4.-2.0.2.4.6.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n����������|��-�-���(a)(b)(c)\n(d)(e)(f)E –EF[eV]—up—down—with SOC\nw/oSOCw/  SOCDOS [a.u.]AHC \t\t\t\t\t\t\t\t\n\t0\t1\t2\t3\t4\t5\t6\n\t0\t1\t2\t3\t4\t5\t6\t0\t0.2\t0.4\t0.6\t0.8\t1\n01234560123456\n!!\"!\"\"01234560123456024624610-1-210-1-20.00.51.0-1.0-0.510-1-2GXWKGLUWLK|UX—sxy syz—szxE –EF[eV]\nE –EF[eV]\n𝑊\"#𝑊\"$𝑊!\"𝑊!#𝑊%#𝑊%$𝑊&#𝑊&$𝑊'$𝑊'#kzkxkxkzahe2\nFIG. 2. (a) Band structure along the high symmetry lines. (b) The density of states for (red) up- and (blue) down-spin states\nand (c) anomalous Hall conductivities \u001bxy,\u001byx,\u001bzxas a function of energy. (d) The Brillouin zone and high symmetry lines\nof the system. (e) Con\fguration of the Weyl points with and without spin-orbit coupling and (f) the Berry curvature with\nspin-orbit coupling on the kx-kzplane.\nWeyl points, the AHCs can be calculated with the dis-\ntances between the Weyl points with opposite chirality\n\u0001Q\u000b= (K+\n\u000b\u0000K\u0000\n\u000b) [3],\n\u001bWeyl\nij =e2\n(2\u0019)2~12X\n\u000b=1X\nk\u000fijk\u0001Q\u000b\nk: (9)\nHereK+(\u0000)\n\u000b is the position of the Weyl points with chi-\nrality +(\u0000). This\u001bWeyl\nxy is being\u00190:75 [e2=ah], which\nis in good agreement with the maximized value obtained\nby the Kubo formula, shown in Fig. 2(c). Therefore,\nthe AHE in our model mainly originates from the Weyl\npoints.\nNext, we discuss the role of SOC for the Weyl points\nand the AHC. In the absence of SOC, the Weyl points\nare distributed symmetrically, as represented by empty\ncircles in Fig. 2(e). This can be anticipated by the\ncrystal symmetry of the system [53]. In this case, the\nsumP\n\u000b\u0001Q\u000bcancels, and thus AHC \u001bWeyl\nxy vanishes.\nOn the other hand, in the presence of SOC, the po-\nsitions of the Weyl points are shifted as represented\nby \flled circles in Fig. 2(e). For the pairs ( W+\n1,W\u0000\n1)\nand (W+\n2,W\u0000\n2), the distance between the Weyl points\u0001Q\u000b=1;2\nz becomes longer. Whereas, for the pairs of\n(W+\n3,W\u0000\n3) and (W+\n4,W\u0000\n4), \u0001Q\u000b=3;4\nz becomes shorter.\nTherefore, the cancellation of the sumP\n\u000b\u0001Q\u000bis bro-\nken, giving rise to the \fnite AHC \u001bWeyl\nxy. We note that\nthe Weyl points in other planes are shifted in a similar\nmanner to those in the kx-kzplane.\nV. SPIN AND ORBITAL MAGNETIZATIONS\nIn this section, we study spin and orbital magnetiza-\ntions in our model. The spin magnetization is calculated\nby the following equation,\nMspin\nz=\u0016BZEF\nE0[D\"(\u000f)\u0000D#(\u000f)]d\u000f: (10)\nHere,\u0016Bis the Bohr magneton, and E0is the lower band\nedge.D\"(\u000f) [D#(\u000f)] is the DOS for the up-spin (down-\nspin). The spin magnetization as a function of energy is\nshown in Fig. 3(a). Recall that the magnetic moments\nof Ti and Mn are parallel to the zaxis.Mspin\nzvanishes\natE=EF, indicating the compensated ferrimagnetic5\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n-1.-0.50.0.51.-2.-1.0.1.\n𝑀!\"#$%𝜇&Å'𝐸−𝐸!eV(a)(b)\n𝑀()*$+\t×10,'𝜇&Å'𝐸−𝐸!eV\n1.00.50.0-0.5-1.0-2-101\n-2-101\n1.00.50.0-0.5-1.0𝑀!\"#$%&𝑀'\"#$%&𝑀(\"#$%&\nFIG. 3. (a) Spin magnetization Mspin\nzand (b) each compo-\nnent of orbital magnetization Morbitas a function of energy\nfor the magnetic moment mparallel to the zaxis.\nphase. Owing to this characteristic DOS, \fnite spin mag-\nnetization may be obtained by tuning EF, using a gate\nvoltage, for instance. When EFis increased (decreased),\n\fnite and positive (negative) spin magnetization might\nbe generated. This indicates that, in Ti2MnAl, one can\ninduce and switch the spin magnetization electrically.\nRecently, the electrical control of ferrimagnetic systems\nhas been studied from viewpoint of a functional magnetic\nmemory, for example in Ref. 57.\nNext we study the orbital magnetization [58{60],\nMorbit\nk =e\n2~Z\nBZd3k\n(2\u0019)3X\ni;j\u000fijk\nImX\nm6=nf(Enk)hn;kj~vijm;kihm;kj~vjjn;ki\n(Emk\u0000Enk)2\n\u0002(Enk+Emk\u00002EF):\n(11)\nFigure 3(b) shows each component of the orbital magne-\ntizationMorbitas a function of energy. We \fnd that\nMorbit\nz can be \fnite, while Morbit\nx andMorbit\ny vanish.\nAlthough orbital magnetization is usually much smaller\nthan spin magnetization, Morbit\nz is \fnite at EF, where\nMspin\nzis fully compensated. As discussed later, this small\nbut \fnite orbital magnetization might be used to switch\nthe directions of magnetic moments on Ti and Mn sites.\nVI. MAGNETIC ISOTROPY\nIn this section, we study the magnetic anisotropy and\nangular dependences of the AHCs and orbital magneti-\nzation. We start to study magnetic anisotropy. Figure 4\n04590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.1\n𝜃!\"[deg]𝜃!\"=𝜃#$𝜃!\"=0\n𝜃!\"𝜃#$\n𝜃#$04590135180225270315360-2.10-2.15-2.20-2.25-2.30-2.35-2.40-2.45Total Energy eVFIG. 4. The magnetization-angle dependence of the total\nenergy of electron, Eq. (12), when one rotates the magnetiza-\ntions of Mn and Ti uniformly (solid red line) or the magneti-\nzation of Mn solely (dotted blue line).\nshows the total energy of electrons, computed as\nEe=1\nNk6X\nn=1X\nkEnkf(Enk); (12)\nwhereNkis the number of k-mesh. Here, we consider the\ntwo tilted con\fgurations in which the magnetic moments\nof (a) Mn only or (b) both Mn and Ti are tilted in x-\nzplane. Their tilting angles are denoted by the angles\n(a)\u0012Mnand (b)\u0012Mn=\u0012Ti. The schematic \fgures for\nthese situations are shown in the insets of Fig. 4. In case\n(a), as the blue dotted line indicates, the electron energy\nis maximized at \u0012Mn= 180\u000e, indicating the stability of\nthe compensated ferrimagnetic ordering. On the other\nhand, in case (b), the electron energy is independent of\n\u0012Mn=\u0012Tias the red solid line indicates. Therefore, in\nour model, the compensated ferrimagnetic ordering does\nnot show the easy-axis magnetic anisotropy. With these\nresults, the ferrimagnetic interaction between Ti and Mn\ncan be estimated as 0 :1t0\u001933 meV, when t0= 0.33 eV\nis assumed.\nThen, we study a correlation between magnetic mo-\nments and orbital magnetization. In the previous sec-\ntion, we showed that, at the energy of the Weyl points,\nour model shows compensated spin magnetization and\n\fnite orbital magnetization. In the following, we tilt the\nmagnetic moments on both Ti and Mn sites in the x-z\nplane, as shown in the inset of Fig. 5(a). Figure 5(a)\nshows each component of the orbital magnetization as\na function of \u0012. We \fnd that the orbital magnetization\nfollows the direction of the magnetic moments. This fea-\nture can be used for the control of the magnetic mo-\nments, as similarly discussed in Refs. 24 and 40. Under\nan external magnetic \feld, magnetic moments on each\nsite and orbital magnetization couple with the \feld via\nZeeman interaction. However, as discussed in the pre-\nvious paragraph, the ferrimagnetic coupling between the\nmagnetic moments on Ti and Mn is stronger than the\ntypical strength of the Zeeman interaction ( \u00190:15 meV\nat 1 T). Thus, the Zeeman interaction of the magnetic6\n04590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.104590135180225270315360-1.5-1.-0.50.0.51.1.5\n04590135180225270315360-0.06-0.04-0.020.0.020.040.06\n04590135180225270315360-2.45-2.4-2.35-2.3-2.25-2.2-2.15-2.1\n𝜃[deg]AHC !!\"#𝜎!\"𝜎#!𝜎\"#𝜃[deg]𝑀$%&'(\t𝜇)Å*𝑀#$%&'(𝑀!$%&'(𝑀\"$%&'((a)\n(b)\n𝜃𝜃04590135180225270315360\n045901351802252703153600.060.040.020.00-0.02-0.04-0.061.51.00.50.0-0.5-1.0-1.5\nFIG. 5. The magnetization-angle dependence of (a) orbital\nmagnetization and (b) AHCs at E=EFwhen one changes\nthe direction of magnetic moment uniformly in the x-zplane.\nmoments on both sites cancels each other. This cancella-\ntion implies that only the orbital magnetization couples\nwith the external magnetic \feld. In addition, owing to\nSOC, the directions of the magnetic moments are lockedwith the orbital magnetization, as shown in Fig. 5(a).\nConsequently, the direction of the magnetic moments can\nbe controlled by an external magnetic \feld, even with-\nout net magnetization. The changes in the direction of\nthe magnetic moments may be probed by the AHE. Fig-\nure 5(b) shows the \u0012dependence of the AHCs. We \fnd\nthe relation \u001bij/\u0000P\nk\u000fijkMorbit\nk. This relation indi-\ncates that the directions of the magnetic moments are\nexperimentally determined by measuring the AHCs.\nVII. CONCLUSION\nIn this paper, we constructed an e\u000bective model of\ncompensated ferrimagnetic Weyl semimetal Ti2MnAl.\nThe Weyl points con\fguration in our model reasonably\nagrees with those obtained by the \frst-principles calcu-\nlations. The \fnite AHC in the presence of SOC can be\nunderstood by the shifting of the Weyl points. 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Rev.\nLett.99, 197202 (2007), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.99.197202 .\n[60] Y. Ominato, S. Tatsumi, and K. Nomura, Phys. Rev.\nB99, 085205 (2019), URL https://link.aps.org/doi/\n10.1103/PhysRevB.99.085205 ."    },    {        "title": "2003.04608v1.Room_temperature_ferrimagnetism_of_anti_site_disordered_Ca2MnOsO6.pdf",        "content": "1 Room-temperature ferrimagnetism of anti-site-disordered Ca 2MnOsO 6 \nHai L. Feng ( 冯海),1,2,3,*# Madhav Prasad Ghimire,4,5,6 Zhiwei Hu,2 Sheng-Chieh Liao,2 Stefano \nAgrestini,2,7 Jie Chen,1,8 Yahua Yuan,9,1 Yoshitaka Matsushita,10 Yoshihiro Tsujimoto,1,8 Yoshio \nKatsuya,11 Masahiko Tanaka,11 Hong-Ji Lin,12 Chien-Te Chen,12 Shih-Chang Weng,12 Manuel \nValvidares,7 Kai Chen,13 Francois Baudelet,13 Arata Tanaka,14 Martha Greenblatt,3 Liu Hao Tjeng,2 \nKazunari Yamaura 1,8* \n1. Research Center for Functional Material, National Institute for Materials Science, 1-1 Namiki, \nTsukuba, Ibaraki 305-0044, Japan  \n2. Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany \n3. Department of Chemistry and Chemical Biology, Rutgers, the State University of New Jersey, 610 \nTaylor Road, Piscataway, New Jersey 08854, United States \n4. Central Department of Physics, Tribhuvan University, Kirtipur, 44613 Kathmandu, Nepal  \n5. Leibniz Institute for Solid State and Materials Research, IFW Dresden, P.O. Box 270116, \nD-01171 Dresden, Germany \n6. Condensed Matter Physics Research Center, Butwal-11, Rupandehi, Nepal \n7. ALBA Synchrotron Light Source, E-08290 Cerdanyola del Vall`es, Barcelona, Spain \n8. Graduate School of Chemical Sciences and Engineering, Hokkaido University, North 10 West 8, \nKita-ku, Sapporo, Hokkaido 060-0810, Japan  \n9. School of Physics and Electronics, Central South University, Changsha 410083, China \n10. Materials Analysis Station, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, \nIbaraki 305-0047, Japan  \n11. Synchrotron X-ray Station at SPring-8, National Institute for Materials Science, Kouto 1-1-1, \nSayo-cho, Hyogo 679-5148, Japan  \n12. National Synchrotron Radiation Research Center, 101 Hsin-Ann Road, Hsinchu 30076, Taiwan, \nROC \n13. Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, 91192 Gif-sur-Yvette Cedex, France \n14. Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8526, \nJapan \n* Corresponding authors E-mail: hai.feng@iphy.ac.cn, Hai.Feng_nims@hotmail.com (HLF); \nYamaura.Kazunari@nims.go.jp (KY)  \n# Current address: The Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China \n  2 Abstract  \nRoom-temperature ferrimagnetism was discovered for the anti-site-disordered perovskite \nCa2MnOsO 6 with Tc = 305 K. Ca 2MnOsO 6 crystallizes into an orthorhombic structure with a space \ngroup of Pnma, in which Mn and Os share the oxygen-coordinated-octahedral site at an equal ratio \nwithout a noticeable ordered arrangement. The material is electrically semiconducting with \nvariable-range-hopping behavior. X-ray absorption spectroscopy confirmed the trivalent state of the \nMn and the pentavalent state of the Os. X-ray magnetic circular dichroism spectroscopy reveals that \nthe Mn and Os magnetic moments are aligned antiferromagnetically, thereby classifying the material \nas a ferrimagnet which is in accordance with band structure calculations. It is intriguing that the \nmagnetic signal of the Os is very weak, and that the observed total magnetic moment is primarily due \nto the Mn. The Tc = 305 K is the second highest in the material category of so-called disordered \nferromagnets such as CaRu 1-xMnxO3, SrRu 1-xCrxO3, and CaIr 1-xMnxO3, and hence, may support the \ndevelopment of spintronic oxides with relaxed requirements concerning the anti-site disorder of the \nmagnetic ions.  \n  3 1. INTRODUCTION \n Double perovskite oxides containing 3d and 4d/5d elements are currently much in focus \nsince promising properties for spintronic applications have been reported. For example, Sr 2FeMoO 6 \nshows a low-field magnetoresistance at room temperature[1], and Sr 2CrReO 6 half-metallic (HM) \ntransport with a remarkably high Curie temperature ( Tc) of 635 K[2,3]. Analogous closely related \nferrimagnetic (FIM) oxides have been synthesized with a wide variety of 3d and 4d/5d elements, \nsuch as A2FeMoO 6 (A = Ca, Ba)[1,4], A2FeReO 6 (A = Ca, Sr, Ba)[5,6], Ca 2CrOsO 6[2,3], and \nCa2FeOsO 6[7,8]. In addition, a ferromagnetic (FM) Dirac–Mott insulating state ( Tc ~100 K) has been \nfound in Ba 2NiOsO 6 [9] and an exchange bias effect in Ba 2Fe1.12Os0.88O6 [10], which may also be \nuseful for further development of spintronic oxides.  \n In the magnetic ground state of Sr 2FeMoO 6 and Sr 2CrReO 6, the 3d magnetic moments are \nordered parallel to each other and antiparallel to those of the 4d/5d[1-3]. Anti-site disorder between \nthe 3d and 4d/5d elements has, however, a significant negative impact on the magnetic properties, \nbecause strongly antiferromagnetic (AFM) Fe–O–Fe and Cr–O–Cr bonds are formed which interfere \nwith the long-range magnetic order[11,12]. Indeed, even a small degree of anti-site disorder \ndramatically decreases Tc, and the spin-polarization is also strongly reduced in Sr 2FeMoO 6[13,14], \naccompanied by a linearly decreasing saturation magnetization[12]. Therefore, accurate control of \nthe anti-site disorder has been a significant issue for the fabrication of practical devices since \ngrowing anti-site-disorder-free materials is highly challenging[11,12]. Alternatively, \nanti-site-disorder tolerant materials with promising properties are in demand.  \n A so-called disordered FM has been reported in a substitutional study of the paramagnetic \nperovskite CaRuO 3, where Ru is partially replaced by a variety of magnetic or nonmagnetic elements \nsuch as Sn, Ti, Mn, Fe, Ni, or Rh[15,16]. Among these, Ca 2MnRuO 6 is particularly of high interest, \nbecause its magnetic moment is relatively large (~1.6 μ B/f.u.) and its Tc is high (~230 K)[17,18]. In 4 the structure of Ca 2MnRuO 6, Mn and Ru are distributed over the perovskite B site at an equal ratio \nwithout an ordered arrangement. This material was revealed to be FIM by neutron diffraction owing \nto a balance between FM (Mn3+ to Mn4+) and AFM (Ru to Mn) interactions[17-19], and the \nexperimental magnetic state was consistent with that obtained by first-principles calculations[19]. \nAnother example is SrRu 0.6Cr0.4O3 which was found to have a high transition temperature of 400 K \nbut a very small saturated moment of 0.15 μ B/f.u.[20]. \n In this study, an anti-site-disordered compound, Ca 2MnOsO 6 was synthesized for the first \ntime by a high-pressure and high-temperature method at 6 GP and 1500 ºC. Ca 2MnOsO 6 shows a \nFIM transition at Tc = 305 K, which is the second highest Tc among the disordered FMs. Here we \nreport the refined crystal structure and bulk magnetic properties of Ca 2MnOsO 6. The results suggest \nthat the compound is useful for further development of anti-site-disorder-tolerant spintronic oxides.  \n \n2. EXPERIMENTAL \n Polycrystalline Ca 2MnOsO 6 was synthesized via a solid-state reaction from powders of \nCaO2 (lab-made from CaCl 2∙2H2O, 99% Wako Pure Chem.), Os (99.95%, Heraeus Materials), and \nMnO2 (99.997%, Alfa-Aesar). The powders were thoroughly mixed at the stoichiometric ratio, \nfollowed by sealing in a Pt capsule. The preparation was conducted in an Ar-filled glove box. The Pt \ncapsule was statically and isotropically compressed in a belt-type high-pressure apparatus (Kobe \nSteel, Ltd., Japan [21]), and a pressure of 6 GPa was continuously applied while the capsule was \nheated at 1500 °C for 1 h, followed by quenching to room temperature in less than a minute. The \npressure was then gradually released over several hours.  \n A dense, black polycrystalline pellet was obtained, and several pieces were cut out from it. \nA selected piece was finely ground for a synchrotron X-ray diffraction (SXRD) study, which was 5 conducted in a large Debye–Scherrer camera in the BL15XU beam line, SPring–8, Japan[22,23]. The \nSXRD pattern was collected at room temperature and the wavelength was confirmed to be 0.65298 Å \nby measurement of a standard material, CeO 2. The absorption coefficient was measured in the same \nline. The Rietveld method was used to analyze the SXRD pattern with the RIETAN–VENUS \nsoftware [24,25].  \n X-ray absorption spectroscopy (XAS) at the Mn- L2,3 and Os-L3 edges of Ca 2MnOsO 6 was \ncarried out at the BL11A and BL07C beamlines using the total electron yield and transmission \nmethod, respectively, in the National Synchrotron Radiation Research Center, Taiwan. The Mn- L2,3 \nspectrum of MnO and the Os- L3 spectrum of Sr 2FeOsO 6 were also measured for energy calibration \npurposes. X-ray magnetic circular dichroism (XMCD) spectra at the Mn- L2,3 and the Os- L2,3 edges \nwere obtained at the BL29 BOREAS beamline of the ALBA synchrotron radiation facility in \nBarcelona and at the ODE beamline of Soleil France, respectively. The degree of circular \npolarization in BOREAS and ODE beamline was close to 100% and 90%, respectively.  The XMCD \nspectra were measured in a magnetic field of 60 kOe at a temperature of 20 K for Mn- L2,3 and in 13 \nkOe at 4 K for the Os- L2,3. The Os L3/L2 edge-jump intensity ratio I(L3)/I(L2) was normalized to 2.08 \n[26], which accounts for the difference in the radial matrix elements of the 2p 1/2–to–5d(L2) and 2p 3/2–\nto–5d(L3) transitions.  \n The electrical resistivity ( ρ) of polycrystalline Ca 2MnOsO 6 was measured by a four-point \nmethod at a gauge current of 0.1 mA in a physical properties measurement system (Quantum Design, \nInc.). Electrical contacts on a piece of Ca 2MnOsO 6 were prepared using Pt wires and Ag paste in the \nlongitudinal direction. The temperature dependence of the specific heat capacity ( Cp) was measured \nin the same apparatus by a thermal relaxation method at temperatures between 2 K and 300 K using \nApiezon N grease to thermally connect the material to the holder stage.  \n The magnetic susceptibility ( χ) of a loosely gathered Ca 2MnOsO 6 powder was measured in a 6 magnetic properties measurement system (Quantum Design, Inc.). The measurement was conducted \nin field cooling (FC) and zero-field cooling (ZFC) conditions in a temperature range between 2 K \nand 390 K. The applied magnetic field was 10 kOe. The magnetic field dependence of the \nmagnetization ( M) was measured between −50 kOe and +50 kOe at fixed temperatures of 2, 200, \n300, and 350 K. The alternative current (ac) χ was measured in the same apparatus between 5 K and \n320 K; the amplitude and frequencies of the ac-magnetic field were 5 Oe and 0.5 –500 Hz, \nrespectively.  \n The density functional theory (DFT) calculation was performed on Ca 2MnOsO 6 with the \nall-electron full-potential local-orbital code[27] using the standard generalized-gradient \napproximation (GGA) [28]. In this study, a linear tetrahedron method was employed for all k space \nintegration with a 12 × 12 × 12 subdivision in the full Brillouin zone for an ordered phase of \nCa2MnOsO 6. The magnetic ground state was obtained by computing the total energy of possible \nmagnetic configurations. Our calculation was double-checked in selected cases using the \nfull-potential linearized augmented plane wave method, as implemented in the WIEN2k code[29]. In \norder to estimate the correlation effects, the GGA+ U (U is the Coulomb interaction) function with \natomic-limit (AL) double-counting correction was used[30,31]. The values of U were chosen to be 5 \neV for Mn-3d and 1.5 eV for Os-5d, which were comparable to the values in the literature [9,32-36]. \nThe self-consistent full-relativistic calculations were carried out with spin-orbit coupling (SOC) \nincluded. This is necessary to determine the magnetic anisotropy energy (MAE) and to check its \ninfluence on the electronic and magnetic properties. The calculation was first performed with the \nexperimental lattice parameters obtained by SXRD, and then geometry optimization was conducted. \nThe energy and charge convergence for the self-consistent calculation was set to 10-8 Hartree and \n10-6 of electron, respectively.  \n3. RESULTS 7 A. Crystal structure \n The double perovskite oxide A2BB'O6 [A: alkali earth, B: 3d, B': 4d(5d) elements] normally \ncrystallizes into a monoclinic ( P21/n), or tetragonal ( I4/m), or cubic ( Fm-3m) structure, in which two \ndistinct octahedra  BO6 and B'O6 align in a rock-salt fashion [37]. The degrees of the octahedral \ndistortion and rotation seem to have a major impact on the lattice symmetry[38]. Based on the \ngeneral trend, we attempted to characterize the crystal structure of Ca 2MnOsO 6 by applying these \nmodels. In particular, the monoclinic model ( P21/n) was very effective to refine the SXRD pattern as \nreported for the analogous compounds such as Ca 2FeOsO 6 [7], Ca 2CrOsO 6 [2], and Ca 2MnReO 6 [39]. \nThe refinement quality was, however, not fully satisfactory: a significant failure was found for the \npeaks marked by an arrow in the inset of Fig. 1. Although the major peaks were well refined by the \nP21/n model (if Mn and Os are fully ordered), several expected peaks such as (0-11), (-101), and \n(101) were not observed above the background level. This failure suggested that the rock-salt-type \nordered arrangement of MnO 6 and OsO 6 octahedra is not formed in Ca 2MnOsO 6[13], namely, Os and \nMn atoms are distributed randomly over the perovskite B-site (i.e., anti-site disorder), as was found \nin Ca2MnRuO 6 (Pnma [17]).  \n \nFig. 1 Rietveld refinement of the powder synchrotron XRD pattern collected at room temperature. \nThe crosses and solid lines show the observed and calculated patterns, respectively, with their \ndifference shown at the bottom. The Os was analyzed simultaneously as the secondary phase. The \nestimated mass proportion was Ca 2MnOsO 6: Os = 0.986: 0.014. (Inset) Analysis by an ordered Mn \n8 and Os atoms model; the arrow indicates an obvious discrepancy between the model and the \nexperiment in the analysis.  \n \n We therefore tested the Pnma model of the SXRD data, and as shown in the main panel of \nFig. 1, the Pnma model successfully characterized the structure of Ca 2MnOsO 6. The final solutions \nof the lattice parameters, atomic coordinates, and temperature factors are listed in Table 1. Note that \nthe orthorhombic model ( Pnma) was proposed for both end compounds, CaOsO 3[40] and \nCaMnO 3[41] as well, which indicates  that Ca 2MnOsO 6 is not a double perovskite, but a solid \nsolution of them.  \nTable 1 Atomic coordinates and temperature factors ( B) for Ca 2MnOsO 6 at room temperature \nobtained from synchrotron XRD  \nAtom Site Occp. x y z B  (Å2) \nCa 4 c 1 0.9523(3) 0.25 0.0102(9) 1.12(3) \nOs/Mn 4b 0.5/0.5 0 0 0.5 0.22(1) \nO1 4 c 1 0.0340(11) 0.25 0.5718(10) 1.03(8) \nO2 8 d 1 0.2106(7) 0.4578(5) 0.1850(7) 1.03 \nSpace group: Pnma; lattice constants a = 5.50669(4) Å, b = 7.60567(6) Å, and c = 5.39322(4) Å; Z = \n2; dcal = 6.1946 g/cm3; and the final R indices are Rp = 2.029% and Rwp = 3.597%.  \n \n The average of the M‒O (M = Mn or Os) bond lengths of Ca 2MnOsO 6 is compared with that \nof related compounds; it is shorter than that of CaOs4+O3[40] and LaMn3+O3[42] and longer than that \nof NaOs5+O3[43] and CaMn4+O3[41] (see Table 2). The comparison suggests that the valence state of \nM is intermediate between the two groups. In addition, we examined the impact of the Jahn–Teller \n(JT) distortion on the average structure through comparison of the distortion factors (defined as the \nratio of the longest to the shortest M–O length of an octahedron) of the compounds. It is 1.084 for 9 Ca2MnOsO 6, which is much smaller than 1.142 for LaMn3+O3 and much greater than 1.004 for \nCaMn4+O3, 1.030 for CaOs4+O3, and 1.004 for NaOs5+O3. Although the comparison indicates that the \nJT distortion could contribute to the average structure of Ca 2MnOsO 6, it is extremely difficult to \nuniquely identify the exact charge distribution on Mn and Os of Ca 2MnOsO 6 from only an analysis \nof the powder SXRD pattern, because of the anti-site disorder.  \nTable 2 Comparison of structural parameters of Ca 2MnOsO 6 and related compounds  \n LaMn3+O3  \n[42]  CaMn4+O3  \n[41]  Ca2MnOsO 6  \n[this work]  CaOs4+O3  \n[40]  NaOs5+O3  \n[43]  \nSpace group Pnma Pnma Pnma Pnma Pnma \na (Å) 5.5367(1) 5.279(1) 5.50666(4) 5.57439(3) 5.38420(1) \nb (Å) 5.7473(1) 7.448(1) 7.60562(6) 7.77067(4) 7.58038(1) \nc (Å) 7.6929(2) 5.264(1) 5.39317(4) 5.44525(3) 5.32817(1) \nV (Å3)  244.8 207.0 225.9 235.9 217.5 \nM–O1 (Å) 1.9680(3) 2 1.895 2 1.945(1) 2 2.003(1) 2 1.946(1) 2 \nM–O2 (Å) 1.907(1) 2 1.900 2 1.917(4) 2 1.978(4) 2 1.939(3) 2 \nM–O2 (Å) 2.178(1) 2 1.903 2 2.077(4) 2 2.037(4) 2 1.940(3) 2 \nDistortion a 1.142 1.004 1.084 1.030 1.004 \n<M–O> (Å) 2.018 1.899 1.980 2.006 1.942 \nM–O1–M (º) 155.48(2) 158.6 149.6(2) 151.7(1) 153.9(2) \nM–O2–M (º) 155.11(5) 157.2 155.7(1) 152.0(2) 155.2(2) \na Defined as the ratio of the longest to the shortest M–O length of the MO6 octahedron  \n \nB. X-ray absorption spectroscopy  \n To investigate the valence states of Mn and Os, we conducted the Mn- L2,3 and Os-L3 XAS \non Ca 2MnOsO 6 and compared the spectra to those of MnO[44], LaMnO 3[44], SrMnO 3[45], and \nSr2FeOsO 6[46] which serve as Mn2+, Mn3+, and Mn4+, and Os5+ references, respectively (see Fig. 2). \nThe spectral weight of the L 3 white line shifts to higher energies by 1 eV or more as the Mn valence 10 state increases by one: from Mn2+ (MnO) to Mn3+ (LaMnO 3) and further to Mn4+(SrMnO 3)[44,45]. \nThe similar energy position and multiplet spectral featur es of the Mn-L2,3 of Ca 2MnOsO 6 and \nLaMnO 3 demonstrate  the trivalent state of Mn in Ca 2MnOsO 6. Analogously, the Os- L3 spectrum of \nCa2MnOsO 6 locates at the same energy as that of Ca 2FeOsO 6, indicating the pentavalent state of the \nOs[7,47], thereby fulfilling the charge balance requirement for the Mn3+/Os5+ state. Consequently, \nthe valence state of Ca 2MnOsO 6 (Mn3+, Os5+) is different from that of the analogous compound \nCa2MnRuO 6, in which the mixed valent states of (Mn3+, Ru5+) and (Mn4+, Ru4+) are degenerate in \nenergy[17].  \n \nFig. 2 (a) Room temperature Mn- L2,3 XAS spectra of Ca 2MnOsO 6 and of MnO[44], LaMnO 3[44], \nSrMnO 3[45] as reference compounds for Mn2+, Mn3+, and Mn4+ valence states. (b) Os- L3 XAS \nspectra of Ca 2MnOsO 6 and of Sr 2FeOsO 6[46]as an Os5+ reference.  \n \nC. Electrical transport \n11  Fig. 3 shows an increasing resistivity ρ(T) of Ca2MnOsO 6 with lowering the temperature, \nexperimentally demonstrating a semiconducting-like behavior. Interestingly, it is qualitatively and \nquantitatively different from the half-metallic behavior observed for the related compound \nCa2MnRuO 6[17,19]. The temperature dependence of ρ(T) is poorly characterized by the Arrhenius \nmodel at low temperatures (< 300 K, Inset of Fig. 3); however, a variable-range-hopping conduction \nmodel better explains the data (inset of Fig. 3)[48]. Although the Arrhenius model does not exactly \nfit the data, we can roughly estimate the lower limit of the thermal activation energy in the \nhigh-temperature region (> 300 K) as ~0.11 eV from the linear fit as shown in the inset of Fig. 3.  \n \nFig. 3 Temperature dependence of ρ for polycrystalline Ca 2MnOsO 6. The inset shows an \nalternative plot of the data. The blue line indicates a fitting to the Arrhenius law at high temperatures \n(> 300 K).  \n \nD. Magnetic properties  \n The magnetic susceptibility χ(T) curves measured in the zero field cooled (ZFC) and field \ncooled (FC) conditions for Ca 2MnOsO 6 are shown in Fig. 4. Upon cooling, there is an increasing \nvalue of χ at around 305 K, indicating the establishment of a FM-like transition. The divergence \nbetween the ZFC and FC curves at Tc is negligible, but it becomes prominent at approximately 200 K, \n12 suggesting a possible formation of magnetic domains, or the like in the measurement process. We \nwould like to note that the range of the high-temperature part (> Tc) of the χ(T) curve is too narrow \nfor a meaningful Curie–Weiss analysis (see the right side of Fig. 4).  \n \nFig. 4 Temperature dependence of χ of polycrystalline Ca 2MnOsO 6 measured in a field of 10 kOe. \nAn alternative plot ( T vs. χ-1) is shown on the right side.  \n \n The AC χ (= χ’ + iχ’’) of Ca 2MnOsO 6 was measured at temperatures between 5 K and 320 K, \nand the χ’ and χ’’ vs. T curves are shown in Fig. 5. A sharp peak is observed at 305 K, which is \nconsistent with the onset temperature of the χ(T) measurement. In contrast to multiple transitions \nfound for Ca 2MnRuO 6, no additional anomaly is detected, indicating a single magnetic transition \nover the temperature range. Moreover, it substantiates that the divergence between the ZFC and FC \ncurves in the χ(T) measurements is not caused by a magnetic transition. In Fig. 6, the temperature \ndependence of the heat capacity Cp shows an anomaly evolving at 305 K in zero magnetic field, and \nit becomes broader when a magnetic field of 70 kOe is applied. This observation supports that this \nanomaly is caused by a magnetic transition. \n13   \nFig. 5 (a) Real and (b) imaginary components of ac-magnetic susceptibility for polycrystalline \nCa2MnOsO 6 measured in an ac-magnetic field ( Hac) of 5 Oe at various frequencies. The inset shows \na horizontal expansion around the magnetic transition temperature.  \n \nFig. 6 Temperature dependence of specific heat capacity of Ca 2MnOsO 6.  \n \n The field-dependence of the magnetization was measured at several temperatures between 2 \n14 K and 350 K (see Fig. 7). The linear feature at 350 K confirms the paramagnetic behavior above Tc. \nIn contrast, magnetic hysteresis appears at temperatures below Tc, suggesting that a FM component is \ninvolved in the magnetically ordered state. The magnetization at 2 K and 50 kOe is 1.40 μ B/f.u. \nwhich indicates the partial cancellation of the Mn and Os magnetic moments.  The separate Mn and \nOs contributions to the total magnetic moment will be disentangled using the element specificity of \nX-ray magnetic circular dichroism ( vide infra ).  \n \n \nFig. 7 Isothermal magnetization of polycrystalline Ca 2MnOsO 6 measured at various temperatures.  \n \nE. X-ray magnetic circular dichroism \n Figure 8a shows the Mn- L2,3 XAS spectrum (green) and the XMCD spectrum (blue) which \nis the difference between circularly polarized light with positive and negative helicities in an applied \nmagnetic field of 60 kOe at 20 K. The XMCD signal can be clearly seen in Fig.8a. In order to extract \nthe Mn moment, we performed the well-established configuration-interaction cluster calculations \nusing the XTLS code[49]. The method uses a MnO 6 cluster, which includes explicitly the full atomic \nmultiplet interaction, the hybridization of the Mn with the oxygen ligands, and the crystal field acting \non the Mn ions. The hybridization strengths and the crystal field parameters were taken from Ref.[50]. \n15 Fig. 8b shows the calculated XAS (green) and XMCD (blue) for the Mn cluster in an exchange field \n(Hex) of 30 meV. With this field (the energy scale of which reflects the experimentally determined Tc of \n305 K) the Mn is fully magnetized having a moment of 3.9 μ B. We observe that the line shape of the \ncalculated XMCD spectrum is very similar to the experimental one, but we also notice that its \nmagnitude is larger than measured. To reproduce the experimental size of XMCD signal in Fig.8a, \nwe need to rescale the calculated XMCD spectrum by a factor of 1/3 as shown in Fig. 8b. This \nimplies that the XMCD experiment presents the net Mn moment of 3.9/3 = 1.3 μ B.   \n \nFig. 8 (a) Mn- L2,3 XAS (green) and XMCD (blue) spectra of Ca 2MnOsO 6 measured at 20 K under \na magnetic field of 60 kOe and (b) the calculated XAS (green) and XMCD (blue) rescaled by a factor \n1/3. \n \n Figure 9 shows the Os- L2,3 XAS spectrum (green curve) together with the XMCD spectrum \n(blue curve) measured below Tc in an applied magnetic field of 13 kOe at 4 K. The XMCD signal of \n16 the Os has an opposite sign as that of the Mn, which indicates an antiparallel alignment of Os \nmagnetic moment with that of the Mn. We have also performed configuration-interaction cluster \ncalculations using an OsO 6 cluster with parameters taken from Ref.[51]. Fig. 9b shows the calculated \nXAS (green curve) and XMCD (blue curve) spectra. With an exchange field of 30 meV ( Tc of 305 K) \nthe Os is fully magnetized having a moment of 2.05 μ B. The line shapes of the calculated spectra match \nvery well those of the experiment. We notice, however, that the magnitude of the experimental XMCD \nis about 400 times smaller than that of the calculated, which suggests that most of the Os ions are \nmagnetically disordered and/or antiferromagnetically aligned. \n \nFig. 9(a) Os- L2,3 XAS (green) and XMCD (blue multiplied by 400) spectra of Ca 2MnOsO 6 measured \nat 4 K under a magnetic field of 13 kOe. (b) the calculated Os- L2,3 XAS (green) and XMCD (blue) \nspectra. \n \nWe would like to note that our finding from XMCD that the net Mn moment is 1.3 μ B and that \n17 the net Os moment is very small (antiparallel aligned) is in excellent agreement with the \nmagnetization measurement (see the previous section) which yielded a total net moment of 1.40 \nμB/f.u.  \nF. Density functional theory (DFT) calculation   \n We investigated the electronic and magnetic ground state of Ca 2MnOsO 6 by DFT \ncalculations. In this theoretical study, we investigated a hypothetically ordered phase ( P21/n; see \nTable S1, Figs. S1, and S2 in supplementary materials [52]). The symmetry of the P21/n structure has \nbeen reduced further to lower symmetry P-1 (space group: 2) which gives rise to two in-equivalent \natoms each of Os and Mn, respectively. The magnetic ground state of the ordered phase was studied \nby calculating the total energy for each of the five different magnetic states, namely FM (FM- ↑↑↑↑; \nOs1: Os2: Mn1: Mn2), two AFM (AFM-1-↑↓↑↓ and AFM-2-↑↓↓↑), and two FIM (FIM-1-↑↑↓↓ and \nFIM-2-↑↑↑↓) states (see Fig. 10). From the total energy calculations, FIM-1 is found to have the \nlowest energy, with an energy difference of 198 meV/unit cell to that of the next lowest order \n(AFM-1). The spin-polarized DFT calculations for the hypothetically ordered Ca 2MnOsO 6 suggest \nthat the magnetic ground state is certainly FIM-1 with the anti-parallel alignment of Mn moments to \nOs. The electronic ground state is HM (the HM gap is approximately 0.87 eV in the spin-up channel) \nwithin the GGA functional. Additionally, SOC effects has been considered where the spins are \nallowed to ordered along the [100] and [001] direction to check its influence on the electronic and \nrelated properties. Significant difference was not found for the net moments for Mn and Os (see \nTable 3), while a reasonable change has been observed in the electronic behavior. 18  \nFig. 10  Possible spin arrangements considered in the theoretical calculation for Ca 2MnOsO 6 with \nordered Mn and Os atoms; ferromagnetic (FM), antiferromagnetic (AFM), and ferrimagnetic (FIM) \nconfigurations.   \n \nTable 3 Spin < ms> and orbital < ml> moments measured by XMCD and two sets of theoretical \nvalues for ordered Ca 2MnOsO 6  \n XMCD a GGA+SO b \nOrdered structure \n[100] Ordered structure \n[001] \nMn Os Mn Os Mn Os \nms (μB/atom) 1.3 -0.005 3.13 -1.38 3.14 -1.41 \nml (μB/atom) 0.0 0.000 0.02 0.02 0.01 0.08 \nms+ml \n(μB/atom) 1.3 -0.005 3.15 -1.36 3.16 -1.33 \na T = 4 K and H = 13 kOe for Os and 20 K and 60 kOe for Mn  \nb Standard generalized-gradient approximation with spin-orbit interaction  \n The electronic band structure and the spin-resolved DOS (total and partial) within GGA \nfunctional are shown in Fig. 11 for the ordered phase, which is HM with an insulating band gap of \n~0.87 eV in the spin-up channel and metallic state in the spin-down channel. The major contributions \n19 to the total DOS around the EF are attributed to the Os-5d orbitals in both spin channels, which \nhybridize strongly with the O-2p orbitals. From the partial DOS and band structure (see Fig. 11 and \nFig. S3a [52]), the Os-5d orbitals in the spin-up channel are found to be fully occupied by 5d- t2g \norbitals, which shows the Os5+ (5d3) state. The six bands at and around the Fermi level ranging from \n-1.5 eV upto ~ +0.7 eV are from the Os-5d -t2g orbitals which hybridizes strongly with the O-2p \norbitals (see Fig. 11 and Fig. S3a [52]) in spin-up and spin down channel. On the other hand, Mn-3d \nstates are found to dominate mostly above +0.7 eV in the conduction region in spin-up channel, \nwhile in spin-down channel three d-orbitals (d- t2g) are fully occupied lying below -1.5 eV while two \nof the d-states (d- eg) cross the Fermi level signaling the partial occupancy in spin-down channel. This \nfeature is close to the ionic picture Mn3+ which should have four d-states occupied. With SOC taken \ninto account (within GGA), band splitting was observed (see Fig. S3a [52]), however, the overall \nband gap did not open. This suggests the necessity of using GGA+U. \n \nFig. 11  (a) Band dispersion and (b) DOS of Ca 2MnOsO 6 with fully ordered Mn and Os atoms \nwithin GGA (optimized).  \n20  \n To further investigate the experimentally observed semiconducting transport, we performed \na GGA+U calculation with U values ranging from 0–5 eV for Mn and 0–2.5 eV for Os[32-35]. The \nHM state changed to a semiconducting state at U(Mn) = 4 eV and U(Os) = 1.25 eV or higher[19]. \nWith U = 5 eV for Mn and 1.5 eV for Os, the calculated DOS and band structure are shown in Fig. \n12. A band gap of 0.11 eV was achieved, which agrees well with the experiment (~0.11 eV). With \nSOC, an indirect band gap of 0.14 eV is found between the high symmetry point Γ and M, and a \ndirect band gap of 0.21 eV at Γ point in the band dispersion shown in Fig. S3b [52]. Splitting of the \norbitals for the easy axis [001] is large with a gap size of 0.14 eV. From the scalar relativistic DOS \nand band structure shown in Fig. 12, and relativistic fat bands in Fig. S3b [52], the Os-5d states are \nfound to fully occupy with three t2g orbitals in the spin-up channel and the remaining eg orbitals are \nunoccupied, while in the spin-down channel, all the d-bands lie above EF. Most of the bands from \nMn-3d around EF hybridize with the O-2p and Os-5d states. The broad band lying just above EF is \nmainly contributed by the Mn-3d orbital; the exchange energy splitting is of the order of ~1.75 eV. \nBesides, MAE has been considered for the FIM-1 ground state. The calculated MAE is 29.6 \nmeV/unit cell with its easy axis along the cubic [100] direction[53] within GGA.  In contrast, within \nGGA+U, the easy axis is found along the out-of-plane with MAE of 3.46 meV/unit cell with all the \nspins aligned along the [001] direction. 21  \nFig. 12 (a) Band dispersion and (b) DOS of Ca 2MnOsO 6 with fully ordered Mn and Os atoms \nwithin GGA+ U (U = 5 eV for Mn and 1.5 eV for Os atoms, respectively). \n \nTo evaluate the impact from anti-site disorder on the electronic ground state, five different \nanti-site-disordered configurations (AS1, AS2, AS3, AS4, and AS5) were created as shown in the \nFig. S4 [52], where a 1 x 1 x 2 supercell has been generated which corresponds to a total of 40 atoms \nin a unit cell. DFT calculations were first carried out on these five different anti-site-disorder cases \nwith FIM spin state. However, in comparison with the ordered FIM-1 case these anti-site-disordered \ncases showed higher energies within ~0.6 to 1.37 meV/unit-cell (see Table S2 [52]), with energies \nthat AS1 < AS2 < AS3 < AS4 < AS5. We further calculated these anti-site-disordered cases with FM \nspin state. Energy differences (∆E) between FM and FIM states in the respective anti-site-disordered \ncases are summarized in Table S3 [52].  FIM states showed lower energies, indicating that AFM \nalignment is favorable between Mn-Os, while FM coupling may be favorable among the Mn-Mn or \nOs-Os, giving rise to long-range FIM order in the anti-site-disordered Ca 2MnOsO 6. The electronic \n22 band gap is noted for AS1 and AS2 cases, while others remain metallic within GGA+U (see table S2 \n[52]). \n \nDISCUSSION \nWe synthesized an anti-site-disordered double perovskite Ca 2MnOsO 6 (equivalent to \nCaMn0.5Os0.5O3) with Mn3+ and Os5+ ions randomly distributed at the B site of the perovskite. It is \nnoteworthy that neighbor compounds Ca 2FeOsO 6 and Ca 2CrOsO 6 all crystallize in ordered double \nperovskite structure with rocksalt arrangement of FeO 6/CrO6 and OsO 6 octahedra. The B and B' site \nordering in double perovskite is generally determined by the charge difference and effective ionic \nsize difference of B and B' ions[54].  Given that Mn3+ has the same effective ionic radii (0.645  Å) as \nthat of Fe3+ in octahedral coordination[55], the reason for the absence of Mn3+ and Os5+ ordering in \nCa2MnOsO 6 is unclear, although perhaps the JT distortions induced around the Mn3+ ions may play \nan important role. Despite the absence of Mn3+ and Os5+ ordering, Ca 2MnOsO 6 is electrically \nsemiconducting and features a remarkably high Tc above room-temperature (305 K). \nTo rationalize the magnetic properties, we first consider the double perovskite Ca 2MnOsO 6 \nwith perfect Mn3+ and Os5+ ordering. Our band structure calculations suggest that this would lead to \na FIM ground state, where the moments of Mn3+ (t2g3eg1) and Os5+ (t2g3) are arranged antiparallel. In \norder to interpret this result, we notice that the nearest-neighbor Mn3+–O–Os5+ super-exchange \ncoupling can either be FM (the Mn3+ eg electrons hop on to the empty Os5+ eg states) or AFM (the \ndown spin Os5+ t2g electrons hop on to the Mn3+ t2g states). In comparison to 3d ions, the 5d ions have \nmuch enhanced crystal field splitting, therefore the FM coupling involving the virtual hopping \nbetween e g orbitals becomes weak. In particular when the crystal structure is distorted this FM \ncoupling can become even weaker[47,56], with the result that the AFM Mn3+–O–Os5+ coupling is the \ndominant interaction and generate the FIM state in the hypothetically ordered Ca 2MnOsO 6.   23 Introducing now anti-site disorder in Ca 2MnOsO 6, we will have also NN Os5+–O–Os5+ and \nMn3+–O–Mn3+ interactions. The Os5+–O–Os5+ coupling is AFM as shown in AFM NaOsO 3[43,57]. \nThe Os5+ moments are therefore expected to compensate each other in such a situation. This would \nthen be consistent with the experimental observation that the Os XMCD effect is very small. The \ncase concerning the Mn3+–O–Mn3+ interactions is more complex. Depending on the orbital \norientation due to the JT distortion of Mn3+, the interaction can be either FM or AFM which has been \nwell discussed in LaMnO 3[42,58-61]. In our crystal structure analysis, the contribution from the JT \neffect of Mn3+ was clearly noticed (see crystal structure part). The moment, 1.3 μ B/Mn3+, obtained \nfrom Mn XMCD is not negligible, but smaller than the theoretically calculated 3.15 μ B/Mn3+ for \nordered Ca 2MnOsO 6 phase (see Table 3) and also smaller than the 3.9 μ B/Mn3+ from the cluster \ncalculations. This indicates that there are competing FM and AFM Mn3+–O–Mn3+ interactions. \nStudies of highly anti-site disordered double perovskite A2Mn3+B'O6 (A = Ca, Sr; B' = Ta, Sb), in \nwhich only Mn3+ is magnetic, found that the FM interactions are dominant in these compounds with \npositive Weiss temperatures varied from 64–107 K [62]. A large FM component with a \nmagnetization of about 1.47 μ B/Mn3+ (at 5 K and 50 kOe) was observed in Sr 2MnTaO 6 despite its \nglassy state at low temperatures [62]. These results are thus similar to our Ca 2MnOsO 6 case. \n  Recent theoretical and experimental studies of the anti-site disorder on the magnetic \nproperties of Sr 2FeMoO 6, Sr2FeReO 6, Sr2CrReO 6, Sr2CrOsO 6, and Sr 2FeRuO 6 suggested that the \nanti-site disorder suppresses Tc or destroys the magnetically ordered state, because the anti-site \ndisorder introduces strong AFM Fe3+–O–Fe3+ and Cr3+–O–Cr3+ interactions to hinder the long-range \nmagnetic order[11-14,63-65]. Different from Fe3+ and Cr3+, the Mn3+–O–Mn3+ super-exchange \ninteractions can be both FM and AFM[42,58-61] depending on the orbital orientation, which may \nhelp to maintain net partial Mn3+ moments and also a high Tc in this anti-site-disordered Ca 2MnOsO 6. \nThis scenario is likely to be different from the ferrimagnetism of anti-site-disordered \nCa2MnRuO 6[17,18], where the presence of mixed valence states of Mn3+/Mn4+–Ru4+/Ru5+ would 24 also lead to a ferromagnetic double exchange mechanism for the Mn3+–Mn4+ and a HM state for \nCa2MnRuO 6[17-19]. \n4. CONCLUSION \n Anti-site-disordered Ca 2MnOsO 6 was synthesized for the first time under high-pressure (6 \nGPa). It crystallizes into an orthorhombic structure (space group: Pnma), in which trivalent Mn and \npentavalent Os share the Wycoff 4 b position without an ordered arrangement. Ca 2MnOsO 6 is \nelectrically semiconducting. XAS measurement confirmed the trivalent Mn and pentavalent Os \noxidation states. The XMCD reveals the antiparallel alignment of the net Mn and Os magnetic \nmoments. Remarkable is that the net Mn moment is only about 1/3 of its full Mn3+ value and that the \nnet Os moment is very small. We have discussed the strength and sign of various inter-site exchange \ninteractions in this material using data from band structure calculations, taking into account also the \npresence of anti-site disorder and JT distortions around the Mn3+ ions.  The Tc = 305 K is the second \nhighest in the material category of so-called disordered ferromagnets and could therefore be useful in \nthe development of oxide spintronic devices that are less sensitive for anti-site disorder during \nfabrication.  \nACKNOWLEDGMENTS  \nMPG thanks the Alexander von Humboldt Foundation for the financial support through HERMES \nprogram and Ulrike Nitzsche for the technical support. This study was supported in part by JSPS \nKAKENHI Grant Number JP16H04501, a research grant from Nippon Sheet Glass Foundation for \nMaterials and Engineering (#40-37), and Innovative Science and Technology Initiative for Security, \nATLA, Japan. We acknowledge support from the Max Planck-POSTECH-Hsinchu Center for \nComplex Phase Materials. The work in Dresden was partially supported by the Deutsche \nForschungsgemeinschaft through SFB 1143 (project-id 247310070). The synchrotron radiation \nexperiments were performed at the NIMS synchrotron X-ray station at SPring-8 with the approval of 25 the Japan Synchrotron Radiation Research Institute (Proposal Numbers 2017A4503, 2017B4502, \n2018A4501, and 2018B4502). MG acknowledges support of NSF-DMR-1507252 grant of USA. \n  26 REFERENCES \n[1] K. I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 (1998). \n[2] R. Morrow, J.R. Soliz, A.J. Hauser, J.L. Gallagher, M.A. Susner, M.D. Sumption, A.A. 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Solid State Chem. 78, 281 (1989). \n \n "    },    {        "title": "1302.2487v2.Phase_stability_of_chromium_based_compensated_ferrimagnets_with_inverse_Heusler_structure.pdf",        "content": "arXiv:1302.2487v2  [cond-mat.mtrl-sci]  28 Feb 2013Phase stability of chromiumbasedcompensatedferrimagnet swith inverseHeusler\nstructure\nMarkusMeinert,ManuelP.Geisler\nThin Films and Physics ofNanostructures, Department ofPhy sics, Bielefeld University, D-33501 Bielefeld, Germany\nAbstract\nChromium based inverse Heusler compounds of the type Cr 2YZ (Y=Co, Fe; Z=Al, Ga, In, Si, Ge, Sn) have been proposed\nas fully compensated half-metallic ferrimagnets. Such mat erials are of large interest for spintronics because they co mbine small\nmagnetic moment with high spin polarization over a wide temp erature range. We assess their thermodynamic stability by t heir\nformationenthalpiesobtainedfromdensityfunctionalthe orycalculations. Allcompoundsunderinvestigationareun stable. Cr 2FeSi\nandCr 2CoAlarestablewithrespecttotheelementalconstituents, butdecomposeintobinaryphases. Cr 2FeGe,Cr 2CoGa,Cr 2FeSn\nandCr 2CoInarefoundtobeunstablewithrespecttotheirelemental constituents. We identifypossiblebinarydecompositions .\n1. Introduction\nFully compensated half-metallic ferrimagnets (also known\nas half-metallic antiferromagnets) [1] have promising pro per-\ntiesforspintronics. Duetotheirinternalspincompensati on,the\ntotal momentis (nearly)zeroat low temperature,yet thepos si-\nbly strong local moments allow for high Curie temperatures.\nHowever, approaching the Curie temperature one expects to\nfindanon-zerototalmagneticmoment[2]. Inconjunctionwit h\npossible appearance of half-metallicity, such materials w ould\nbe ideal electrode materials for spin transfer torque switc hing\ndevicesandotheradvancedapplications[3, 4].\nAmongtheinverseHeuslercompounds(spacegroup F¯43m,\nprototype Hg 2CuTi) [5], several compositions with 24 valence\nelectrons have been identified which could have these unusua l\nproperties,in particularCr 2CoGa and Cr 2FeGe [6]. In contrast\nto regularHeusler compounds,the inverse Heusler compound s\ndo not possess inversion symmetry, such that the Cr atoms be-\ncomenearestneighbors,whichleadstoanantiparallelcoup ling\nof their magnetic moments due to direct exchange interactio n.\nThesecompoundsprefertheinvertedHeuslerstructureover the\nregular one and the local moments are predicted to be quite\nlarge. In consequence, high Curie temperatures of 1520K and\n748K areexpected,respectively[6].\nExperiments indicate that Cr 2CoGa crystallizes in a cubic\nstructure,whichmightbetheinverseHeuslerstructure[7] . How-\never,magnetizationmeasurementsrevealthattheCr 2CoGacom-\npoundissusceptibletoatomicdisorder,whichgeneratesan on-\nzero moment [8, 9]. Further, the Curie temperature was ob-\nserved to be only about 300K and the electrical resistivity w as\nsemimetallic [8]. Both experimental findings indicate that se-\nverestructuraldisordermustbepresentandthattheactual crys-\ntal structure of Cr 2CoGa cannot simply be portrayedas the in-\nEmailaddresses: meinert@physik.uni-bielefeld.de (Markus\nMeinert), mgeisler@uni-bielefeld.de (Manuel P.Geisler)verse Heusler structure. Recent experiments on Cr 2FeGe indi-\ncatethatthiscompoundcrystallizesinamorecomplextetra go-\nnalstructure[9].\nTheseexperimentalresultspromptedustoperformastabil-\nity analysis of these compounds and their relatives of the ty pe\nCr2YZ (Y=Co, Fe; Z=Al, Ga, In, Si, Ge, Sn) with 24 valence\nelectrons. Our analysis is based on formation energies com-\nputed within density functional theory (DFT) [10, 11], simi -\nlar to recent analyses for a large number of Heusler and in-\nverse Heusler compoundsgiven by Gilleßen and Dronskowski\n[12,13].\n2. Computationalapproach\nTheDFTcalculationswiththePerdew-Burke-Ernzerhof(PBE )\n[14] exchange-correlation functional were performed with the\nElkcode[15],animplementationofthefull-potentiallineari zed\naugmentedplane-wavemethod(FLAPW).Themu ffin-tinsphere\nradii were set to 2.0bohr for all elements and the augmented\nplane-wave expansion was taken to kmax=4.0bohr−1.k-point\nmesheswith 20×20×20pointswereused.\nWe obtained formation energies (zero-temperature forma-\ntionenthalpies)withrespecttotheelementalsolidsfromt heto-\ntalenergiesofthecompoundsandthetotalenergiesperatom of\nthe constituent elements in their respective ground-state struc-\ntures:\n∆E0(Cr2YZ)=ECr2YZ\ntot−/parenleftBig\n2ECr\ntot+EY\ntot+EZ\ntot/parenrightBig\n.(1)\nThe total energydi fferenceswere convergedto within 10meV.\nStructuralrelaxationsoftheconstituentelementswerepe rformed\nwith the GPAW code [16, 17]. Decompositions into binary\nphaseswerestudiedwiththehelpofthe MaterialsProject database\n[18,19].\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials April12,2021a0∆E0∆E0mY(A)mCr(B)mCr(C)mtot\n(Å) (eV/f.u.) (kJ/mol) (µB) (µB) (µB) (µB)\nCr2FeSi 5.62 −0.78−75.6 0.00 0.00 0.00 0.00\nCr2FeGe 5.72 0.15 14.1 −0.22 1.17−0.92 0.04\nCr2FeSn 6.07 1.13 108.8 −0.14 2.11−1.93 0.03\nCr2CoAl 5.78 −0.27−26.0 0.30 1.36 −1.49 0.01\nCr2CoGa 5.78 0.09 8.8 0.40 1.48 −1.64 0.09\nCr2CoIn 6.06 1.06 101.8 0.59 2.06 −2.32 0.13\nTable 1: Equilibrium lattice constants, formation energie s in eV per formula unit and kJ per mole compound. Site resolve d local magnetic moments given in µB,\nwhere Y=Fe,Co and A,B, C,are described in the text.\n/s45/s48/s46/s56/s53/s45/s48/s46/s56/s48/s45/s48/s46/s55/s53/s45/s48/s46/s55/s48/s45/s48/s46/s54/s53/s45/s48/s46/s54/s48/s45/s48/s46/s53/s53/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48\n/s32/s70/s105/s77\n/s32/s78/s77\n/s52\n/s51\n/s50\n/s49\n/s48/s109/s97/s98/s115/s32/s40/s181/s66/s41\n/s53/s46/s56 /s53/s46/s55 /s53/s46/s54 /s53/s46/s53\n/s97/s32/s40/s197/s41/s52\n/s51\n/s50\n/s49\n/s48\n/s53/s46/s57 /s53/s46/s56 /s53/s46/s55 /s53/s46/s54\n/s97/s32/s40/s197/s41/s67/s114/s50/s70/s101/s83/s105 /s67/s114/s50/s67/s111/s65/s108\nFigure 1: Total energies for ferrimagnetic (FiM) and nonmag netic (NM) con-\nfigurations(toprow)andabsolutemagneticmoments(bottom row)asfunctions\nof the lattice constant for Cr 2FeSi and Cr 2CoAl. Dotted lines represent third-\ndegree polynomial fits.\n3. Results\nThe equilibrium lattice constants and formation energies\nand magnetic moments of the six compounds with 24 valence\nelectronsaregatheredinTable1. Thetotalmomentsarenear ly\nzero in all cases. The sites A, B, C, D, denote the internal co-\nordinates (0,0,0), (1\n4,1\n4,1\n4), (1\n2,1\n2,1\n2), and (3\n4,3\n4,3\n4). Note that\nthe values of the local magnetic moments depend strongly on\nthe chosen muffin-tin sphere radii and may thus di ffer from\nvalues found in the literature. We find that only Cr 2FeSi and\nCr2CoAl are possibly stable in the inverse Heusler structure,\nsince only these two compounds have negative formation en-\nergy. WhileCr 2CoAlisferrimagnetic,Cr 2FeSiisnot. Cr 2FeSn\nandCr 2CoInareclearlythermodynamicallyunstableduetotheir\nlargepositiveformationenergy.Cr 2FeGeandCr 2CoGaarecrit-\nical cases, with the magnitudeof the energyofformationclo se\nto zero. Thus, the quality of the prediction may su ffer from\nthe imperfect exchange-correlationfunctional and may be d if-\nferent for another functional. Gilleßen and Dronskowski ha ve\nestimated the accuracy of their calculations (which are sim i-lar to ours) by comparison with experimental data obtained o n\nNi3Al, Ni3Ga, and Ni 3Ge with AuCu 3structure. They found\ndeviationsof−4to−12%(−4to−18kJ/mol)fromexperiment,\nwhich are quite small. Experimentally, the trend is found th at\nCr2CoGa is just stable with a disposedness to chemical disor-\nder, while Cr 2FeGe is indeed not stable in the cubic phase and\nprefersa tetragonaloneasdescribedinthe introduction.\nFor comparison we have calculated the formation energies\noftheCo 2FeSi,Co 2FeGe, Co 2FeSnseriesoffullHeuslercom-\npounds. We find−1.38,−0.59,and+0.05eV/f.u.,respectively.\nIndeed, Co 2FeSn is known to be thermodynamically unstable\nand to disintegrate into binary phases, but it can be synthe-\nsized in the Heusler phase in a non-equilibrium process [20] .\nIts binary decomposition into CoFe +CoSn has a formation\nenergy of−0.32eV/f.u. In contrast, Co 2CrSi has a formation\nenergy of−0.85eV/f.u., yet it is found to be unstable and to\nformdisorderedandbinaryphases[21]. Themostfavorableb i-\nnary decomposition is Co 2Si+Cr, with a formation energy of\n-1.35eV/f.u.\nSearchingforpossiblebinarydecompositionsoftheCrbase d\ncompounds, we find that also Cr 2CoAl and Cr 2FeSi are unsta-\nble. Cr 2CoAl would most likely decompose into 2Cr +CoAl\n(∆E0=−1.20eV/f.u.), whereas 3Cr 2FeSi would decompose\ninto2Cr 3Si+Fe3Si(∆E0=−1.35eV/f.u.). Themostfavorable\nbinaryreactionsforthesix compoundsaregivenin Table2.\nForcompleteness,we studythe magnetismasa functionof\nthe lattice constant of Cr 2FeSi and Cr 2CoAl in Fig. 1, where\nthe absolute momentdenotesthe sum of the absolute valuesof\nmuffin-tin moments. The energy zero is chosen as the sum of\nthe elemental total energies, i.e., the second term on the ri ght-\nhand side of Eq. 1. Cr 2FeSi is nonmagnetic at its equilibrium\nlatticeconstant,butturnsintoaweakferrimagnetatsligh tlyin-\nreaction ∆E0(eV/f.u.)\n3(2Cr+Fe+Si)→2Cr3Si+Fe3Si -1.35\n3(2Cr+Fe+Ge)→2Cr3Ge+Fe3Ge -0.45\n2Cr+Fe+Sn→2Cr+Fe+Sn 0.00\n2Cr+Co+Al→2Cr+CoAl -1.20\n2Cr+Co+Ga→2Cr+CoGa -0.56\n3(2Cr+Co+In)→In3Co+6Cr+2Co -0.05\nTable 2: Most favorable reactions forming binary phases and their reaction\nenergiesaccording tothe MaterialsProject database. Therearenostablebinary\nphases in the Cr-Fe-Sn system.\n2creased volume. Cr 2CoAl has a ferrimagnetic ground-state in\nthe inverse Heusler structure. Due to the similarly large ma g-\nnetic moments as in Cr 2CoGa, a high Curie temperature may\nbe expectedforthiscompound[6].\n4. Conclusions\nInsummary,wehavestudiedtheenergyofformationforsix\nchromium based ternary materials with inverse Heusler stru c-\nture, which potentially could form compensated ferrimagne ts.\nWe haveshownthat all of these compoundsare thermodynam-\nically unstable,havingpositiveformationenergyor becau seof\nenergeticallymore favorablebinarydecompositions. Howe ver,\nthe inverse Heusler structure may be metastable at finite tem -\nperature,eventuallyopeningapathtothesynthesisofthec om-\npoundswe havestudied.\nAcknowledgements\nWe thank the developers of the E lkand GPAW codes and\nof theMaterials Project for their efforts. We further thank\nG¨ unterReissforhissupport. FinancialsupportbytheDeut sche\nForschungsgemeinschaft(DFG) isgratefullyacknowledged .\nReferences\nReferences\n[1] H.van Leuken, R. A.deGroot, Physical Review Letters 74 ( 1995) 1171.\n[2] E.S ¸as ¸io˘ glu, Physical Review B 79 (2009) 100406.\n[3] W.E.Pickett, J.S.Moodera, Physics Today 54 (2001) 39.\n[4] X.Hu,Advanced Materials 24 (2011) 294.\n[5] S.Skaftouros, K. ¨Ozdo˘ gan, E.S ¸as ¸io˘ glu, I. Galanakis, Physical Review B\n87 (2013) 024420.\n[6] I.Galanakis, E.S ¸as ¸io˘ glu, Applied Physics Letters 9 9 (2011) 052509.\n[7] T. Graf, F. Casper, J. Winterlik, B. Balke, G. H. Fecher, C . Felser,\nZeitschrift f¨ ur Anorganische und Allgemeine Chemie 635 (2 009) 976.\n[8] J. F. Qian, L. Feng, W. Zhu, E. K. Liu, X. D. Tang, W. H. Wang, G. H.\nWu, F. B. Meng, H. Y. Liu, H. Z. Luo, Acta Physica Sinica 60 (201 1)\n056402.\n[9] M. Hakimi, M. Venkatesan, K. Rode, K. Ackland, J. M. D. Coe y, Jour-\nnalofApplied Physics,12thJointMMM /Intermag Conference 2013pro-\nceedings.\n[10] P.Hohenberg, W.Kohn, Physical Review 136 (1964) B865.\n[11] W.Kohn, L.J.Sham,Physical Review 140 (1965) A1133.\n[12] M. Gilleßen, R. Dronskowski, Journal of Computational Chemistry 30\n(2009) 1290.\n[13] M. Gilleßen, R. Dronskowski, Journal of Computational Chemistry 31\n(2010) 612.\n[14] J.P.Perdew, K.Burke, M.Ernzerhof, Physical Review Le tters 77 (1996)\n3865.\n[15] TheElk FP-LAPWcode, http: //elk.sourceforge.net,Version 1.4.22.\n[16] J. J. Mortensen, L. B. Hansen, K. W. Jacobsen, Physical R eview B 71\n(2005) 035109.\n[17] S. R. Bahn, K. W. Jacobsen, Computing in Science and Engi neering 4\n(2002) 56.\n[18] A.Jain, G. Hautier, C. Moore, S. P.Ong, C. Fischer, T. Mu eller, K. Pers-\nson,G.Ceder, Computational Materials Science 50 (2011) 22 95.\n[19] S.P.Ong,A.Jain,G.Hautier, M.Kocher,S.Cholia,D.Gu nter,D.Bailey,\nD.Skinner, K.Persson, G.Ceder, http: //materialsproject.org /.\n[20] M. A. Tanaka, Y. Ishikawa, Y. Wada, S. Hori, A. Murata, S. Horii, Y.\nYamanishi, K. Mibu, K. Kondou, T. Ono, S. Kasai, Journal of Ap plied\nPhysics 111 (2012) 053902.\n[21] M. Aftab, G. Hassnain Ja ffari, S. K. Hasanain, S. Ismat Shah, Journal of\nApplied Physics 110 (2011) 053903.\n3"    },    {        "title": "0711.3002v1.Probing_phase_coexistence_and_stabilization_of_the_spin_ordered_ferrimagnetic_state_by_Calcium_addition_in_the_YBa__1_x_Ca__x_Co__2_O__5_5__layered_cobaltites_using_neutron_diffraction.pdf",        "content": "arXiv:0711.3002v1  [cond-mat.str-el]  19 Nov 2007Probing phase coexistence and stabilization of the spin–or dered ferrimagnetic state by\nCalcium addition in the Y(Ba 1−xCax)Co2O5.5layered cobaltites using neutron\ndiffraction\nG. Aurelio,∗J. Curiale,†and R. D. S´ anchez‡\nComisi´ on Nacional de Energ´ ıa At´ omica – Centro At´ omico B ariloche,\nAv. Bustillo 9500, 8400 S. C. de Bariloche, RN, Argentina\nG. J. Cuello\nInstitut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, Fr ance\n(Dated: November 30, 2018)\nIn this article we study the effects of a partial substitution of Ba with the smaller cation Ca in\nthe layered cobaltites YBaCo 2O5+δforδ≈0.5. Neutron thermodiffractograms are reported for\nthe compounds YBa 0.95Ca0.05Co2O5.5(xCa= 0.05) and YBa 0.90Ca0.10Co2O5.5(xCa= 0.10) in the\ntemperature range 20 K ≤T≤300 K, as well as high resolution neutron diffraction experim ents at\nselectedtemperatures for thesamples xCa= 0.05,xCa= 0.10 andtheparentcompound xCa= 0. We\nhave found the magnetic properties to be strongly affected by the cationic substitution. Although\nthe “122” perovskite structure seems unaffected by Ca additi on, the magnetic arrangements of\nCo ions are drastically modified: the antiferromagnetic (AF M) long–range order is destroyed, and\na ferrimagnetic phase with spin state order is stabilized be lowT∼290 K. For the sample with\nxCa= 0.05 a fraction of AFM phase coexists with the ferrimagnetic on e belowT∼190 K, whereas\nforxCa= 0.10 the AFM order is completely lost. The systematic refinemen t of the whole series\nhas allowed for a better understanding of the observed low–t emperature diffraction patterns of the\nparent compound, YBaCo 2O5.5, which had not yet been clarified. A two–phase scenario is pro posed\nfor thexCa= 0compoundwhichis compatible withthe phasecoexistence o bserved inthe xCa= 0.05\nsample.\nI. INTRODUCTION\nDuring the last decade, cobaltites have gained in-\ncreased attention. A great effort is being made to clarify\nand systematize the extremely rich variety of phenom-\nena they exhibit. Initially, they were expected to show\nsimilar properties to other perovskite–family members,\nsuch as manganites and cuprates,1–5but soon it was\nfound that they present additional tunable features, as\nthe cobalt spin state, that add to their complexity but\nalso make them even more fascinating and challenging.\nAmong cobaltites, the layered compounds RBaCo2O5+δ\n(Rbeing a rare earth) are currently being intensively\nstudied.3–6The oxygen content in these compounds can\nbe modified in a wide range (0 ≤δ≤0.9 depending\non theRcation and the synthesis conditions),6which in\nturn controls the mixed valence state of Co ions. Several\nfactors strongly influence the physical properties of these\ncobaltites: the non–stoichiometry, the Rcation size, the\nvacancies structural order, and — as we will show in the\npresent work— also the structural disorder introduced\nby doping the Ba site with small quantities of smaller\ncations with the same valence state.\nFrom a structural point of view, RBaCo2O5+δis\nformed by a stacking sequence of [CoO 2]–[BaO]–[CoO 2]–\n[ROδ]planesalongthe c−axis,2,7theusuallycalled“112”\nstructure derived form the aPxaPx2aPcell, being aP\nthe perovskite unit cell constant. The symmetry may\nbe tetragonal or orthorhombic, depending on the oxygen\ncontent and the Rcation. The oxygen vacancies have\na strong tendency to become ordered, which results inseveral superstructures.8\nOf particular interest is the case δ= 0.5, for which Co\nis expected to be completely in the +3 valence state. In\nthis case, a particular order of oxygen vacancies leads to\nthe “122” superstructure, consisting of an ordered array\nof 50% Co atoms in octahedral oxygen coordination and\n50% in a pyramidal environment. This, in turn, favors a\nmetal–insulator (MI) transition just above room temper-\nature (the TMIdepends again on the Rcation) which can\nbefoundonlyfor δvaluesverycloseto0.5.4Whendoping\nwith holes (Co4+,δ >0.5) these compounds behave as\nmetals above the TMItransition, but when doping with\nelectrons (Co2+,δ <0.5), these do not seem to partic-\nipate in charge transport, which has been explained in\nterms of a spin blockade.9There has arisen a big contro-\nversy regarding the physical phenomena which occur at\nTMI. Regardless of the Rcation, cobaltites with δ∼0.5\nall show a jump in resistivity and a concomitant lattice\ndistortion with a sudden volume collapse. The distortion\nis associated with specific changes in the Co–O distances\nin pyramids and octahedra, and became the subject of\ndifferent interpretations. A possible driving force for the\nMI transitionhasbeen proposedto be aspin state transi-\ntionfromthe Colow–spinstate(LS: t6\n2ge0\ng)toahigh–spin\nstate (HS: t4\n2ge2\ng) occurringonly at the octahedral sites.10\nFor the particular cases of R= Pr11and Gd,10,12this\nhypothesis would also be supported by a change in the\nslope of the inversesusceptibility curveat TMI, which has\nbeen analyzed in terms of the Curie–Weiss model. Fur-\nther support to this scenario was given by Maignan et\nal.9forR= Ho based on thermoelectric measurements,2\nshowing that the spin blockade mechanism is fully com-\npatible with this picture. Other authors have proposed\nthat at2g−eghybridization is enhanced in the metal-\nlic phase by the lattice distortion, such that the metallic\nor insulating behavior would be determined by the in-\ntersite mixing of the itinerant 3d electrons between the\noctahedral and pyramidal sites.13ForR= Tb the dis-\ntortions of pyramids and octahedra were interpreted as a\nd3x2−r2/d3y2−r2orbital ordering transition accompanied\nby a intermediate–spin (IS: t5\n2ge1\ng)to HS spin state transi-\ntion.7Furthermore, Pomjakushina et al.14proposed that\nthe observed volume collapse at the transition tempera-\nture and the existence of an isotopic effect are indicative\nof a charge delocalization breaking the orbital order of\nthe insulating phase, which could be compatible with a\nspin state switch, but occurring in pyramids, not in octa-\nhedra. It seems obvious that this issue is far from being\nclarified and some effort must be made to systematize\nthe study of the MI transition in cobaltites.\nA second controversy, closely related to the one men-\ntioned in the previous paragraph, concerns the low tem-\nperature ordering of the magnetic moments at the Co\nsites. Again, the feature which seems to be common to\nallRcobaltites is the existence of a spontaneous mag-\nnetization in a more or less narrow temperature range,\ndepending on R, below room temperature. Above this\nrange they are paramagnetic, and below this range they\ntransformtoanantiferromagnetic(AFM) state. Itisnow\nwell established that the origin of the spontaneous mag-\nnetization is not a ferromagnetic order, but among the\ntwo remaining possibilities, i.e., a ferrimagnetic phase or\na canted AFM phase, there are various different mod-\nels which have been proposed. Some of these models\ninvolve a so–called spin state ordering (SSO) in which\nnot only the spin state may be different between Co\natoms located at pyramids and Co atoms located at oc-\ntahedra, but also among the pyramidal15or the octahe-\ndral sites16,17a SSO may arise leading to a doubling of\nthea−axis in the unit cell. Indeed, a theoretical work\nby Khomskii and L¨ ow18showed that such spin super-\nstructures can be energetically favorable. These mod-\nels would correspond to a ferrimagnetic phase. On the\nother hand, the proposers of canted–AFM models argue\nthat there are no structural evidences for the doubling\nof thea−axis, and adopt the canted models which also\nexplain the neutron diffraction data, with a doubling of\nthea−axis just in the magnetic cell.11,19However, some\ncare must be taken when comparing all these experimen-\ntal data. For instance, there may be no evidence of a\n“222”superstructure in cobaltites with R= Gd and Pr11\nbut the case might be different for other lanthanides. In\nfact, some studies using NMR techniques showed that\nforR= Y, there are four non–equivalent Co sites at low\ntemperature,21and forR= Eu there are three,22which\nis compatible with the SSO scenario. It should be em-\nphasized, too, that the “222” superstructure is very hard\nto detect from diffraction measurements unless an ex-\nceptionally high signal–to–noise ratio is attained. Usingtransmission geometry, Chernenkov et al.23have shown\nthat the superstructure can indeed be observed in sin-\ngle crystals with R= Gd using X–ray diffraction. In all\ncases, there seems to be consensus on the IS character\nof pyramidal Co atoms,5,10,24–26although the spin state\n—or spin states— at octahedral sites remains uncertain\nor may, at least, depend on the Rsize.\nMoststudies oflayeredcobaltiteswereconducted for R\namongthelanthanides,butthecompoundwith R=Y3+,\nwhich is a small, non–magnetic ion, is a good candi-\ndate to isolate the intrinsic properties of Co and ex-\nplore the small– Rregion of the phase diagram. It is\nnow well documented, for instance, that the MI tran-\nsition temperature decreases with the Rsize. To gain\nmore insight into the possible role of disorder, we have\nintroduced a second source of distortion, by substitut-\ning the Ba–site with Ca, which has a smaller atomic\nradius. In addition, it has recently been postulated on\nthe basis of density–functional theory calculations, that\na smaller cation substitution in the Ba–site of small lan-\nthanide cobaltites could be a promising compound to\nexhibit enhanced giant magnetoresistance properties.27\nThe present work is aimed at characterizing and corre-\nlating the Ba–substituted compounds when compared to\nthe parent YBaCo 2O5.5cobaltite. We have performed a\nstructural characterization using neutron powder diffrac-\ntion (NPD) to study the interplay between the structures\nand their magnetic order, and correlate this information\nwith our previous magnetic studies.28,29\nII. EXPERIMENTAL METHODS\nThree polycrystalline samples were prepared by\nsolid–state reaction. High–purity powders of Y 2O3,\nBaCO 3, CaCO 3and Co 3O4were mixed at stoichio-\nmetric weights to prepare the compounds YBaCo 2O5+δ\n(xCa= 0), YBa 0.95Ca0.05Co2O5.5(xCa= 0.05) and\nYBa0.90Ca0.10Co2O5.5(xCa= 0.10). After a de–\ncarbonationprocessat1173Kfor18h, themixtureswere\npressed into pellets and annealed. The samples were an-\nnealed together during 25 h at 1273 K and slowly cooled\nat 1 K/min in oxygen flow. After a regrinding of the re-\nsulting pellets, the compression and annealing at 1273 K\nin oxygen processes were repeated. A single batch was\nused for all the samples to guarantee identical synthesis\nconditions, which resulted in samples of about 1.5 g.\nThe oxygen content in our samples has been deter-\nmined by refinement of our NPD data. In addition, we\nhave compared the macroscopic magnetization and resis-\ntivity of our xCa= 0 sample with a very detailed study\nof the parent compound YBaCo 2O5+δearly reported by\nAkahoshi and Ueda30. Their work presents the existing\ncorrelation between oxygen content and magnetic and\ntransport properties. In particular, the magnetization\ncurve for our sample (Fig. 5) reveals an excellent quan-\ntitative agreement with their results for δ= 0.5, and a\nclear disagreement outside the range 0 .44< δ≤0.52.3\nMoreover, the resistivity measurements in our samples29\nshow the characteristic sharp jump of the MI transition,\nwhich has been shown to occur only for 0 .45< δ≤0.65\nbut only to be sharp for δ≃0.54. These limits give\nus confidence in the refined values from our NDP data.\nThe global oxygen contents refined independently from 8\nhigh resolutiondiffractograms,correspondingto different\nsamples and temperatures —always below 350 K— were\nin mutual agreement within experimental error, yield-\ning an average value of δ= 0.46±0.02. In the fol-\nlowing we shall refer to the samples using the notation\nY(Ba,Ca)Co 2O5.5.\nNeutron thermodiffraction data were collected on the\nhigh–intensity two–axis diffractometer D20 located at\nthe High Flux Reactor of Institute Laue–Langevin ILL,\nGrenoble, France. Samples with xCa= 0.05 andxCa=\n0.10werecooledin astandardorangecryostatfromroom\ntemperature down to 20 K, and diffraction patterns were\nthen collected every two minutes at a warming rate of\n1 K/min from 20 K to 320 K. A wavelength of ∼2.41˚A\nwas used to highlight the magnetic diffraction and was\ncalibrated using a Silicon sample.\nIn addition, high–resolution NPD data were collected\nat diffractometer Super–D2B of ILL for samples with\nxCa= 0,0.05 and 0.10. A wavelength of ∼1.594˚A was\nused to collect patterns at selected temperatures for ap-\nproximately 3 h. It is worth noting that the volume of\nsampleavailablewasnotasmuchastheidealforthiskind\nof experiment, so we looked for a compromise between\nthe collection time, the available beamtime, and our ca-\npabilities for preparing all the samples in a single batch.\nThe NPD patterns were processed with the full–pattern\nanalysis Rietveld method, using the program FULLPROF\n31for refining the crystal and magnetic structures.\nIII. RESULTS\nA. Description of structures and refinement\nstrategy\nThe room temperature structures of the parent com-\npound YBaCo 2O5+δwere first reported by Akahoshi and\nUeda,30who showed that for δ= 0.5 there may form\ntwo competing structures. One of them is orthorhombic,\nand corresponds to the space group Pmmm having the\n“122” superstructure characteristic of similar cobaltites\nwithδ= 0.5.2,3,7,24A schematic representation of this\nphase is shown in Fig. 1. The vacancies order consists of\nalternating[CoO 6]octahedrachainsalongthe c−axisand\ncorner–sharing[CoO 5] pyramidsalongthe b−axis,result-\ning in alternating octahedral and pyramidal layers in the\na−cplane. This produces a doubling of the cell along\ntheb−axis, with a unit cell aPx2aPx2aP. The second\nstructure that may stabilize in this system for δ= 0.5\n(and other values as well) has a tetragonal symmetry\nand no doubling of the b−axis,i.e., no ordering between\npyramids and octahedra. In this case, the space groupBa YOCo \nabc\n(a) (b) \n(c) (d) \nFIG. 1: (Color online) (a) The structure of YBaCo 2O5.5. Half\nCo atoms are in square–pyramidal coordination and the other\nhalf in octahedral coordination. (b) Magnetic model adopte d\nfor the SSO ferrimagnetic phase after Ref. 17. The nuclear\n“122” unit cell and the magnetic “222” unit cell are indicate d,\nand light atoms correspond to octahedral coordination. In\nthisschematicrepresentation, onlyCoandOatomsare shown\nfor clarity. (c) Magnetic model adopted for the AFM– O2\nphase after Ref. 17. (d) Magnetic model adopted AFM– O1\nphase.\nisP4/mmmand the unit cell aPxaPx2aP. Recently,\nFrontera et al.6have shown that, although the order of\nvacancies may not always be perfectly achieved, the or-\nder between the R–cation layer and the Ba layer is well\nestablished and there is no mixing between them. Nei-\nther are there significant oxygen vacancies in the [BaO]\nlayers. The “122” structure admits a certain degree of\ndisorder, consisting of misplaced pyramids or octahedra,\nbut keeping the long–range “122” order.\nIn the present refinements, the Ca cations were ran-\ndomly introduced in the structure at the crystallographic\nsiteoccupied byBa, in appropriateproportions. We have\nfound no evidence of Ca segregation nor the formation of\nadditional phases, so we believe that Ca has been suc-\ncessfully incorporated into the cobaltites structure. The\nstrategyforthe Rietveldrefinement wasasfollows. First,\nthe high resolution data from D2B were refined to ob-\ntain an accurate nuclear structure for each sample. The\nraw data coming from the Super–D2B detector were pro-\ncessed using the LAMP software32to obtain two sets of\ndata: one of them having a better angle resolution at\nthe expense of losing some neutron counts, the other one\nhaving all neutron counts collapsed into a single diffrac-\ntogram. The first set was used to determine the lattice4\nparameters, while the second set was used to refine the\natomic positions and temperature factors, and both sets\nwere iteratively refined until convergence to the struc-\nture. The magnetic structures were also included in the\nrefinements. The models we have used will be discussed\nin the following sections. At a second step, the struc-\ntural data obtained were used to refine sequentially the\nneutron thermodiffractogramsobtained at D20. Temper-\nature scans where divided into different ranges according\nto the structural and magnetic order, and for each range\nthe atomic positions and occupations obtained at D2B\nwere kept fixed, while lattice parameters, temperature\nfactors and magnetic moments were allowed to vary.\nThe objective of this work is to focus on the role\nof Ca addition to the parent compound YBaCo 2O5.5.\nOur results will show that there is a clear logical se-\nquence between the three samples studied, corresponding\ntoxCa= 0,xCa= 0.05 andxCa= 0.10. Surprisingly, the\nsample with greater Ca content, xCa= 0.10, turned out\nto be the simplest one, and as Ca is removed the com-\nplexity increases, resulting in a quite complicated tem-\nperature evolution of the parent compound. This fact\nprobably explains why this compound has not yet been\nfully reported in such detail as other Rcobaltites, except\nfor the structural study by Akahoshi and Ueda,8and a\nrecent neutron diffraction study by Khalyavin et al.33fo-\ncusing on the ferrimagnetic to paramagnetic transition.\nFor the above reasons, we will present our results follow-\ning the decreasing Ca sequence.\nB. Sample with xCa= 0.10\nIn Fig. 2 we present two different sections of the pro-\njected thermodiffractograms for the sample xCa= 0.10.\nFig. 2(a) corresponds to the low–angle range, in which\nmost reflections are of magnetic nature, and disappear\nsimultaneously at ∼295 K on warming from 20 K to\n300 K. In Fig. 2(b) we focus on the 2 θrange where the\nBragg reflections (2 0 0) and (0 4 0) clearly show a dis-\ntortion occurring at room temperature.\nThe high resolution data were refined using a nuclear\nphase with the “122” structure, as described in sec-\ntion IIIA. We do not discard the possibility of the actual\nstructure being “222”, with a doubling of the a−axis and\nfour different crystallographicsites for the Co atoms,17,33\nin line with the magnetic model adopted. However, as we\nare interested in the temperature evolution of rather low\nresolutiondatafromD20,andgiventhecomplexityofthe\nother samples, we have decided to refine the whole series\nwith the averaged “122” structure. Moreover, following\nFrontera et al.11we have fixed the zcoordinate of the oc-\ntahedral Co site to 1/4, in orderto obtain centrosymmet-\nric octahedra. With these assumptions, we reduced the\nnumber of free parameters which is critical when dealing\nwith multiphasic systems. We have nevertheless allowed\nfor disorder between pyramids and octahedra, by refin-\ning the occupation of apical oxygen sites (site 1gmostlyTABLE I: Structural parameters refined from the high reso-\nlution D2B data for the compound YBa 0.90Ca0.10Co2O5.5at\nT= 70 K, 230 K and 350 K. Atomic fractional coordinates\ncorrespond to space group Pmmm in the following Wyck-\noff positions: Y ( 2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct\n(2r)=(0,1\n2,1\n4); CoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2\n(1e)=(0,1\n2,0); O3 ( 1g)=(0,1\n2,1\n2); O3’ ( 1c)=(0,0,1\n2); O4\n(2s)=(1\n2,0,z); O5 (2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 230 K T= 350 K\nY y0.2714(5) 0.2717(5) 0.2684(5)\nBa, Ca y0.253(1) 0.254(1) 0.248(1)\nCoPyr z0.260(1) 0.260(1) 0.262(1)\nO4 z0.3112(7) 0.3108(7) 0.3107(7)\nO5 z0.276(1) 0.276(1) 0.271(1)\nO6 y0.2481(8) 0.2479(8) 0.2416(8)\nO6 z0.2954(6) 0.2947(5) 0.2980(5)\nO3 Occ 0.94(2) 0.93(2) 0.89(2)\nO3’ Occ 0.0 0.0 0.02(2)\na(˚A) 3.8423(1) 3.8438(1) 3.8254(1)\nb(˚A) 7.7947(2) 7.8118(2) 7.8503(2)\nc(˚A) 7.4835(2) 7.4965(2) 7.5217(2)\nRB 7.4 7.4 7.1\nRmag 13.6 14.7\nχ27.4 5.8 5.2\noccupied, and site 1cmostly unoccupied in the Pmmm\nspace group). We remark that even for different degrees\nof vacancies disorder, the total occupation of sites always\nsummed up to the sameoxygencontentin all samplesbe-\nlow 350 K. In Table I we present the details of the refined\nstructure for the sample with xCa= 0.10 from D2B data\ncollected at 70 K, 230 K and 348 K.\nThe model we have adopted for refining the mag-\nnetic phase is the SSO ferrimagnetic model proposed by\nKhalyavin,17,33which leads to the best agreement with\nour NPDdata, macroscopicmagnetization data and high\ntemperature susceptibility data.28,29It is schematized in\nFig. 1(b). Although the model involves Co atoms at half\nthe octahedral sites being in low–spin state, and there-\nfore having a magnetic moment equal to zero, after a\nfirst step in the refinement we allowed this site to adopt\na non–zero magnetic moment, which resulted in a small\nvalue compatiblewith the fact that the apical oxygensite\n1gis not completely occupied, and therefore some octa-\nhedra are, in fact, misplaced pyramids.11In Fig. 3(a)\nwe present the evolution with temperature of the lattice\nparameters refined in the Pmmm “122” structure. The\nsequential D20 results are presented together with the\nresults from D2B at the studied temperatures. The char-\nacteristic structural distortion occurring at TMI∼295 K\nis clearlyobserved. We havealreadyreportedthat in this\nsystem, the MI transition occurs almost simultaneously\nwith the paramagnetic–ferrimagnetic transition,28which\nseems to be only a coincidence. Therefore, no further5\nanomaly is observed in the lattice parameters down to\n20 K, as there is in this sample no further magnetic tran-\nsition. In Fig. 4(a) we present the magnetic moment of\nCo atoms in each crystallographicenvironment. We have\nnot included in the figure the small magnetic moment of\nmisplaced pyramids, which remains always less than 0.4\nµB.\nC. Sample with xCa= 0.05\nOur preliminary X–ray diffraction pattern of the as–\nsynthesized sample xCa= 0.05 at room temperature re-\nsulted identical to xCa= 0.10with just a slight difference\nin the lattice parameters, indicating that the room tem-\nperaturestructure is the same in both samples. However,\nthe macroscopic magnetization data in Fig. 5 show that\natlowtemperature theybehavedifferently. Below200K,\non cooling, the magnetization of the xCa= 0.05 sample\nstarts dropping but the sample retains a net magnetiza-\ntion down to 5 K, in contrast with the parent compound\nxCa= 0 which shows an AFM behavior. Moreover, the\nbig hysteresis between the data collected on cooling and\nwarming in the xCa= 0.05 sample suggests that there is\nacompetitionbetweentwostates. TheNPDexperiments\nreveal the nature of these two states.\nIn Fig. 6 (c) we present a projection of the thermod-\n T[K] T[K] \n2θ2θ(a) \n(b) \nFIG. 2: (Color online) Projection of two selected sections\nof the thermodiffractograms corresponding to sample with\nxCa= 0.10. (a) 16◦<2θ <44◦and (b) 75◦<2θ <80◦\nshowing the (2 0 0) and (0 4 0) Bragg reflections of the “122”\nstructure. Data were collected at D20 with λ∼2.41˚A be-\ntween 20 K and 320 K.0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(a) b1\n2a1\nc1\n  Lattice Parameters [Å] \nT [K] \n0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(b) b2b12a2\n2a1\nc2c1\n  Lattice Parameters [Å] \nT [K] \n0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(c) cb\n2a Akahoshi & Ueda [ref. 8] \nb2b1\n2a2\n2a1\nc1=c2\n  Lattice Parameters [Å] \nT [K] \nFIG. 3: (Color online) Thermal evolution of the lattice pa-\nrameters 2 a,bandcfor theO1 (dark symbols) and O2 (light\nsymbols) phases in samples with xCa= 0.10 (a),xCa= 0.05\n(b) and xCa= 0 (c), determined from data collected at D20\nand D2B. In (c) the data from D2B (solid symbols) are com-\npared with data reported by Akahoshi and Ueda,8plotted\nusing diamonds.\niffractograms collected at D20 for sample xCa= 0.05. In\n(a) and (b) we show the temperature evolution of the in-\ntensity of the magnetic reflections (0 1 0) and (1/2 1 1)\nrespectively, indexed in terms ofthe “122”nuclear phase.\nThese reflections correspond to the ferrimagnetic order\ndiscussed in the previous subsection. Figure 6 (a) and6\n0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n Octahedral site 2 \n Pyramidal site \n(a) \n  m [ µB/Co] \nT [K] \n0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n(b) \n  m [ µB/Co] \nT [K] \nFIG. 4: (Color online) Temperature dependence of the mag-\nnetic moments in pyramids (square symbols) and half the oc-\ntahedra (triangular symbols) for the samples xCa= 0.10 (a)\nandxCa= 0.05 (b), obtained from the Rietveld refinements\nof neutron data collected at D20. Moments were refined along\n[100].\n(b) compare the intensities of these reflections in sam-\nplesxCa= 0.05 andxCa= 0.10. We can observe that\nthey both behavealmostidentically from200K to320K,\nbut below 200 K the ferrimagnetic reflections are lower\nin thexCa= 0.05 sample. Moreover, at 200 K additional\nfeatures are evidenced in the thermodiffractograms. The\nsmall arrows in Fig. 6 (c) mark two reflections which ap-\npear only below 200 K. They are indicative of the pres-\nence of a second —magnetically ordered— phase which\nmay be indexed with a further doubling of the cparame-\nter as reported by various authors in the AFM region of\nlayered cobaltites.11,15–17,19But here we also observe si-\nmultaneouschangesin the nuclearstructureand not only\na magnetic phase separation, or gradual magnetic transi-\ntion to a different magnetic arrangement.11,15,16,20This\nis illustrated, for instance, by a peak arising at 200 K\nin 2θ∼77.6◦lying in between the (2 0 0) and (0 4 0)\nreflections of the “122” nuclear phase. In Fig. 7 we show\nthe evolution with temperature of the collected intensity\nat 2θ∼77.6◦, for this sample as well as for the sample\nwithxCa= 0.10 for comparison. When warming, there\nis a sudden drop at 200 K, to a value corresponding to0 50 100 150 200 250 300 012345678910 \n0 50 100 150 200 250 300 012345678910 \n0 50 100 150 200 250 300 012345678910 \nxCa =0.10 \nxCa =0.05 \nFCW FCW \nFCC FCC \nxCa =0 \n  \n  M [emu/g] \nT [K] \n  \nFIG. 5: (Color online) Low–field magnetization as a func-\ntion of temperature for samples with xCa= 0,xCa= 0.05\nandxCa= 0.10. Empty symbols represent the magnetization\nmeasured on cooling under a magnetic field of 5 kOe (FCC),\nand solid symbols represent the magnetization subsequentl y\nmeasured on warming (FCW).\nthe overlap of the neighboring (2 0 0) and (0 4 0) reflec-\ntions, and a further drop to background values above the\nMI transition, when these reflectionsaresuddenly shifted\napart by the distortion. We first evaluated the possibil-\nity of this behavior being a result of a distortion of the\n“122” nuclear phase at 200 K. Such hypothesis gave no\nsatisfactory results when trying to refine simultaneously\nthe D2B and D20 data at 70 K. Additional attempts to\nmodel the 70◦<2θ <80◦region using Gaussian peaks\nof just one orthorhombic phase, even allowing for unreal-\nistically wide peaks, could not reproduce the triple–peak\nshape observed in the D20 spectra (inset in Fig. 7).\nWe also considered other structural models to account\nfor the low temperature data coming from D2B and D20.\nNeither the PccanorPmmaspace groups proposed by\nPlakhty et al.15and Khalyavin,33together with the re-\nspective magnetic models proposed in their work, gave\nsatisfactory results for the simultaneous refinement of all\nour data.\nAnother possible explanation is the presence of a sec-\nond structural phase. This is suggested by the poor\nresults obtained refining the D2B data using the same\n(single-phase)modelasforthe xCa= 0.10sampleat70K\n(Pmmmspace group), as well as with other single-phase\nmodels (Pcca,Pmma). Based on the D20 data, we con-\nsidered a possible second phase with a strong tetragonal\ndistortion, which would not be unreasonable considering\nthat a two-phase mixture of orthorhombic and tetrago-\nnal phases for YBaCo 2O5.5had already been reported\nby Akahoshi and Ueda,30both phases occurring compet-\nitively. It is also worth noting that in the same batch as\nthe present samples, we have also synthesized a series of\nsamples where Barium is replaced by Strontium, a sub-7\nT[K] Intensity [arb. units] x=0.10 \nx=0.05 \nT[K] Intensity [arb. units] x=0.10 \nx=0.05 \n2θT[K] (a) (b) \n(c) \nFIG. 6: (Color online) Thermal evolution of the intensity of Bragg reflections (0 1 0)(a) and (1/2 1 1)(b) indexed in terms o f\nthe “122” phase for samples with xCa= 0.05 (triangles) and xCa= 0.10 (diamonds) from data collected at D20. (c) Two–\ndimensional projection of the neutronthermodiffractogram s collected at D20 for sample xCa= 0.05 in the range 16◦<2θ <44◦.\nThe small arrows indicate theonset ofmagnetic reflections f rom theAFM “224” phase. The longarrows between graphs indic ate\nthe position in the thermodiffractogram of the two reflection s whose intensity is plotted in (a) and (b).\nstitution which clearly favors a tetragonal phase in which\nthe position of the (2 0 0) Bragg reflection is almost ex-\nactly coincident with this new peak in the xCa= 0.05\nsample.34Consequently, we tried to refine both the D2B\nand D20 sets of data using a mixture of the “122” phase\nand a tetragonal ( P4/mmm) phase. This gave no satis-\nfactoryresultseither, andmoreover,the proposedtetrag-\nonal symmetry could not account for the observed mag-\nnetic supercell.\nWe finally proceeded to adopt for the second phase an\northorhombic ( Pmmm) cell with the constraint b= 2a,\nand at a further step we allowed the bandalattice pa-\nrameters to vary independently. The diffractogram at\n70 K was therefore refined with one orthorhombic phase\nwith ferrimagnetic order ( O1 phase) plus a second or-\nthorhombic phase ( O2) with AFM order which is con-\nsistent with a “224” supercell. This finally gave muchmore satisfactory results for the refinement. The mag-\nnetic peaks arising below 200 K, although weak, could\nbe accounted for using the AFM model proposed by\nKhalyavin,17schematized in Fig. 1(c), with the con-\nstraint of a single magnitude for the magnetic moment of\nall Co atoms in pyramidal positions, and once again (as\nin the ferrimagnetic SSO model), two possible spins for\noctahedral Co. At 230 K, the O1 phase with ferrimag-\nnetic order was enough to refine the D2B diffractogram.\nIn TableII we presentthe details ofthe refined structures\nfor the sample with xCa= 0.05 from D2B data collected\nat 70 K and 230 K. Figure 8 shows the Rietveld refine-\nment (solid line) of the high resolution data (symbols)\ncollected at 70 K. The four sets of Bragg reflections indi-\ncated at the bottom by vertical bars correspond to each\nof the above mentioned phases. The difference pattern\nbetween observed and calculated data is also shown.8\nIntensity [arb. units] \nT[K] 2θxCa = 0.05 \nxCa = 0.10 \n T\n[K] \n2θ\nT\n[K] \nFIG. 7: (Color online) Thermal evolution of the intensity\nin the thermodiffractograms at 2 θ∼77.6◦, in the position\nbetween the (2 0 0) and (0 4 0) Bragg reflections of the\n“122” phase for the sample with xCa= 0.05 (diamonds) and\nxCa= 0.10 (squares). The insets show the three–dimensional\nthermodiffractograms for the relevant 2 θrange.\nThe presented scenario for the evolution with temper-\nature of the xCa= 0.05 sample yields consistent results\namong the D2B and D20 data, as well as it accounts\nfor other experimental facts. When cooling from room\ntemperature, theintensityoftheferrimagneticreflections\nstarts dropping because there is a second phase develop-\ning in the sample, so that the volumefraction ofthe ferri-\nmagneticphaseisreduced. Inaddition, theobservedhys-\nteresis among the FCC and FCW magnetization curves\nin Fig. 5 are also indicative of a possible phase separa-\ntion, as well as the hysteresis in the resistivity curves\npresented previously.29The presence of a second nuclear\nphase has become more evident when performing ther-\nmodiffractograms with λ= 2.52˚A. This scenario would\nbe very difficult to infer just from D2B data collected at\nlower wavelengths and at isolated temperatures, due to\npeak overlap and to a lack of perspective of the continu-\nous thermal evolution of the sample.\nThe evolution with temperature of the lattice param-\neters refined in the O1 andO2 phases is shown in\nFig. 3(b). The sequential D20 results are presented to-\ngether with the results from D2B at the studied temper-\natures. For the O1 phase, the distortion at TMI∼295K\nis again observed as in the sample with xCa= 0.10.\nFor the O2 phase, we observe a tetragonal distortion\nabove 170 K: above that temperature the a2andb2lat-\ntice parameterscouldonlybe refinedusing the constraint\n2a2=b2. The volume per atom of both phases is prac-\ntically the same, although it is observed that the lat-\ntice parameter relation is different (2 a1< b1, 2a2> b2).\nIt would be interesting to stabilize the O2 phase to get\nmore detailed information of its structural properties.34\nInFig.4(b)wepresentthemagneticmomentofCoatomsTABLE II: Structural parameters refined from the high reso-\nlution D2B data for the compound YBa 0.95Ca0.05Co2O5.5at\nT= 70 K and 230 K. Two sets of atomic fractional coordi-\nnates are given for the O1 andO2 phases, which correspond\nto space group Pmmm in the following Wyckoff positions: Y\n(2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct ( 2r)=(0,1\n2,1\n4);\nCoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2 ( 1e)=(0,1\n2,0);\nO3 (1g)=(0,1\n2,1\n2); O3’ (1c)=(0,0,1\n2); O4 (2s)=(1\n2,0,z); O5\n(2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 230 K\nO1 O2 O1\nY y0.2757(8) 0.264(2) 0.2731(5)\nBa, Ca y0.254(1) 0.233(3) 0.254(1)\nCoPyr z0.262(2) 0.267(2) 0.261(1)\nO4 z0.317(1) 0.291(5) 0.3125(7)\nO5 z0.261(2) 0.287(5) 0.274(1)\nO6 y0.2460(9) 0.243(2) 0.2473(7)\nO6 z0.2941(6) 0.301(1) 0.296(1)\nO3 Occ 1.0 0.84(6) 0.92(2)\nO3’ Occ 0.0 0.0 0.00(2)\na(˚A) 3.8460(2) 3.8845(3) 3.8468(1)\nb(˚A) 7.7887(4) 7.7099(6) 7.8075(2)\nc(˚A) 7.4827(5) 7.4819(7) 7.4959(2)\nf(%) 64(4) 36(3) 100\nRB 7.0 7.0 7.3\nRmag 15 32 16\nχ24.48\nin the ferrimagnetic phase for each crystallographic en-\nvironment. We have not included in the figure the small\nmagnetic moment of misplaced pyramids, which remains\nalwayslessthan 0.5 µB. Unfortunately, the qualityofour\nD20 data and the two-phase scenario do not allow for a\nconfident determination of the magnetic moments in the\nO2 AFM phase. These were constrained to be aligned\nalong [100] following the model described above and to\nadopt values similar to those in the O1 phase, in order\nto obtain reliable phase fractions which were comparable\nto the values refined from D2B data.\nFigure 9 shows the net spontaneous magnetization of\nsamples with xCa= 0.05 andxCa= 0.10, obtained\nasM=µCo·fO1, whereµCorepresents the net mag-\nnetic moment per Co atom in the ferrimagnetic phase\n(=µCoOct/4) andfO1is the refined phase fraction of\ntheO1 phase. These results can be compared with the\nmacroscopic determination of the magnetization of the\nsamples as a function of temperature (Fig. 5), always\nconsidering these were collected under an applied field of\n5 kOe. The overall similarity between the curves is in\nexcellent agreement.9\n2θ(deg) Intensity[arb. units]\nFIG.8: (Color online)Rietveldrefinementfor thesamplewit h\nxCa= 0.05 from data collected at D2B at T= 70 K. Vertical\nbars at the bottom indicate Bragg reflections from the phases\nincluded in the refinement: the nuclear phases O1,O2 and\nthe magnetic phases SSO “222” and AFM “224”.\nD. Sample with xCa= 0\nWe finally turn to the parent compound. This sample\nwas only studied in the high resolution instrument, so we\ncannot present continuous temperature scans in the low–\ntemperature range as in the other samples. We collected\nthree diffractograms at T= 70 K, 273 K and 348 K. A\nsimilar study on this system has been very recently re-\nported,33focused on the paramagnetic to ferrimagnetic\ntransition, and a verygood agreementis found. It should\nbe emphasized that those authors have refined the pat-\nterns using the expanded “222” cell for the nuclear phase\nin the ferrimagnetic region in the Pmmaspace group, an\nhypothesis which —as we mentioned before— we do not\ndiscard but prefer to use the “122” Pmmm cell to be\nconsistent along the whole series. For the lowest temper-\nature, however, those authors did not present any refine-\nment of their data. The intriguing fact that the “222”\nmagneticreflectionsreappearedat190Kafterhavingdis-\nappeared at 265 K was left unexplained. In the present\nwork, we show that our diffractogramat 70 K can be sat-\nisfactorily refined in the framework of the analysis pre-\nsented for the xCa= 0.05 sample. Therefore, the nuclear\ndiffraction was accounted for by using two orthorhombic\nphases,O1 andO2, and the whole set of magnetic reflec-\ntions could then be assigned to each of these phases. As\ninxCa= 0.05, theO2 phase presents an AFM ordering\nwith a “224” magnetic cell. The O1 phase, on the other\nhand, cannot keep its ferrimagnetic SSO order because0 50 100 150 200 250 300 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 \nxCa =0.10 \nxCa =0.05 \n  MH=0  [ µB/Co] \nT [K] \nFIG. 9: (Color online) Spontaneous magnetization of the fer -\nrimagnetic phase in samples with xCa= 0.05 andxCa= 0.10,\nobtained from our NPD refinements as M=µCo·fO1,\nwhereµCorepresents the net magnetic moment per Co atom\n(=µCoOct/4) andfO1is the phase fraction of the O1 phase.\nthis would not be compatible with the macroscopic mag-\nnetization measurements. In addition, a close inspection\nof the magnetic peaks reveals that not all the reflections\nfrom the 273 K ferrimagnetic phase are present, but that\ncontributionstointensityrelatedtoFMplanesareabsent\nat 70 K. Therefore, we have used a second AFM model\nfor theO1 phase at 70 K, with a “222” magnetic cell,\nand a G–type–like ordering, although we have allowed\nfor different magnetic moment values in pyramids and\noctahedra. The model is schematized in Fig. 1(d). This\nscenariois also supported by our data on a different sam-\nple, substituted with 5 % Sr ( Y(Ba 0.95Sr0.05)Co2O5.5)\nfor which we have also performed neutron thermodiffrac-\ntion scans and apparently behaves almost the same as\nthe parent compound. These results will be published in\na separate paper.34\nIn Table III we present the details of the refined struc-\ntures for the sample with xCa= 0 from D2B data. The\ndiffractogram at 70 K was refined with the O1+O2 mix-\nture, plus two AFM phases associated to each of them,\none having a “224” supercell and the other one with a\n“222” supercell. At 230 K, only the O1 phase with fer-\nrimagnetic order was enough to refine the diffractogram,\nwhereas at 348 K just the nuclear O1 phase was refined.\nIt is worth noting that the parent compound seems to\nhave a higher degree of misplaced octahedra when com-\npared to the Ca–substituted samples. This is evidenced\nby the non–zero occupation of the O3’ site, which cor-\nresponds to the empty apical oxygen position of pyra-\nmids. At the highest temperature, however, there is a\nslight rearrangement of vacancies among the O3 and O3’\nsites. When comparing the low temperature phases O1\nandO2 in the xCa= 0 and xCa= 0.05 samples, we10\nTABLE III: Structural parameters refined from the high reso-\nlutionD2BdatafortheparentcompoundYBaCo 2O5.5atT=\n70 K, 273 K and 348 K . Two sets of atomic fractional coordi-\nnates are given for the O1 andO2 phases, which correspond\nto space group Pmmm in the following Wyckoff positions: Y\n(2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct ( 2r)=(0,1\n2,1\n4);\nCoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2 ( 1e)=(0,1\n2,0);\nO3 (1g)=(0,1\n2,1\n2); O3’ (1c)=(0,0,1\n2); O4 (2s)=(1\n2,0,z); O5\n(2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 273 K T= 348 K\nO1 O2 O1 O1\nY y0.2809(9) 0.271(1) 0.2745(6) 0.2692(5)\nBa y0.255(2) 0.238(2) 0.254(1) 0.244(1)\nCoP z0.251(3) 0.270(3) 0.261(1) 0.260(1)\nO4 z0.321(2) 0.300(2) 0.3127(9) 0.3126(7)\nO5 z0.251(2) 0.276(3) 0.274(1) 0.267(1)\nO6 y0.2449(9) 0.243(1) 0.2495(9) 0.2437(8)\nO6 z0.299(1) 0.300(1) 0.2935(7) 0.2984(5)\nO3Occ 1.0 0.67(6) 0.86(3) 0.81(2)\nO3’Occ 0.0 0.18(4) 0.05(2) 0.12(2)\na(˚A) 3.8515(2) 3.8819(2) 3.8496(1) 3.8221(1)\nb(˚A) 7.7785(5) 7.7156(4) 7.8085(2) 7.8581(3)\nc(˚A) 7.4859(6) 7.4845(6) 7.5032(2) 7.5250(2)\nf(%) 42(3) 58(3) 100 100\nRB 7.7 7.5 8.1 6.4\nχ23.98 3.97 3.53\nobserve that the O1 phase seems to prefer a more per-\nfect order of pyramids and octahedra, while the excess\noxygen vacancies accommodate in the O2 phase. There\nis also consistency among the structural parameters in\nboth samples for the O1 andO2 phases. The results\nfor the refined lattice parameters at the three temper-\natures studied are shown in Fig. 3(c). Our data are\ncompared with those published by Akahoshi and Ueda\n(diamonds).30The difference at 70 K is due to the use\nof one or two phases to refine the data. We have found\nno way of indexing the whole set of magnetic reflections\nbased on a single nuclear structure. It has been shown\nin other Rcobaltites that there could be two coexisting\nmagnetic arrangementson a single nuclearstructure,11,16however, the evidence found in the xCa= 0.05 thermod-\niffractograms, and the consistency obtained in the whole\nseries when adopting such a phase separationmodel, give\nus confidence in the proposed scenario. The complexity\nof the systems seems to be related to the small size of the\nRcation.\nIV. CONCLUDING REMARKS\nAlthough the layered cobaltites RBaCo2O5.5have re-\nceived great attention in the past five years, much of\nits behavior remains still controversial and unclear. In\nthis paper, an attempt is made to get some insight into\nthe role of cationic disorder by substituting the Ba–site,\na topic that has not yet been investigated to the best\nof our knowledge. Interestingly, the systematics of this\nsubstitution led us to clarify and to propose a model\nthat describes the behavior at low temperature of the\nundoped parent compound. Even though Ca addition\ndoes not lead to severe structural distortions, it has nev-\nertheless dramatic effects on the magnetic arrangement\nand stability of the ferrimagnetic phase on detriment of\nthe AFM long–rangeorder. Our results open up the pos-\nsibility of studying Ca–doped cobaltites in order to iso-\nlate the intrinsic properties of the “122” ferrimagnetic\nphase in monophasic samples, avoiding spurious effects\nin the analysis of macroscopic properties. Further work\nis in progress to investigate the role of different cations\nsubstitution and the systematics of the Ba–site disorder\neffects.\nAcknowledgments\nThis work is part of a research project supported by\nAgencia Nacional de Promoci´ on Cient´ ıfica y Tecnol´ ogica\n(Argentina), under grant PICT 17-21372 and 20144,\nby CONICET (Argentina) under grant PIP 5250/05\nand 5657/05, and by SECTyP, Universidad Nacional de\nCuyo. JC acknowledges a fellowship from CNEA and\nCONICET. 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Kato, E. Nishi-\nbori, M. Takata, M. Sakata, A. Nakamura, J. Phys. Soc.\nJapan70, 3577 (2001).\n25M. Pouchard, A. Villesuzanne, and J. P. Doumerc, J. Solid\nState Chem. 162, 282 (2001).\n26V. Pardo, and D. Baldomir, Phys. Rev. B 73, 165117\n(2006).\n27Hua Wu, Phys. Rev. B 64, 092413 (2001).\n28G. Aurelio, J. Curiale, R. D. S´ anchez and R. E. Carbonio,\nPhysica B 384, 106 (2006).\n29G. Aurelio, J. Curiale, R. D. S´ anchez and G. J. Cuello,\nPhysica B, in press.\n30D. Akahoshi and Y. Ueda, J. Phys. Soc. Japan 68, 736\n(1999).\n31J. Rodr´ ıguez–Carvajal, FULLPROF: A Program for Ri-\netveld Refinement and Pattern Matching Analysis , Ab-\nstracts of the Satellite Meeting on Powder Diffraction\nof the XV Congress of the IUCr, 127, Toulouse, France\n(1990).\n32LAMP, the Large Array Manipulation Program: http:\n//www.ill.fr/data_treat/lamp/front.html\n33D.D. Khalyavin, D.N. Argyriou, U. Amann, A.A. Yarem-\nchenkoandV.V.Kharton, Phys.Rev.B 75, 134407 (2007).\n34Aurelio et al., in preparation.0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(a) b1\n2a1\nc1\n  Lattice Parameters [Å] \nT [K] 0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(c) cb\n2a Akahoshi & Ueda [ref. 8] \nb2b1\n2a2\n2a1\nc1=c2\n  Lattice Parameters [Å] \nT [K] 0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n Octahedral site 2 \n Pyramidal site \n(a) \n  m [ µB/Co] \nT [K] 0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n(b) \n  m [ µB/Co] \nT [K] "    },    {        "title": "1506.06585v3.Exchange_scattering_as_the_driving_force_for_ultrafast_all_optical_and_bias_controlled_reversal_in_ferrimagnetic_metallic_structures.pdf",        "content": "arXiv:1506.06585v3  [cond-mat.str-el]  28 Feb 2016Exchange scattering as the driving force for ultrafast all- optical\nand bias-controlled reversal in ferrimagnetic metallic st ructures\nA. M. Kalashnikova and V. I. Kozub∗\nIoffe Physical-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia\n(Dated: today)\nExperimentally observed ultrafast all-optical magnetiza tion reversal in ferrimagnetic metals and\nheterostructures based on antiferromagnetically coupled ferromagnetic d−andf−metallic layers\nreliesonintricateenergyandangularmomentumflowbetween electrons, phononsandspins. Herewe\ntreat the problem of angular momentum transfer in the course of ultrafast laser-induced dynamics in\na ferrimagnetic metallic system using microscopical appro ach based on the system of rate equations.\nWe show that the magnetization reversal is supported by a cou pling of d−andf−subsystems to\ndelocalized s−orp−electrons. The latter can transfer spin between the two subs ystems in an\nincoherent way owing to the ( s;p)−(d;f) exchange scattering. Since the effect of the external\nexcitation in this process is reduced to the transient heati ng of the mobile electron subsystem, we\nalso discuss possibility to trigger the magnetization reve rsal by applying a voltage bias pulse to\nantiferromagnetically coupled metallic ferromagnetic la yers embedded in point contact or tunneling\nstructures. Weargue thatsuchdevices allow controlling re versal withhigh accuracy. Wealsosuggest\nto use the anomalous Hall effect to register the reversal, thu s playing a role of reading probes.\nPACS numbers: 75.78.Jp, 75.40.Gb, 75.50.Gg\nI. INTRODUCTION\nThe possibility of ultrafast control of the magnetic\nstate of nanostructures is an important constituent of\nferromagnet-based spintronics. Due to the problem of\nnon-localityand difficulty to createstrongyet shortmag-\nnetic field pulse,1–3the natural idea to use the latter is\nbecoming incompatible with the request for further in-\ncrease of storage densities and operation speed of novel\nspintronic devices.4Thus a great enthusiasm arose fol-\nlowingthe proposal5,6to usespininjection forcontrolling\nthe magnetization state of ferromagnetic specimen with\nthe help of an applied voltage. Later such a possibil-\nity was profoundly studied both theoretically and exper-\nimentally (for the critical review see e.g. [7]). However,\ntwo important obstacles were found. First, the switch-\ning time of the magnetization reversal by spin injection\nis defined by magnetization precession damping and is\nrather long (around ∼10−9s). Second, relatively large\ncurrents required for effective switching inevitably lead\nto unreasonable heat losses.\nThus a great attention8was attracted by recent exper-\niments demonstrating extremely fast ( ∼10−12s) magne-\ntization reversal triggered by a single femtosecond laser\npulse in ferrimagnetic metallic rare-earth (RE) - tran-\nsition metal (TM) alloy GdFeCo.9,10Very recently, ex-\nperimental observation of ultrafast laser-induced switch-\ning was reported in a variety of the engineered ferrimag-\nnetic structures, showing that this process is not specific\nfor the RE-TM single phase alloys, but can be realized\nin exchange coupled RE-TM multilayers, as well as het-\nerostructures comprised by two TM layers antiferromag-\nnetically coupled through 0.4nm nonmagnetic metallic\ninterlayers.11\nMost importantly, experimental studies have demon-\nstrated that the all-optical reversal of magnetizationis not precessional and relies solely on subpicosec-\nond quenching of the magnetizations of RE and TM\nsublattices.10Furthermore, as it was revealed by the\ntime-resolved X-ray experiments and supported by the\natomistic simulations,12the laser-induced quenching of\nthe TM and RE sublattice magnetizations occurs on dis-\ntinct time-scales. As a result, the magnetization rever-\nsal proceeds via non-equilibrium transient ferromagnetic\nstate.13Suchnon-equilibriumdynamicsallowsforthede-\nterministic magnetization reversal, without any need for\nother stimuli defining the magnetization direction. We\nnote, that circularly polarized laser pulse polarization\nwas mostly used for triggering the all-optical magnetiza-\ntion reversal.9–11,14–18However, it has recently been pro-\nposed that the difference in the magnetization reversal\nprocesses triggered by left- and right-handed laser pulses\ncan be explained to a large extent by a magnetic circular\ndichroism possessed by the studied samples.19\nNaturally, microscopical mechanism underlying such\nunconventional response of magnetization of a ferrimag-\nnetic metallic system to a femtosecond laser pulse is\nthe subject of intense discussion nowadays. In Refs.\n13,14,20–22 atomistic and multiscale calculations based\non the Landau-Lifshitz-Bloch equation23for the ensem-\nble of the exchange-coupled spins have been successfully\nemployed to account for the main features of the all-\noptical magnetization reversal. This approach allowed\ndescribing the all-optical reversal in both single phase al-\nloys and exchange-coupled multilayers.24In Ref.25 com-\nprehensive phenomenological model based on the On-\nsager’s relations suggested by Baryakhtar26was devel-\noped in order to account for the reversal via transient\nferromagnetic-like state. This theoretical study intro-\nduced the exchange-dominated regime of laser-induced\ndynamics in a ferrimagnet, which allows the reversal of\nmagnetization solely due to the ultrafast heating. This2\nwork highlighted the importance of the angular momen-\ntum exchange between the sublattices. Understanding\nmicroscopical processes responsible for this angular mo-\nmentumexchangebecame, therefore,thekeyissueinthe-\noretical studies of the laser-induced magnetization rever-\nsal. In Ref.22the two-magnonbound statewasproposed\nto mediate the angular momentum transfer. In Ref. 27\ndissipationless energy and angular momentum exchange\nbetween TM and RE sublattices mediated by 5 d-4fex-\nchange coupling in RE ions was analyzed as the driv-\ning mechanism for the all-optical magnetization rever-\nsal. The role of the exchange electron-electron scattering\nin the magnetization reversal was recently discussed in\nRef.28.\nHere we consider the problem of the angular momen-\ntum exchange between two nonequivalent magnetic sub-\nlattices in a metal in the frameworks of a general micro-\nscopic model based on the rate equations. This model\ndescribes evolution of the occupation numbers of two\ndifferent ferromagneticsublattices coupled antiferromag-\nnetically. They are formed either by nearly localized d-\nelectrons in a case of a transition metal sublattice or lo-\ncalizedf-electrons in a case of a rare-earth metal sublat-\ntice. The coupling between the sublattices is mediated\nby delocalized s- orp-electrons. In the frameworks of\nthis model we demonstrate that the spin exchange be-\ntween the localized ferromagnetic subsystems is medi-\nated by delocalized electrons and is triggered by ultra-\nfast increase of the temperature of the latter. This leads\nto the switching of the net magnetization without any\nadditional stimuli, such a external magnetic field or cir-\ncular polarization of light. Importantly, the model we\npropose is not restricted to the case of RE-TM alloys or\nheterostructures, but is also applicable for the case of the\nstructures composed by two different transition metals.\nFurthermore, since the laser pulse only plays a role\nof the stimulus supplying energy to the delocalized elec-\ntrons, we consider feasibility of the magnetization re-\nversal triggered by a short pulse of external electric\nbias in the ferrimagnetic system either imbedded into\nmetallic point contact or sandwiched between two tun-\nnel junctions. We show that, first, this alternative ap-\nproachfor driving the magnetic system into the strongly-\nnonequilibrium state enables one to tune the demagne-\ntization times by variation of the bias. This is impor-\ntant since the reversal depends on a delicate interplay\nbetween demagnetization time and cooling time of the\nmobile electrons. Second, in this case one deals with a\ncompact nanoscale device compatible with existing spin-\ntronics applications.\nThe paper is organized as follows. In Section II we\nintroduce the microscopical model describing the evolu-\ntion of the ferrimagnetic metallic system in response to\nthe rapid increase of the delocalized electrons tempera-\nture. In Section III we discuss the applicability of the\nproposed model to the process of the all-optical reversal\ndemonstrated experimentally. In Section IV we consider\nthe electric bias induced reversal either in point contactsor in tunnelling structures.\nII. THEORETICAL MODEL OF\nMAGNETIZATION REVERSAL IN A METALLIC\nFERRIMAGNET\nA. Model of a metallic ferrimagnet\nWe start our consideration by introducing three inter-\nacting electronic subsystems (Fig.1). We denote two fer-\nromagnetic sublattices as A and B. For a sake of clarity\nA is the transition metal d-electrons subsystem, while B\nis either d-electrons or the rare-earth metal f-electrons\nsubsystem. The A and B subsystems could comprise ei-\nther single phase alloy or exchange-coupled layers. The\nthird subsystem is the mobile s- orp-electrons(e). In the\nstructures where both A and B sublattices are based on\nthe TM elements these mobile electrons do not give an\nimportantcontributiontotheferromagnetismofeitherof\nA and B sublattices. By contrast, mobile electrons sup-\nport ferromagnetism of the rare-earth sublattice B via\nthe indirect exchange with the ferromagnetic TM sublat-\ntice A. In our model these are the mobile electrons that\nplay a decisive role in energy and angular momentum ex-\nchange within the sample. In particular, we assume that\ntheir coupling to d- andf- electrons controls the energy\ndistribution in the corresponding subsystems. For a sake\nof convenience in the following discussion we consider s-\nelectronsasthemobileelectrons,whilealltheconclusions\nare valid for the case of mobile p-electrons as well.\nFIG. 1: (Color online) The A, B and s-subsystems comprising\nferrimagnetic metal. ∆0\nA,Bare the exchange splittings in the\nequilibrium. NA,B\n↑↓,n↑↓are the occupation numbers of the\nsubsystems A, B and e. Subscripts ↑,↓denote up and down\nspin states with respect to the initial net magnetization di rec-\ntion.γexare the exchange constants between corresponding\nsubsystems\nForasakeofsimplicityweconsiderthespinsubsystems\nAandBcharacterized by pronounced peaks in energy\ndistribution. Both subsystem are assumed to be strong\nferromagnets and, thus, the exchange splittings ∆0\nAand\n∆0\nBforthesesubsystemsarelargerthanthe widths ofthe3\ncorresponding energy peaks, as shown in Fig.1. ∆0\nA,B\ndescribe exchange between neighboring ions comprising\nthe subsystems A and B, and are equal to the averageex-\nchangecouplingscorrespondingtothe Weissfield. Under\nassumptionofAandBbeingstrongferromagnets,theoc-\ncupation numbers NA\n↓,NB\n↑= 0, as illustrated in Fig.1.\nHere the subscripts ↑,↓correspond to the up and down\nspin directions with respect to the net magnetization di-\nrection. Here we consider the case, when the magnetiza-\ntion of the A subsystem is larger than the magnetization\nof the B subsystem.\nB. Rate equations for the ferrimagnetic metallic\nsystem\nThe excitation of the described above system is intro-\nducedin ourmodelasarapidincreaseofthe temperature\nTeof the mobile electrons. We consider instantaneous\nincrease of Teat the time t= 0 followed by the slow de-\ncrease, governed by the processes specific for the system\nin consideration.\nIn order to simulate response of the A and B subsys-tems to the rapid increase of the temperature of the mo-\nbile electrons we exploit the fact that at temperatures\nclose to the critical ones the suppression of ferromag-\nnetism of the TM sublattice A occurs mainly via the\nStoner excitations which are created by a transfer of\nthed-electron from a majority band to a minority band\n(Fig.2(b)). Such a transfer leads to a decrease of the\nsubsystem magnetization and is naturally related to an\nenergy and angular momentum cost which is supplied by\nthe mobile s-electrons. The decrease of magnetization\nof the A subsystem is compensated by the spin reversal\n↓→↑of thes-electron mediating the excitation. If the\ns-electrons are simultaneously coupled to both A and B\nsubsystems, they can effectively lead to spin exchange\nbetween A and B subsystems. Thus there is the indirect\ninteraction between total spins of A and B subsystems,\nwhich is spin conserving in a natural way.\nTo describe this interaction we write down the rate\nequationsfortheoccupationnumbersofsitescorrespond-\ning to subsystems A ( NA\n↓↑), B (NB\n↓↑) and occupation\nnumbers of s-electrons states ( n↑↓) participating in the\nexchange scattering:\ndNA\n↓↑\ndt=−/integraldisplaydε\nTe/bracketleftbigg1\nτAe/parenleftbig\nn↑↓(1−n′\n↓↑)NA\n↓↑(1−NA\n↑↓)−n↓↑(1−n′\n↑↓)NA\n↑↓(1−NA\n↓↑)/parenrightbig/bracketrightbigg\n; (1)\ndNB\n↓↑\ndt=−/integraldisplaydε\nTe/bracketleftbigg1\nτBe/parenleftbig\nn↑↓(1−n′\n↓↑)NB\n↓↑(1−NB\n↑↓)−n↓↑(1−n′\n↑↓)NB\n↑↓(1−NB\n↓↑)/parenrightbig/bracketrightbigg\n; (2)\ndn↑↓\ndt=−[n↑↓(1−n′\n↓↑)/parenleftbigg1\nτeANA\n↓↑(1−NA\n↑↓)+1\nτeBNB\n↓↑(1−NB\n↑↓)/parenrightbigg\n+n↓↑(1−n′\n↑↓)/parenleftbigg1\nτeANA\n↑↓(1−NA\n↓↑)+1\nτeBNB\n↑↓(1−NB\n↓↑)/parenrightbigg\n]+1\nτs(n↓↑−n↑↓). (3)\nThe r.h.s of the Eqs.1-3 are the standard collision inte-\ngralsoftheBoltzmannequationsdescribingthe exchange\nscattering between three components of the electronic\nsystem. Each equation corresponds to the pair of the in-\nteracting subsystems. nandn′are the functions of ener-\ngiesεandε′, respectively. The values of εandε′are con-\nnected by the energy conservation relations and include,\nin particular, the exchange splittings ∆ A,Bwithin the\nsubsystems A and B. Here we take into account that, in\ncourse of demagnetization the exchange splittings ∆ A,B\ndiffer from the equilibrium values ∆0\nA,B.\nIn the Eqs.1,2 integration is performed only over the\nenergy of the delocalized electrons energies. The inte-\ngration over the energies of the states within the A- and\nB-subsystems distributions is omitted under the assump-\ntion that A and B subsystems are the strong ferromag-\nnets. By contrast, the subsystem of mobile s-electrons\nhas a broad energy distribution with Fermi energy much\nlarger than Te. Nevertheless, Eq.3 is written only forthoses-electrons which are effectively coupled to A and\nB subsystems and their energy is within the band of a\nwidth∼TearoundtheFermilevel. The latterfact means\nthat the phase volume of the s-electrons involved in the\nexchange scattering is smaller than the phase volume of\neither of the ferromagnetic subsystems A and B. Fur-\nthermore, it allows to assume the s-electrons densities of\nstates within corresponding energy interval to be nearly\nconstant. In addition, in what follows we do not not\ntake into account the energy dependencies of nas well as\nenergy dependencies of relaxation times τ.\nτA,B;eare the effective electron-electron relaxation\ntimes, characterizing A−sandB−sexchange scat-\ntering processes. The factors 1 /Teτ(A,B)ein Eqs.1,2 are\nthe probabilities of the exchange scattering involving s-\nelectronsnormalizedwithrespecttoenergy ε. Thevalues\n1/τ(A,B)eareofthe orderoftotalscatteringprobabilities,\nsince the integration over εis only within the energy in-\nterval∼Te.4\nThe values 1 /τe(A,B)describe the probabilities of ex-\nchange scattering of s−electrons by AandBsubsys-\ntems. The effective exchange scattering probability of\ns-electrons including both relaxation channels is given\nby 1/τee= 1/τeA+1/τeB.\nCharacteristic time τsdescribes the angular momen-\ntum exchange between the s-electrons and the external\nbath.\nEqs.1-3 take into account spin balance within the sys-\ntem only and thus do not include processesleading to the\nthermal equilibrium. We assume that the characteristic\ntimes for electron-electron processes, responsible for the\nthermalization within considered subsystem are smaller\nthan spin relaxation times. The evolution of the temper-\natureTefollowingthe instantaneousincrease, is governed\nbyelectron-phononprocessesandheatwithdrawal,which\nis specific for different systems. These processes are con-\nsidered to be slower than the introduced above charac-\nteristic times τresponsible for the angular momentum\ntransfer.\nFinally, we stress that in this model the effect of direct\nA-A and A-B or indirect B-B exchange couplings is not\nincluded. The processes involving these interactions are\nexpected to be related to spin reversals leading to ferro-\nmagnetic or antiferromagnetic ordering in corresponding\nsubsystems. We believe that at highly-nonequilibrium\nstate the spin-conserving processes considered above are\nmore efficient and fast than the ones including spin dis-\nsipation and, thus, are the dominant mechanism of the\nspin-redistribution. In order to set the criterion for a\nrange of the mobile electrons temperatures Tewhere the\nexchange scattering governs the evolution of a particular\nferromagnetic system subsystem, we take into account\nthat this process is effective only for Te>∆0\nA,B. Con-\nsequently, we introduce effective partial critical tempera-\ntures ofthe A and B subsystems, which arerelated to the\ncorrespondingexchangesplittings TA;B\nC∼∆0\nA,B. Strictly\nspeaking the concept of critical temperature holds only\nforthermodynamicallimit. Intheequilibriumthecritical\ntemperatures of A and B sublattices coupled via mobile\nelectrons should be considered equal, in agreement with\nthe experimental data.29In the strongly non-equilibrium\nstate of the medium, if the rate of electron-electron in-\nelastic scattering within given subsystem is higher than\nthe rate of the corresponding sublattice magnetization\nevolution, one can introduce the partial electron temper-\nature. Electron-electron inelastic scattering, responsible\nfor the electron thermalization is, typically, in the range\nof 50-300fs.30–33In the non-equilibrium state the A and\nB subsystems can be also considered as partly decoupled\nfrom each other and, therefore, we can discriminate be-\ntween partial values of the critical temperatures TA;B\nCof\nthese subsystems. We will consider the Curie tempera-\ntures for uncoupled A (pure TM metal) and B (pure TM\nor RE metals) systems as the partial critical tempera-\nturesTA\nCandTB\nC, respectively.C. Exchange scattering probabilities and\nrelaxation times\nFrom the Eqs.1-3 it follows that the efficiency of spin\ndecay within a given subsystem is related to the purely\nspin-conserving s−dors−fexchange scattering and\nis controlled by relaxation rates 1 /τ(A,B)e. Correspond-\ningly, the decay rate is higher for a subsystem where\nthis parameter is larger, i.e. the exchange coupling with\nmobiles-electrons is stronger. One expects that the\nexchange scattering between s-electrons and the corre-\nsponding ferromagnetic subsystem is more pronounced if\nthe latter possesses strong exchange interaction within\nitself. Although the evolution of magnetization in any of\nthe subsystems includes not only spin transferbetween A\nor B subsystem and s-subsystem, controlled by 1 /τ(A,B)e\nbutalsothespindecaywithin s-subsystem(1 /τs), weex-\npect that it is the difference between the values of τ(A,B)e\nthat leads to the distinct times of spin decay within A\nand B subsystems.\nNumerical estimation of the relaxation rates 1 /τ(A,B)e\nand 1/τe(A,B)requires knowledge of the electron spec-\ntrum of all involved systems. Here we use simplified es-\ntimates. As it is known, for electron-electron scattering\nin standard metals relation 1 /τ∼T2\ne/εF¯hholds, where\nεFis the Fermi energy (see, e.g., Ref.34). This expres-\nsionmakesuseofthe momentum conservationlawforthe\nelectron system where Te<< εF. In the case considered\nherethe situation is differentsince the electronscattering\ntakes place between two different electronic subsystems,\nwith one of them (A or B) characterized by very narrow\nenergy band, and, thus, the effective mass of electrons in\nwhich is much larger than in the subsystem of the mo-\nbile electrons. In this case the momentum conservation\nlaw can be disregarded and thus the electron-electron\nscattering time has a form close to the one for electron\nscattering by impurities. Therefore, one can estimate the\nrelaxationtime as1 /τ∼σneffvrwhereσisthescattering\ncross-section, neffistheeffective concentrationofscatter-\ners, and vris the relative velocity of scattered electron\nwith respect to the scatterer (see, e.g., Ref.34). Tak-\ning these considerations into account, one obtains for the\ncharacteristic time of the exchange scattering of the A\nor B subsystem electrons by the s−electrons with the\nspherical Fermi surface:\n1\nτ(A,B)s∼γ(A,B)e\nex¯h2nTe\nε3/2\nFm3/2, (4)\nwherenis the concentration of the s-electrons, εFis the\nFermi energy of the s-electrons, mis their mass. γ(A,B)e\nex\nis the dimensionless A−sandB−sexchange constant,\nwhich absorbs the dependence of the exchange scattering\nprobability on the exchange splittings ∆0\nA,B. Here the\nestimates σ(A,B)e∼¯h2/mεF,vr∼ε1/2\nF/m1/2,neff∼\nnTe/εFare used.\nInits turn, forthe probabilityofexchangescatteringof\ns−electrons by the ones of A or B subsystems we obtain5\nfor the case Te>∆A,B:\n1\nτs(A,B)∼γ(A,B)e\nex¯h2NA,B\nε1/2\nFm3/2, (5)\nwhereNA,Bis the concentration of the subsystem A or\nB. Thus, according to these expressions, the values of\nτ(A,B)sandτs(A,B)are different for the same value of the\nexchange scattering crossections and exchange constants\nbetween the A(B) subsystem and the mobile electrons.\nAs Eqs.4,5 demonstrate, the relaxation probabilities\n1/τ(A,B)eand 1/τee= 1/τeA+ 1/τeBpossess different\ntemperature dependencies. For the exchange scattering\ntimeτ(A,B)ewe have1 /τ(A,B),e∝Te. The exchangescat-\ntering time τeeis, in turn, temperature -independent.\nThis is in contrast to standard electron-electron scatter-\ning where 1 /τ∼¯hT2\ne/εF. The later relation holds for\nelectron-electron scattering between s-electrons respon-\nsible for thermalization at the initial stage.\nD. Evolution of the ferrimagnetic system with\nnearly quenched sublattice magnetizations\nUsing Eqs.1-3 we consider the evolution of the mag-\nnetizations of the A and B sublattices triggered by an\ninstantaneous increase of the temperature of the mobile\nelectrons in the metallic ferrimagnet. If one would deal\nwith asingle ferromagneticsubsystem, e.g. A, coupled to\nthe mobile electrons, the rapid increase Te> TC\nAwould\ntrigger the decrease of the sublattice magnetization, i.e.\ndecrease of NA\n↑and increase NA\n↓due to spin transfer\nto the mobile electrons via exchange scattering and the\nfollowing spin decay within s-subsystems. This would\nfinally lead to total suppression of magnetization.\nIt is important to stress that the values of ∆ A,Bde-\ncrease in the course of the demagnetization process. In\nparticular, when the averagemagnetization of the A sub-\nlattice tends to zero, the same holds to the average ex-\nchange fields. As a result, in the mean field approxima-\ntion ∆ Atends to zero as well. However, locally, given\nmagnetic ions from the A sublattice is exchange coupledto the nearest neighboring ions. The distribution of local\nmagnetic moments does not possess any long range or-\nder, and the sum magnetization of the neighboring ions\nfluctuates depending on the spatial position. Since in the\nequilibrium ∆0\nA∝ NA, whereNAis the number of neigh-\nbours, the mean exchange splitting ∆ A∼∆0\nA/(N)1/2\nA\nwhen the average magnetization of the A sublattice is\nzero.\nFor the antiferromagnetically coupled A and B subsys-\ntems the evolution of their magnetizations is somewhat\nmoredelicate than in the caseofthe single sublattice sys-\ntem. The exchange scattering with the mobile electrons\nleads to the redistribution of the total spin between the\nsubsystems according to the factor τA,e/τB,e. The sup-\npression of total magnetization would occur only via the\nspin non-conserving process, which is described by the\nterm (n↓↑−n↑↓)/τsin Eq.3. Without this term the total\nsuppression of magnetization in the system is possible\nonly if the magnetizations of A and B subsystems are\nequal initially.\nIn order to illustrate the evolution of the magneti-\nzations of the A and B subsystems which follows the\ninstantaneous increase of the s−electrons temperature\nTe> TA,B\nCwe consider the situation when the creation\nof Stoner excitations completely suppresses magnetiza-\ntion of the one of the subsystems. We consider the case,\nwhenτA,e< τB,e, which is consistent with the nota-\ntions we accommodated, i.e. A and B subsystems are\nformed by d−andf−electrons, respectively. Then mag-\nnetization of the A sublattice is quenxhed, NA\n↑=NA\n↓,\nwhile the magnetization of the B subsystem remains fi-\nnite,NB\n↓> NB\n↑(Fig.2(c)). In this case the rate equation\nfor the A subsystem takes a form:\ndNA\n↓↑\ndt|0=1\n4τAe(n↓↑−n↑↓). (6)\nHere the notation |0means the configuration where\nNA\n↓=NA\n↑. The further calculations give that at NA\n↑≃\nNA\n↓≃1/2,|n↑−n↓|<< n ↑the rate equation of the\ns-electrons takes a form:\ndn↑↓\ndt=1\nτs(n↓↑−n↑↓)+1\n4τee(n↓↑−n↑↓)+1\n4τBe(NB\n↑↓−NB\n↓↑)+1\n4τAe(NA\n↑↓−NA\n↓↑), (7)\nHere we assumed n↑∼n↓∼1/2.\nThen we recall that the phase volume of the s-\nsubsystem is smaller than the phase volumes related to\nthe subsystems A and B. Therefore, the characteristic\ntime of evolution of the magnetic system as a whole is\nmuch larger than the characteristic relaxation times τeA,\nτeB,τs, relevant for the mobile electrons. Consequently,we neglect time derivative in the l.h.s. of Eq.7. This\nleads to the expression\n(n↓↑−n↑↓)≃ −(τBe)−1\n(4/τs)+(1/τee)(NB\n↑↓−NB\n↓↑).(8)\nSince at the time moment when NA\n↓=NA\n↑the magne-\ntization of the subsystem B is non-zero and the r.h.s. of6\nEq.8 is positive, Eqs.8,6 show that the magnetization of\nthe subsystem A changes its sign. Therefore, after this\nmoment the total configuration of the A-B system starts\nto beferromagnetic ((Fig.2(d))). Note that this happens\nin course of decay of total magnetization of the system,\nprovidedthat spin non-conservingprocessesdescribed by\nτsaretakenintoaccount. Wedenotethemomentoftime,\ncorresponding to reversal of the magnetization of the A\nsubsystem, as the ”reversal point” tr. It is important to\nnote that at the times t > trthe former minority elec-\ntrons of the A subsystem become to be majority and vice\nversa. Correspondingly, the reference for the Stoner ex-\ncitations is changed - now they are referred with respect\nto the ”new” orientation of magnetization. Therefore,\nthe exchange with B subsystem via s-electrons leads to a\ndecrease of the excitations number in the A subsystem.\nIf the electron temperature Tewould be kept constant\naftertr, the magnetizations in both subsystems would\nvanish, provided finite τs. SinceTegradually decreases\naftert=trat some moment it reaches the critical tem-\nperature TA\nC, while is still above TB\nC(Fig.2(e)). At this\nmoment, asshownabove,themagnetizationoftheAsub-\nsystem is non-zero and is aligned along the initial mag-\nnetization direction of the subsystem B. Then, the pres-\nence of a gap between new majority and minority bands\nin the A subsystem is restored, and the electron-electron\ns−dexchange scattering can not support anymore some\npairs of the Stoner excitations with the ”new” reference.\nThis leads to the increase of the magnetization of the A\nsubsystem aligned to the direction of initial magnetiza-\ntion of B subsystem. We note that, simultaneously, the\ninter-A exchange interactions are also restored. To the\ncontrary, magnetization of the B subsystem, character-\nized by the smaller critical temperature TB\nCcontinues to\ndecrease due to the Stoner excitations supported by the\ns-electrons subsystem, according to Eqs.1-3.\nWhen the electron temperature Tedecreases down to\nthe value TB\nCthe subsystem A already acquired the mag-\nnetization large enough to force the subsystem B to re-\nconstruct its magnetization state according to the new\nmagnetization state of the subsystem A via indirect an-\ntiferromagnetic exchange (Fig.2(f)).\nThe critical value of Tecorresponding to irreversible\nswitching can be estimated from ∆0\nAorTC\nA. At this crit-\nical value of Te=TC\nAthe self-consistent character of ex-\nchangeis restoredand the standardWeiss field is formed.\nSuch an estimate is mostly a semi-qualitative one since\nthe process of transition from strongly non-equilibrium\nregimetoanequilibriumhasacomplexcharacter.Itcould\nbe calculated with a help of numerical methods provided\none has a detailed information concerning temperature\nbehavior of τs, the heat withdrawal processes etc.\nWe would like to emphasize that at the strongly non-\nequilibrium state considered above the main processes\ndefining the spin kinetics within the system are related\ntospin-conserving exchange scattering. With lowering\nthe temperature below critical temperatures TA,B\nCof the\nsubsystems A and B this scattering becomes suppressed,\ns\nA B\ns\nA BSE SE\ns\nA Bs\nA B\nsinitial state\nfinal states\nA Bferromagnetic\nstateT=T0\nT>TCA;B\nT>TCA;BT>TCA;B\nTC<T<TCA B\nT=T0\nA B\nreversal point tr(a)\n(b)\n(c)(d)\n(e)\n(f)\nFIG.2: (Color online)EvolutionoftheA,Band s-subsystems\nfollowing the rapid increase of Te. Snapshots of the subsys-\ntems at six distinct time moments are shown: (a) the initial\nstate; (b) Stoner excitations (SE) decrease the magnetizat ion\nof the A and B subsystems; (c) reversal point tr, where the\nmagnetization of the A subsystem vanishes; (d) the magneti-\nzationofthesubsystemAispartlyrestoredinanewdirectio n,\nwhile the magnetization of the B subsystem is remains finite\n- transient ferromagnetic-like state; (e) the magnetizati on of\nthe A subsystem is restored sufficiently for reversing the mag -\nnetization of the B subsystem via the direct A-B exchange\ninteraction; (f) the final state. Corresponding temperatur es\nof the mobile electrons subsystem with respect to the partia l\ntemperatures TA;B\nCare shown for each snapshot. See text for\ndetails.\nwhile the standard exchange interactions implying spin\nnon-conserving processes start to play the decisive role.\nHowever, close to the reversal point trthe direct A-B\nexchange is still suppressed since it is proportional to\nthe product of total spins of the A and B subsystems.\nThis explain the fact that, while the demagnetization of\nsubsystems A and B takes relatively short time below\n1ps, the total equilibrium is established at times of tens\nof picoseconds.10\nFinally, wewouldliketonotethat, despite theassump-7\ntions and simplifications made above, we expect that\nthe main conclusions drawn on the base of our simpli-\nfied model hold also for more realistic ferromagnetic sys-\ntems with smaller exchange splitting and for more com-\nplex nature of the demagnetization process. Indeed, in\nour model the Stoner excitations are considered to be\nthe mechanism of the quenching of magnetization of the\nsublattices A and B. The physical picture of the demag-\nnetization process is more complicated and may include\nspin fluctuations,35arising e.g. from magnons. Magnons\nwere recently suggested to mediate the angular momen-\ntum transfer in the course of all-optical magnetization\nreversal.22Qualitatively, the scheme we applied could be\nused to include these types of excitations in addition to\nthe Stoner’s ones as the mechanism of quenching of mag-\nnetization of the sublattices. These excitations could be\nmagnons of different types including both AandBsub-\nsystems. However, under assumption of narrow distribu-\ntion of energies within AandBsubsystems, the relevant\nmagnons should have specific frequencies satisfying the\nenergy conservation. Since the subsystems AandBare\ndifferent, the magnons coupled to these subsystems have\ndifferent frequencies, and thus can not simultaneously be\ncoupled to AandBsubsystems and can not transferspin\nbetween the two subsystems while this transfer is an im-\nportant ingredient of the reversal process. Therefore, in\norder to include in the suggested model the magnons me-\ndiating the angularmomentum exchange, one cannot use\nanymore the simplification of narrow distribution of en-\nergies within AandBsubsystems.\nAnother simplifications included to our model is that\nboth subsystems A and B are considered to have spin\n1/2, since only two projections of spins are considered\nin Eqs.1-3. Thus in this case we neglect a possible role\nof intermediate spin projections in the process of demag-\nnetization of the A and B sublattices. Strictly speak-\ning, it does not hold for rare-earth ions. Nevertheless\nwe believe that our picture gives at least a qualitative\nexplanation of the effect in a general case. Indeed, in\nany case the switching between the two magnetization\norientations imply transition between the two extreme\nspin projections and, correspondingly, the gap between\nthe two projections can be considered as ∆0\nA,B.\nIII. MAGNETIZATION REVERSAL INDUCED\nBY A FEMTOSECOND LASER PULSE\nThe ultrafast all-optical switching based on the effect\nof exchange electron-electron scattering was recently dis-\ncussed in Refs.28,36. According to Fig.2 of Ref.28, the\npossibility of magnetization switching in the model is\nmainly related to the following. (i) The total numbers\nof localized and itinerant spins were equal. (ii) The spin\nof localized electrons was twice as large as the spin of the\nitinerant carrier. In this case the exchange scattering at\na high level of excitation can force the average spin of\nthe itinerant electrons to the opposite direction dictatedby the local spins to the extend sufficient for the d−d\nexchange to ”fix” this ”new” direction in the course of\nthe electron cooling. The weaker exchange between lo-\ncalized and itinerant carriers in this case can only affect\nthe direction of local spins. However, to our opinion, this\nscenario can not be applied to the case of multilayered\nsystems,11since the penetration length of the d-electrons\nthrough such structures including interlayers is question-\nable.\nIn Ref. 36 the sp-dmodel of ultrafast\ndemagnetization37was applied in order to describe\nthe all-optical magnetization reversal scenario in a\nferromagnet. In this case, however, the optical pumping\nresultingin itinerant spin polarizationhad to be included\nin the model in order to facilitate the magnetization\nreversal.\nWe also note that dissipationless energy and angular\nmomentum exchange between TM and RE sublattices\nas the driving mechanism for the optical magnetization\nreversal has been explored in Ref.27. There the 5 d-4f\nexchange coupling in RE ions was proposed to be deci-\nsive for the reversal, which naturally limits the applica-\nbility of the model to the case of magnetization reversal\nin RE-TM alloys and heterostructures, but not to the en-\ngineered ferrimagnets based on transition metals solely.\nNow we compare the magnetization reversal scenario\ngiven by our model with the experimental findings re-\nported in the papers on ultrafast all-optical magnetiza-\ntion reversal. When the metallic film is subjected to a\nintense femtosecond laser pulse the temperature of the\nmobile electrons in the film is increased on a time-scale\nof the laser pulse.33Therefore, when describing the in-\nteraction of laser pulses with metal, one typically in-\ntroduces a rapid increase of the temperature Teof the\n”bath” formed by the mobile electrons.13,24,33Estimates\nbased on the heat capacity of mobile electrons in a tran-\nsition metal give that the 100fs laser pulse of ∼1mJ/cm2\nfluence leads to the increase of Teup to∼1200K.14.\nThis is followed by the decrease of Tegoverned by the\nelectron-lattice relaxation times. Calculations in frames\nofthe two-temperaturemodel38performede.g. in Ref.14\nyielded that the temperature of the electronic bath in a\nconducting metal equilibrates with the lattice one on a\ntime scale of a few picoseconds. On this time scale Te\ndrops to the values comparable to the equilibrium Curie\ntemperature of a medium, with exact value being deter-\nmined by the amount of energy transferred to the system\nby the laser pulse.39Therefore, the evolution of the elec-\ntronic temperature during and after the excitation by\nthe femtosecond laser pulse corresponds well to the con-\nditions implied in our model.\nThe magnetization reversal scenario yielded by the\nproposed model agrees well in the main details to the\nresults of various experiments aimed at revealing the the\nunderlying mechanism of the all-optical magnetization\nreversal in ferrimagnetic RE-TM alloys.\nIn particular, studies of the laser-induced dynamics\nof magnetizations of Fe and Gd sublattices in GdFeCo8\nalloy have shown that they demagnetize on different\ntime scales on 140 and 400fs respectively.12This results\nin occurrence of the transient ferromagnetic-like state,\nwhich is described by Eqs.6-8 in the model considered in\nSectionII. We note, that the numerical calculations re-\nported in Ref.12 suggested that there is a proportion-\nality between the demagnetization times observed for\nthe TM and RE sublattices and corresponding ratios be-\ntween their magnetic moments and Gilbert damping con-\nstantsµTM;RE/λ. In our model distinct demagnetization\ntimes are accountedby distinct exchangerelaxationrates\n1/τAe>1/τBe, which are correlated to the exchange\nsplittings ∆0\nA,Bpossessed by the A and B subsystems.\nIn the presented model the magnetization reversal is\ndriven by the exchange of angular momentum between\nthe A and A sublattices mediated by the mobile elec-\ntrons, while the transfer of the angular momentum to\nother reservoirs (lattice) is only responsible for overall\ndecay of magnetization of the whole system. Earlier, it\nhas been suggested, based on the studies of the ultrafast\nlaser-induced demagnetization in GdFeCo alloys, that\nthe angular momentum transfer from TM to RE sublat-\ntice plays an important role in the process.40Spatially-\nand element-resolved studies of the reversal dynamics in\nGeFeCo have shown that there is the angularmomentum\ntransfer between Gd-rich and Fe-rich nanoscale areas in\nthe GdFeCo sample which accompanies the reversal.41\nRecently, has been reported that there is a transfer of\nthe angular momentum between RE and TM sublattices\nof the metallic ferrimagnetic alloys CoGd and CoTb in-\nduced by the action of the laser pulse and monitored by\nthe spin- and orbital-resolved X-ray technique.42The re-\nsults of the element-specific studies of the laser-induced\ndemagnetizationand reversalinTbCoalloys43supported\nfurther the involvement of the exchange between the RE\nand TM sublattices in these processes.\nWhen introducing our model we did not specify\nwhether the A-e-B ferrimagnetic system should be sin-\ngle phase one or comprised by coupled A and B layers.\nThus, we argue that this model accounts well for the re-\nsults reported in Ref.11, where the all-optical reversal\nwas observed in four distinct types of single-phase and\nmultilayered synthetic metallic ferrimagnets.\nTherefore, the model consideredherecaptures the gen-\neralpictureofthelaser-inducedmagnetizationreversalin\na metallic ferrimagnet. However, due to a number ofsim-\nplifications applied and since our model does not include\nthe realistic band structure of a ferrimagnetic metal, it\ncannot account for a number of experimental evidences,\nwhich we discuss below.\nOngoing studies of the magnetization reversal reveal\nvery diverse and even contradictory features of the pro-\ncess in the RE-TM metallic alloys of various composi-\ntions. The important issue of the role of the magne-\ntization compensation point possessed by ferrimagnets\nhas been studied experimentally in both alloys9,13,14,17,44\nandengineeredmultilayerstructures.11Numberofexper-\niments have demonstrated,13,14,44that the reversal canbe realized for ferrimagnets which equilibrium tempera-\nture either below compensation point or above it, which\nagrees well with the proposed model. On the other hand,\nexperiments reported in Ref.17 suggest that for the re-\nversal it is essential that the compensation point is above\nthe equilibrium sample temperature. The recent study\nof the reversal in the series of specially engineered fer-\nrimagnetic structures showed that this condition holds\nfor the majority but not for all structures.11Despite of\nthese controversies, all the studies confirm that the re-\nversal does not occur in TM-RE alloys, which are either\nTM-richorRE-rich. Castingthelightonthis problem, in\nRef.45 the importance of the low remanence, possessed\nby ferrimagnets in a vicinity of the compensation point,\nhas been revealed. This is in agreement with the earlier\nreporteddata,14showingthat the closerthe sample to its\ncompensation point, the less laser fluence is required for\nthe reversal. Our model does not treat such details of the\nferrimagnetic metal as the equilibrium ratio between the\nsublatticemagnetizationsandthereforeitcannotaccount\nfor the role of the magnetization and angular momentum\ncompensation points.\nAnother issue which is still to be comprehended is the\nlaser pulse duration required for the reversal. In Ref.14\nthe reversal in GdFeCo alloys of certain compositions\ncould not be realized by the pulses longer than 1.7ps,\nwhile in Ref.17 the reversal by the laser pulses as long as\n10ps was reported. Based on the present knowledge, the\nreversal scenario treated in the frameworks of our model\nrequires femtosecond laser pulses which could bring the\nRE-TM alloys, studied in the reported experiments,14,17\ntothehighlynon-equilibriumstateonthetimescalecom-\nparable to the relaxation times τAE< τBE<1ps. We\nnote, however,thatthemaximalpulselengthrequiredfor\nthe reversalis dependent on the balance between the rate\nandthedegreeoftheelectronictemperatureincrease, the\nexchangerelaxationtimes τAe,τBeandtherateoftheen-\nergy and angular momentum withdrawal τs. Therefore,\nthe knowledge of the details of the spin-conserving and\nspin-nonconserving relaxation processes in a particular\nferrimagnetic samples for the particular pulse durations\nisessentialforunderstandingthe restrictionsonthe max-\nimal pulse duration. We are not aware about the time-\nresolved studies of the laser-induced reversal by pulses\nlonger than 100fs.\nFinally, we note that recently the switching effects for\npurelyferromagnetic structures were reported.46In this\ncaseonly the laser-pulsehelicity dependent switching has\nbeen reported, reopening the discussion about the role\nof the light polarization open. We believe that further\nexperimental studies, clarifying this issues are required\nbefore any conclusions regarding the mechanism of the\nreversal in the ferromagnets can be evaluated.9\nIV. MAGNETIZATION REVERSAL INDUCED\nBY AN ELECTRIC BIAS PULSE\nAccording to the model described in Sec.II the rapid\nheating of the mobile electrons is sufficient for triggering\nthe magnetization reversal. Therefore, we suggest that\nan electric bias pulse used as an alternative to femtosec-\nond optical pulses and can drive the ferrimagnetic metal-\nlic system into strongly-nonequilibrium state, where the\nmagnetization reversal can be realized. We consider a\npossibility of switching within the A-e-B system formed\neither the two ferromagnetic layers AandBor by the\nA-B metallic alloy imbedded within the metallic point\ncontact (see Fig.3(a)).\nNM\nNMIV\nI(a)\n(b)M\nFIG. 3: (Color online) (a) Structure formed by the two ferro-\nmagnetic layers, A and B, separated by normal metal inter-\nlayer NM, which is imbedded into point contact on the base of\nnormal metal NM. The interlayer NM is chosen in a way that\nit supports antiferromagnetic coupling between the layers A\nand B. (b) Structure formed by two ferromagnetic islands A\nand B, separated by normal metal interlayer NM, imbedded\nbetween two normal metal electrodes NM and separated from\nthem by two tunneling interlayers I. The normal metal inter-\nlayer NM supports antiferromagnetic coupling between A and\nB.\nAgain, for a simplicity we consider the model of strong\nferromagnetswhere the minority spins do not exist in the\nequilibrium (Fig.1). Furthermore, we neglect the energy\ndistribution ofboth spin subsystems thus reducingA and\nB subsystems to the two spin sublevels separated by the\nexchange energy ∆0\nAand ∆0\nB, respectively. The energy\ndistribution of the s-electrons is controlled by a voltage\napplied to the point contact. Namely, if the width of\nthe contact Lis smaller than the diffusive length with\nrespect to energy relaxation, the distribution of the s-\nelectrons is formed as a mixture of electrons coming from\nthe left bank of the contact and those coming from the\nright bank, and is controlled by corresponding chemical\npotentials. In the center of the contact the distributionFhas a double-step form:47\nF=1\n2(F0(εF−eV/2)+F0(εF+eV/2)),(9)\nwhereVis the applied voltage, εFif the Fermi energy,\nandF0(ε) = [1+exp( ε−εF)]−1. At some distance from\nthe center to the left or to the right the weight of the cor-\nresponding”left” or”right”contributionincreasesandat\nlargedistances the equilibrium distributions of”left” and\n”right” types are restored. Note that the distribution\n(9) holds near the center of the contact even for diffu-\nsive transport provided that inelastic mean free path is\nsmaller than the size of the contact L.\nAs it is seen, the energy eV, defined by the applied\nvoltageV, can play a role of effective temperature of the\ns-electron system, and, therefore, trigger the magnetiza-\ntion dynamics described in Sec.II. In contrast to the case\nofall-opticalreversal,in this casefollowingthe excitation\nthenon-equilibriumspin occupationsof s-electronsdecay\ndue to ballistic or diffusive transport from the contact to\nthe banks. Thus in this case the spin relaxation time τs\nin Eq.8 is the escape time, which is defined as\nτs=τb\nesc∼L\nv;τs=τd\nesc∼L2\nD, (10)\nfor the case of ballistic and diffusive transport, respec-\ntively. Here Lis the characteristic size of the contact,\nv∼108cm/s is the electron velocity, and Dis the diffu-\nsion constant.\nIn this point-contact device one can control both the\nexcitation intensity (by the value of the bias V) and\nthe parameters of the excitation pulse (including the\nswitching-on/off times). The switching-on time - if small\nenough - is of no great importance. In contrast, the\nswitching-off time should be comparable with the time\nscale of the magnetization reversal. The latter, as we\ndiscussed in Sec.II is controlled by electron-electron ex-\nchangerelaxationtimes τ(A,B)e, which areexpected to be\nof the order of 10−12s. Importantly, as we discuss below,\nthe time requiredforthe magnetizationreversal trcanbe\nincreased both by the choice of the bias and by a proper\nposition of the layers with respect to the contact center,\nsince an increase of the corresponding distance decreases\nthe phase volume of the electron-electron scattering and,\nthus suppressing the magnetization reversal.\nLet us consider an effect of the s-electron distribution\ngiven by Eq.9 on the ferromagnetic layer A imbedded\ninto the contact near its center. We consider the case of\nzero equilibrium temperature, T0= 0. IfeV >∆0\nA, than\nthe occupation NA\n↓of the minority level ofthe subsystem\nA is described by an equation48\ndNA\n↓\ndt+τ−1\nAe\n2∆A(∆A+eV)NA\n↓=τ−1\nAe\n4∆A(eV−∆A).(11)\nNow we take into account that the occupation of the\nminority level leads to a decrease of the exchange field,10\ni.e. ∆ A= (α/2)(NA\n↑−NA\n↓), where αis the propor-\ntionality factor. As we discussed above, although the\naverage exchange splitting vanished at NA\n↑=NA\n↓, the\nexchange splitting of the given ion is ∆ A∝ NAdue to\nlocal fluctuations. Then NA\n↑−NA\n↓= 1−2NA\n↓and thus\n∆A= ∆0\nA−αNA\n↓, whereα= 2∆0\nA.\nIfwedenote eV= 2∆0\nA+δ,thenEq.11canberewritten\nin a form:\ndNA\n↓\ndt=−τ−1\nAe\n4∆0\nA(1−2NA\n↓)\n×/parenleftbig\n4∆0\nA(NA\n↓−1/2)2+2δ(1/2−NA\n↓)/parenrightbig\n.(12)\nForδ <−∆0\nA, i.eeV <∆0\nA, there is no non-zero so-\nlution of the Eq.12. The occupation of the minority\nlevel starts naturally at the threshold value eV= ∆0\nA, or\nδ=−∆0\nA. The gradual increase of NA\n↓with an increase\nofeVterminates at the value NA\n↓= 1/2 which is reached\natδ= 0. Thus one concludes that the transition to the\nferromagnetic-like state with parallel A and B magneti-\nzations takes place at eV= 2∆0\nA. ForeV >2∆0\nAthe\nsteady state of the layer A corresponds to normal metal.\nClose to the critical value of the bias Vc= 2∆0\nA/e,\nthe evolution of NA\n↑↓(NB\n↑↓) and of the exchange field ∆ A\n(∆B) with time is controlled by the difference V−Vc,\nsince the non-vanishing part of r.h.s. of Eq.12 scales with\nthis difference. In the vicinity of NA\n↓= 1/2the difference\n1/2−NA\n↓can be considered as a variable. In the r.h.s. of\nEq.12theleadingtermislinearin1 /2−NA\n↓, andthecoef-\nficient at this term gives the rate of the evolution. Such a\nslow evolution is expected, roughly speaking, in the case\nwhen (1/2−NA\n↓)<|eV−eVc|/∆0\nA. Correspondingly, the\nevolution of ferromagnetic order parameter near critical\npoint is defined by a characteristic time\ntr∼τAe∆0\nA\n|eV−eVc|. (13)\nThus the time required for suppression the ferromagnetic\nstate of the sublattice A can be tuned by a proper choice\nof the bias.\nNow we would like to note that a specific feature of the\npoint contact is a possibility to apply voltage pulses with\na sharp form. As a result, one can operate the device in\na threshold way which was demonstrated above. Thus\nwe believe that our predictions including Eq.13 hold also\nfor more realistic models including finite energy width of\nthe ferromagnetic subsystem.\nAnother design of the bias-controlled switching device\ncanbe basedonthe tunnel junctions(Fig.3(b)). Namely,\nwe assume that the ferrimagnetic structure A−s−Bdis-\ncussed above is fabricated on the base of thin film tech-\nnology, and the corresponding thin film device is sand-\nwiched between two tunnel junctions. Note that for the\ncase of the device formed by two ferromagnetic region\nseparated by normal metal all three components are con-\nsidered to be fabricated within the same plane, as shown\nin Fig.3(b).In this case the external bias is applied to external\nmetallic electrodes of the tunnel junctions. If the trans-\nparencycoefficient, assumed to be the same for both tun-\nneling barriers, is k, than the effective time spent by a\nnon-equilibrium electron within the device, which is the\nmeasure of τs, is\nτs=τd∼d\nvFk−1, (14)\nwheredis athickness ofthe film forming the device while\nvFis the Fermi velocity within the metal structure. Thus\nwe conclude that if τdis comparable to the characteristic\nelectron-electron relaxation time and is larger than the\ncharacteristic electron-phonon relaxation time, then the\nelectron distribution function within the device is given\nin a same way as in Eq.9.\nWe note that for such a design the picture is to some\nextent similar to the one corresponding to optical exci-\ntation In particular, here we also deal with vertical ge-\nometry of excitation. Then, in contrast to the point-\ncontact design (Fig.3(a)), here we have no limitation\nin horizontal size of the device. The important differ-\nence is that in the tunnel junction-based device the sharp\nform of the electron distribution allows effective control\nof the switching process by controlling the form of the\nbias pulses.\nThe point-contact or tunnel junctions scheme where\nthe reversal is driven by the electric bias allows to avoid\na number of drawbacks which are often considered as\nthe limitations for the all-optical reversal. As it fol-\nlows from the considerations in the Sec.II, the magne-\ntization reversal depends on a balance between magne-\ntization decay time τA,B;einTe→ ∞limit and the de-\ncay time of the electron temperature Te. In a case of\noptical excitation achieving such a balance may require\na delicate choice of the pulse duration and pulse inten-\nsity and sample characteristics.14Furthermore, bringing\nthe optical excitation to the nanoscale is a challenging\ntask.49,50An additional factor which requires accurate\ncontrol for application of the all-optical magnetization\nreversalis related to a rate of the cooling time, controlled\nby a heat withdrawal from the laser-excited spot. Thus,\nin Ref.10, where the record-short all-optical write-read\ntime ofτw−r=30ps has been reported, in experiments\nthe residual heating resulted in only 83% magnetic con-\ntrast restored at τr−w. In the case of the bias driven\nmagnetization reversal, better control of the decay of the\nelectron temperature and the cooling can be achieved by\ntuning the duration and the recovery time of the voltage\npulse.\nNaturally, the question arises about an approach al-\nlowing generating bias pulses of required strength and\nduration. The most plausible solution for this problem\nis the photoconducting switch,51employing the illumina-\ntion of the semiconductor by a femtosecond laser pulse.\nThe rise time of such switches is mostly defined by the\nrise time of the laser pulse, and the RCcharacteristic of\nthe circuit. The controllable and short decay time of the11\nvoltage pulse is, however, the challenging issue. We are\naware of the reports where the voltage pulse durations\ndown to several hundreds femtoseconds were achieved by\ndesigningspecialphotoswitchcircuitsandusingthesemi-\nconducting media characterized by short electron decay\ntimes.52–54\nAs for the decay time, they can be very small in\nmetal-based structures. In the limit of ballistic trans-\nport estimates based on Eq.10 give the escape time as\nsmall as τs∼10−14s for s-electrons in the point contact\n(Fig.3(a)) of a size of L∼10nm. This value is the esti-\nmateoftheupperlimitofasharpnessofanyequilibration\nprocesswithintheballisticpointcontact. Inparticular,it\ndescribes the cooling rateafter the externalbias has been\nswitched off. In the diffusive point contact (Fig.3(a)) of\na sizeL∼30nm the escape time will be τs∼10−12s for\ntheelectronmeanfreepathof3nm. Forthe tunnelstruc-\nture presented on Fig.3(b) one can control τs(Eq.14) by\na proper choice of the tunneling transparency coefficient\nk. Certainly, to ensure effective control of the magne-\ntization the time τsshould not be smaller than the ex-\nchangescatteringtimes τe(A,B). Thus, weemphasizethat\nthe advantage of the devices suggested is a possibility to\ncontrol the process of switching in rather broad region\nby a proper choice of the device parameters including its\ngeometry and the bias applied.\nTherefore, we can conclude that the mechanism of for-\nmation of very short electric pulses seems to be the only\nrestriction of operation times for the metal-based devices\nin question. However the same restriction concerns any\nelectronicdevice while many otherrestrictionstypical for\nsemiconductor-based devices seem to be lifted.\nAn important ingredient of any spintronic device is a\npossibility of read-out of the magnetization controlled by\nsome external stimulus. We believe that such a read-out\nin the case of the electric bias driven reversal can be pro-\nvided by the well-known anomalous Hall-effect55(AHE).\nIndeed, the Hall voltage VHis related to the current I\nthrough the structure as VH/I=RH+R1M, whereR\nandR1are some material-dependent parameters, His\nan external magnetic field , and Mis the sample magne-\ntization. In the absence of external field the magnitude\nand the sign of VHis controlled by the sample magneti-\nzation. Thus a presence of Hall probes attached to the\ncorresponding ferromagnetic layer allows to detect the\nstate of the layer magnetization. Note that, although the\nAHE is often used to detect a presence of ferromagnetic\nordering, when other technique possessed poor sensitiv-\nity for the decisive conclusion. However, to the best of\nour knowledge it was not used to detect the sign of mag-netization since typically the samples had multidomain\nstructure.\nV. CONCLUSIONS\nTo conclude, we proposed the general microscopical\nmodeloftheultrafastmagnetizationreversalinantiferro-\nmagnetically coupled ferromagnetic metallic subsystems.\nIn the proposed model the rapid increase of the temper-\nature of mobile s−orp−electrons triggers effective ex-\nchange scattering between these electrons and the ferro-\nmagneticsubsystems oflocalized f−and nearlylocalized\nd−electrons. Then incoherent spin exchangebetween the\ntwo (nearly-)localized ferromagnetic subsystems is medi-\nated by the mobile electrons. Owing to the different ex-\nchange relaxation times for two involved ferromagnetic\nsubsystems, there is a moment after the excitation, when\none of the subsystems is completely demagnetized, while\nanother one still possess finite magnetization. It is this\nmomentwhen the reversalofthe ”faster”sublatticetakes\nplace. The model succeeds to explain most of the main\nfeatures of the all-optical magnetization reversal in ferri-\nmagneticmetallicsinglephaseormultilayeredstructures,\nreported recently by several groups. An important argu-\nment in favor of the model is the fact that the switching\nwas observed only for conducting structures inevitably\ncontaining mobile carriers. Since the effect of the exter-\nnal excitation in the considered here process is limited\nto a transient heating of the mobile electron system, we\nalso analyze a possibility to trigger the magnetization re-\nversal by application of the voltage bias. The relevant\nstructures are metallic point contacts or tunneling struc-\ntures with embedded ferrimagnetic metallic systems. It\nisshownthatsuchdevicesallowtocontrolswitchingwith\na great accuracy. We also suggest to use the anomalous\nHall effect to register the switching thus playing a role of\nreading probes.\nVI. ACKNOWLEDGEMENTS\nWe thank Dr. A. V. 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Phys. 82, 1539 (2010)."    },    {        "title": "2101.07300v1.Quantum_fluctuation_effects_on_the_ordered_Moments_in_a_two_dimensional_frustrated_ferrimagnet.pdf",        "content": "arXiv:2101.07300v1  [cond-mat.str-el]  18 Jan 2021Quantum Fluctuation effects on the Ordered Moments in a two di mensional\nfrustrated Ferrimagnet\nKingshuk Majumdar∗\nDepartment of Physics, Grand Valley State University, Alle ndale, Michigan 49401, USA\nSubhendra D. Mahanti†\nDepartment of Physics and Astronomy, Michigan State Univer sity, East Lansing, Michigan 48824, USA\n(Dated: January 20, 2021)\nWe propose a novel two-dimensional frustrated quantum spin -1/2 anisotropic Heisenberg model\nwith alternating ferromagnetic and antiferromagnetic mag netic chains along one direction and anti-\nferromagnetic interactions along the other. The (mean-fiel d) ground state is ferrimagnetic in certain\nrange of the interaction space. Spin-wave theory analysis o f the reduction of ordered moments at in-\nequivalent spin sites and the instability of the spin waves s uggest a quantum phase transition which\nhas the characteristics of both the frustrated two-dimensi onal antiferromagnetic S=1/2 ( J1,J2)\nmodel and 1D S 1=1, S 2=1/2 quantum ferrimagnetic model.\nPACS numbers: 71.15.Mb, 75.10.Jm, 75.25.-j, 75.30.Et, 75. 40.Mg, 75.50.Ee, 73.43.Nq\nI. INTRODUCTION\nLow-dimensional quantum spin systems are excellent\nexamples to explore the physics of strongly interacting\nquantum many-body systems.1In addition to the inher-\nent quantum nature of the interacting elements (for ex-\nample localized spins with S=1/2), these systems pro-\nvide an array of choices where the effects of competing\ninteractions, non-equivalent nearest neighbor bonds, and\nfrustration on quantum fluctuations of the long range\norder parameter and on quantum phase transitions at\nT= 0K (no thermal fluctuations) can be explored. Al-\nthough extensive studies using different theoretical ap-\nproaches and using different spin models have been done\nover the last severaldecades we will first discuss two sim-\nple models relevant to our present work.2,3They are (i)\ntwo-dimensional (2D) antiferromagnetic S=1/2 Heisen-\nberg model on a square lattice with nearest (NN) and\nnext nearest neighbor (NNN) antiferromagnetic interac-\ntions(J1,J2)[ModelI]and(ii)one-dimensional(1D)spin\nchainconsistingofalternatingS 1=1andS 2=1/2spinsin-\nteracting antiferromagnetically [Model II]. The classical\ngroundstate (GS) ofmodel I in certain ( J1,J2) domain is\nlong range ordered (LRO) antiferromagnet whereas that\nofmodelIIisalongrangeorderedferrimagnet. Quantum\nspin fluctuations (QSF) dramatically affect the physical\nproperties of these systems, which we review briefly after\nfirst introducing a new model [Model III] below.\nWe propose a novel 2D Heisenberg model at T= 0K\nconsistingofonlyS=1/2spins,whichcombinestheessen-\ntial features of the two above models, extreme anisotropy\nof the NN bonds (some ferro and some antiferro) and\nfrustration. The classical ground state (discussed in de-\ntail later in the paper) is a four-sublattice ferrimagnet in\ncertain parameter space. Our focus in this paper is on\nthe stability of this ferrimagnetic ground state and effect\nof QSFs at T= 0K on the long range ordered sublattice\nmagnetizations.II. REVIEW OF MODELS I AND II\nA. Model I\nThe classical ground state of the 2D S=1/2 ( J1,J2)\nHeisenberg model on a square lattice depends on the\nfrustration parameter η=J2/J1.1,2Forη <0.5, the GS\nis a Ne´ el state with ordering wave vector ( π,π), similar\nto the unfrustrated case whereas for η >0.5 the GS is\nthe degenerate columnar antiferromagnetic state (CAF)\nwith ordering wave vectors ( π,0) and (0 ,π). There is a\nfirst-order phase transition from the Ne´ el state to CAF\nstate at η= 0.5. Effects of QSF on this phase transi-\ntion and other properties of this model have been inves-\ntigated using a large number of methods.4–13Here we\nreview the main results obtained within linear spin wave\ntheory (LSWT). Sublattice magnetization, mis reduced\nby QSF from its classical value of 0.5 to 0.303 at η= 0\nand then decreases monotonically with increasing ηand\napproaches zero at the first critical point ηc1= 0.38.\nSimilarly m= 0.37 atη= 1 and then steadily decreases\nto zero at the second critical point ηc2= 0.49. LSWT\nclearly indicates that QSF effects are enhanced in the\npresence of frustration. Also it suggests that in the re-\ngionηc1< η < η c2, the classical GSs are not stable. The\nnature of the GS (e.g. spin-liquid, valence bond) and low\nenergy excitations in this region have been extensively\nstudied during past several years and continue to be of\ngreat current interest.\nB. Model II\nThe second model deals with ferrimagnets. Ferrimag-\nnets are somewhere between ferromagnets and antifer-\nromagnets.14–25It is well known that for 1D quantum\nS=1/2 ferromagnet, the ground state is long-range or-\nderedandQSFsdonotreducethe classicalvalue of m. In2\ncontrast, in a 1D quantum S=1/2 antiferromagnet (AF),\nQSFscompletelydestroythe classicalLRO.Thequestion\nwhat happens forferrimagnetsdrew considerableinterest\nin the late 90’s and several interesting works were done\nusingasimpleisotropicNNantiferromagneticHeisenberg\nmodel with two types of spins, S 1= 1 and S 2= 1/2 in a\nmagnetic unit cell (MUC).17,18,20–26Following Refs. [17]\nand [21] we discuss some of the interesting physical prop-\nerties of this 1D system and point out how our proposed\n2D model differs from this.\nThe Hamiltonian of the 1D system is given by\nH=/summationdisplay\nn/bracketleftbig\nJ/parenleftbig\nS1n·S2n+S2n·S1n+1/parenrightbig\n−hSz\nTn/bracketrightbig\n,(1)\nwhereS1nandS2nare spin-1 and spin-1/2 operators\nrespectively in the nthunit cell (UC), effective field h(=\ngµBHwithggyro-magnetic ratio, µBBohr magneton,\nandHexternal magnetic field) and Sz\nTn=Sz\n1n+Sz\n2n.\nAccording to the Lieb-Mattis theorem27, forH= 0,\nthe GS is long range ordered as the system has total\nspinST=N/2, where Nis the number of UCs in GS,\n/angbracketleftSz\n1n/angbracketright= 1 and /angbracketleftSz\n2n/angbracketright= 0.5. The problem of looking at\nthe excitations is well suited for the LSWT approach.\nSince the elementary magnetic cell consists of two spins,\nLSWT predicts two types of magnons: a gapless “acous-\ntic” or “ferromagnetic” branch with Sz\nT=N/2−1, and\na gapped “optical” or “antiferromagnetic” branch with\nSz\nT=N/2+1. The optical magnon gap for this model\nhas been numerically found to be ∆ opt= 1.759J.23An\nintriguing property of this 1D quantum ferrimagnet is\nthat when one turns on the magnetic field H, the acous-\ntic branch opens up a gap but the optical gap decreases\nand at a critical value of the field Hc1this gap van-\nishes, the system then enters into a quantum spin liq-\nuid (QSL) phase, where the GS is dominated by QSFs\nwith spinon-like excitations.17,18,21With further increase\nin the strength ofthe field this QSL phasegoesinto asat-\nurated ferromagnetic phase.\nBrehmer et al. [17] calculated the sublattice magneti-\nzation for the S=1 sublattice ( mA) and found it to be (1-\nτ) withτ≈0.305. The sublattice magnetization of the\nS=1/2 sublattice can be calculated using their method\nand is found to be mB=−0.5+τ. The ordered moment\nof the S=1/2 sublattice is reduced by a factor of ∼2.5\ndue to QSF. There are two important points worth not-\ning here: (1) the total magnetization (ferromagnetic) per\nmagnetic unit cell is mA+mB= 0.5, the classical value\nand (2) QSF reduction of the S=1/2 sublattice is larger\nthan the 2D S=1/2 Heisenberg model for a square lat-\ntice where η∼0.2. Point (1) is consistent with the fact\nthat the ferromagneticlong rangeorder is not affected by\nQSF. Also mAandmBare independent of the magnetic\nfield.III. MODEL III\nAs mentioned in the beginning, we introduce a new 2D\nHeisenberg model which incorporates different aspects\nof the two models discussed above, anisotropic bonds\nand frustration. Also, instead of two types of spins and\nsingle exchange parameter, our model consists of only\nS=1/2 spins interacting with Heisenberg exchange cou-\nplings of different signs (both ferro and antiferro). The\nunit cell consists of four types of spins which we denote\nasS(µ)(µ= 1..4), it is a Bravais lattice. The lattice\nvectors for the four spins in a rectangular lattice with\nparameters ( a,b) along the xandydirections are given\nbyRiµ=ixaˆ x+iybˆ y+τµwhereτ1= (0,0),τ2=\n(0,b/2),τ3= (a/2,b/4) andτ4= (a/2,3b/4) (see Fig. 1).\nAs we will show, the ground state is ferrimagnetic in cer-\ntain range of exchange parameter space. Three spins\ncombine to form the S=3/2 sublattice. In contrast to\nthe 1D S=(3/2,1/2) model, where the magntitudes of\nthe spins in each sublattice are fixed, in our model, the\nS=3/2 sublattice can undergo amplitude fluctuations. In\nfact, the present model was inspired by recent inelastic\nneutron scattering experiments on a quasi 2D spin sys-\ntems containing Cu+\n2ions, Cu 2(OH)3Br.28However, in\nthis system the effect of orbital ordering of active mag-\nnetic orbitals driven by the ordering of the Br+ions on\nthe exchange parameters is such that the ground state is\nan antiferromagnet with eight spins per unit cell.\nThe Heisenberg spin Hamiltonian ( H) for model III is\ndivided into two parts, intra-chain ( H1) and inter-chain\n(H2):\nH=H1+H2, (2)\nwhere\nH1=−J1/summationdisplay\ni/bracketleftBig\nS(1)\ni·S(2)\ni+1\n2/parenleftBig\nS(1)\ni·S(2)\ni−bˆy+S(2)\ni·S(1)\ni+bˆy/parenrightBig/bracketrightBig\n+J2/summationdisplay\ni/bracketleftBig\nS(3)\ni·S(4)\ni+1\n2/parenleftBig\nS(3)\ni·S(4)\ni−bˆy+S(4)\ni·S(3)\ni+bˆy/parenrightBig/bracketrightBig\n,\n(3a)\nH2=1\n2J3/summationdisplay\ni/parenleftBig\nS(1)\ni+S(2)\ni/parenrightBig\n·/parenleftBig\nS(3)\ni+S(3)\ni−aˆx/parenrightBig\n+1\n2J4/summationdisplay\ni/bracketleftBig\nS(1)\ni·/parenleftBig\nS(4)\ni−bˆy+S(4)\ni−aˆx−bˆy/parenrightBig\n+S(2)\ni·/parenleftBig\nS(4)\ni+S(4)\ni−aˆx/parenrightBig/bracketrightBig\n+1\n2J3/summationdisplay\niS(3)\ni·/parenleftBig\nS(1)\ni+S(1)\ni+aˆx+S(2)\ni+S(2)\ni+aˆx/parenrightBig\n+1\n2J4/summationdisplay\niS(4)\ni·/parenleftBig\nS(2)\ni+S(2)\ni+aˆx+S(1)\ni+bˆy+S(1)\ni+aˆx+bˆy/parenrightBig\n.\n(3b)\nAll exchangeparameters Jµarepositive (see Fig. 1 foran\nillustrative long range ordered ferrimagnetic). We refer\nto this model as ( J1,J2,J3,J4) model.3\n1JJ\n2 J J4J\nJ4\n3S3J3J3\n3J11\nJ\nSS 2J4J4\nS4\nFIG. 1. (Color online) Classical ferrimagnetic ground stat e\nof 2D F-AF chain. The basic magnetic unit cell comprises\nof three up-spins S1,S2,S4and one down-spin S3. The in-\nteraction strengths J1,J2,J3are all positive and J4is the\nfrustrated bond.\nClassical Ground State: The basic model consists of\nalternating 1D ferro (strength J1= 1) and antiferro-\nmagnetic (strength J2=η2J1) S=1/2 chains (along the\ny-axis). The nearest chains interact with interaction\nstrengths J3(=η3J1) andJ4(=η4J1) which are an-\ntiferromagnetic. Before discussing the excitations and\nquantum spin fluctuations, we first consider the ground\nstate of our model when the spins are treated classically\n(mean field state). With J3=J4= 0, the ground state\n(G0) with broken global symmetry consists of decoupled\nalternating ordered F chains ( S1andS2spins) and AF\nchains (S3andS4spins). Due to the time reversal sym-\nmetry, the F chains can be either up or down (chosen\narbitrarily) and the AF chains can be in one of the two\nNe´ elstates. Thedegeneracyofthe G0is 22M, whereMis\nthenumberofF(orAF) chains. For J3>0andJ4= 0, if\nwe fix the orientation of one F chain, the nearest two AF\nchain orientations are fixed by the J3bond. The neigh-\nboring F chain orientations are then fixed. In this way,\nwe have the exact ground state Gas each bond takes its\nminimum energy value. When η3>0 andη4<0 (ferro-\nmagnetic), the system is not frustrated and the classical\nGS is a collinear ferrimagnetic state as shown in Fig. 1.\nHowever, for η3>0 andη4>0, spinS4is frustrated.\nFor weak frustration i.e. η4<< η3,Gis most likely the\nexact ground state and with increasing frustration ( J4)\nthe system will undergo a phase transition to a new state\nwhich may or may not be long range ordered. One ap-proach to attack the problem is to use the generalized\nLuttinger Tisza method [29] first proposed by Lyons and\nKaplan.30It turns out that for our Bravais lattice with\nfour-spin/unit cell system the calculations are quite dif-\nficult. So in the absence of the knowledge of the exact\nground state for large J4, we have used a different ap-\nproach. We study the local stability of Gwith increasing\nstrength of the frustrating bond ( J4). As we will show\nlater, depeding on the strength of J2/J1, there is a criti-\ncal value of J4/J3where the ground state Gis no longer\nlocally stable. Thus in our current analysis of the phase\ndiagram and excitations of the model using spin-wave\ntheory we use Gas the ground state.\nIV. SPIN-WAVE THEORY\nIt is well-known that spin-wave theory is best suited to\ntreat the dynamics of long range-ordered states in quan-\ntum spin system with large spin S. In the leading order\n(linear spin wave theory - LSWT), the excitations are\nmagnons. When magnon-magnon interaction effects are\nnegligible (for example for S >>1/2 and three dimen-\nsions), LSWT provides a very good description of the\nquantum spin fluctuation effects, one example being the\nreduction of the ordered moment in Heisenberg quantum\nantiferromagnets. However, for S=1/2 systems in 2D,\nmagnon-magnon interactions are not negligible and one\nmust incorporate higher order spin (1/S) corrections to\ntreat the system.6,7,31,32Even for these systems, LSWT\nprovides qualitatively correct physics. For example, for\n2D Heisenberg spin systems with nearest neighbor (NN)\nantiferromagnetic (AF) coupling [( J1,J2) model with no\nfrustration i.e. J2= 0] on a square lattice, the ordered\nmoment (average sublattice spin /angbracketleftSz/angbracketright) reduces due to\nQSF from 0.5 to 0.303 as given by LSWT.4,5When one\nincludes the higher order magnon-magnon interaction ef-\nfects using (1/S) expansion theory /angbracketleftSz/angbracketright= 0.307,7,32in-\ndicating that LSWT is very reasonable. For the gen-\neral (J1,J2) model, the effect of frustration is much more\nsubtle. Frustration tends to destabilize long range order.\nWith increase in the strength of frustration, /angbracketleftSz/angbracketright= 0\nat a critical value of J2=J2c. LSWT gives J2c= 0.38\nwhereas including the magnon-magnon interaction one\nfindsJ2c= 0.41,6,7againindicating the reasonablenessof\nLSWT in providinga measureof the QSF induced reduc-\ntion of the magnetization M. In a recent work (Ref. [12])\nresults for this model is obtained using a four-spin bond\noperator technique where it is found that /angbracketleftSz/angbracketright= 0.301\nforJ2= 0 and J2c= 0.36, which are close to the LSWT\nresults. We should mention here that all these method\nfail in the spin disordered state i.e. when J2> J2c.\nIn view of the abovediscussion, we opted to use LSWT\nto analyze the effect of QSF on the averagemagnetic mo-\nment and the critical strength of the frustration where\nthe ordered moments vanish. Unlike the ( J1,J2) model\n(two sublattice with same value of the ordered moment)\nour2D frustrated( J1,J2,J3,J4) model has a 4-sublattice4\nstructure as shown below and different sublattice mo-\nments are affected differently by QSF.\nFor our analysis we only consider the parameter space\n(η2,η3,η4) of the Hamiltonian H[Eq. (2)] where the GS\nis stable and is long range ordered collinear ferrimagnetic\nstate. The spin Hamiltonian in Eq. (3) is mapped onto a\nHamiltonian of interacting bosons by expressing the spin\noperators in terms of bosonic creation and annihilation\noperators a†,afor three “up” spins (spins 1, 2, and 4)\nandb†,bfor one “down’ spin (spin 3) using the standard\nHolstein-Primakoff representation33\nS+i\nin≈√\n2Sain, S−i\nin≈√\n2Sa†\nin, Szi\nin=S−a†\ninain,\nS+j\njn≈√\n2Sb†\njn, S−j\njn≈√\n2Sbjn, Szj\njn=−S+b†\njnbjn,\nand expand the Hamiltonian [Eq. (3)] perturbatively in\npowers of 1 /Skeeping terms only up to the quadratic\nterms. The resulting quadratic Hamiltonian is given as:\nH=Ecl+H0+···, (4)\nwhere\nEcl=−2J1NS2/bracketleftbig\n1+η2+2(η3−η4)/bracketrightbig\n(5)\nis the classical GS energy and\nH0= 2SJ1/summationdisplay\nk∈BZ/bracketleftBig\n(1+η3−η4)/parenleftBig\na(1)†\nka(1)\nk+a(2)†\nka(2)\nk/parenrightBig\n−γy/parenleftBig\na(1)\nka(2)†\nk+a(1)†\nka(2)\nk/parenrightBig\n+(η2−2η4)a(4)†\nka(4)\nk\n+(η2+2η3)b(3)†\n−kb(3)\n−k+η2γy/parenleftBig\nb(3)†\n−ka(4)†\nk+b(3)\n−ka(4)\nk/parenrightBig\n+η3γx/parenleftBig\neikyb/4a(1)†\nkb(3)†\n−k+e−ikyb/4a(2)†\nkb(3)†\n−k+h.c./parenrightBig\n+η4γx/parenleftBig\ne−ikyb/4a(1)†\nka(4)\nk+eikyb/4a(2)†\nka(4)\nk+h.c./parenrightBig/bracketrightBig\n(6)\nwithγx= cos(kxa/2) andγy= cos(kyb/2).\nIn the absence of inter-chain coupling ( η3=η4= 0)\nthemagnonspectrumcanbeobtainedusingthestandard\nBogoliubov transformations.34We find four modes for\neachky(−π/b < k y< π/b) independent of kx(−π/a <\nkx< π/a): two from the F-chains ( α-branches) and two\nfrom the AF-chains (one αand one β). The quadratic\nHamiltonian takes the following form:\nH0=/summationdisplay\nk∈BZ/bracketleftBig\nǫ(1)\nkα(1)†\nkα(1)\nk+ǫ(2)\nkα(2)†\nkα(2)\nk\n+ǫ(3)\nk/parenleftBig\nα(4)†\nkα(4)\nk+β(3)†\n−kβ(3)\n−k/parenrightBig/bracketrightBig\n+/summationdisplay\nk∈BZ/parenleftBig\nǫ(3)\nk−2J1S/parenrightBig\n,\n(7)\nwhere\nǫ(1,2)\nk= 2J1S[1∓γy], (8a)\nǫ(3)\nk= 2J2S/radicalBig\n1−γ2y= 2J2S|sin(kyb/2)|.(8b)\nThe last term in Eq. (7) are the LSWT corrections to\nthe classical ground state energy Eclin Eq. (5) for the\nspecial case η3=η4= 0.With inter-chain coupling (i.e. η3,η4>0), we have\nnot been able to find the analytical Bogoliubov trans-\nformations that transforms the bosonic spin operators to\nBogoliubov quasiparticle operators that diagonalize the\nHamiltonian H0[Eq. (6)]. For the special case kx=π/a\ni.e.γx= 0, we use the equation of motion method\n(see Appendix A) and obtain analytical solutions for the\nmagnon dispersion which are:\nǫ(1,2)\nk= 2J1S/bracketleftbig\n(1+η3−η4)±γy/bracketrightbig\n, (9a)\nǫ(3,4)\nk= 2J1S/vextendsingle/vextendsingle(η3+η4)±/radicalBig\n(η3−η4+η2)2−η2\n2γ2y/vextendsingle/vextendsingle.\n(9b)\nWhenη3=η4= 0 the above dispersions reduce to\nEq. (8) as expected.\nFor the general case we use an elegant method devel-\noped by Colpa to obtain both the eigenenergies (magnon\ndispersions) and eigenvectors (required for the calcula-\ntion of magnetization).35,36First we write the 8 ×8\nHamiltonian [Eq. (6)] in a symmetrized form:\nH0=J1S/summationdisplay\nk∈BZ8/summationdisplay\ni=1X(i)†\nkhkX(i)\nk\n−2J1SN[1+η2+2(η3−η4)],(10)\nwith the eigenvectors\nXk= [a(1)\nk,a(2)\nk,a(4)\nk,b(3)\nk,a(1)†\nk,a(2)†\nk,a(4)†\nk,b(3)†\nk]. The\nhermitian matrix hkis:\nhk=\nA1−B1C∗\n20 0 0 0 C1\n−B1A1C20 0 0 0 C∗\n1\nC2C∗\n2A20 0 0 0 B2\n0 0 0 A3C1C∗\n1B20\n0 0 0 C∗\n1A1−B1C20\n0 0 0 C1−B1A1C∗\n20\n0 0 0 B2C∗\n2C2A20\nC∗\n1C1B20 0 0 0 A3\n,(11)\nwhere the constants are given in Eqs. (A2).\nTheCholeskydecompositionhastobeappliedon hkto\nfindthecomplex Kmatrixthatfulfillsthecondition hk=\nK†K. However, the Cholesky decomposition only works\nif the matrix hkis positive definite (i.e. the eigenvalues\nare all positive).35In case the spectrum of the Hamilto-\nnianH0containszeromodes,onecanaddasmallpositive\nvalue to the diagonal of hkto make the matrix positive\n“definite”. We find that the criterion for the Cholesky\ndecomposition to work for all kisη4≤η2η3/(η2+2η3).\nAs an example, with η2= 3.0,η3= 0.4,η4≤η4c, where\nη4c= 0.316. Ifη4> η4cthe matrix hkis not positive\ndefinite and the procedure fails. As we discuss later, this\nis precisely the same condition for the stability of the fer-\nrimagnetic state. After obtaining the matrix K, we solve\nthe eigenvalue problem of the hermitian matrix KgK†,\nwheregis a diagonal paraunitary matrix with elements\ngii= diag(1 ,1,1,1,−1,−1,−1,−1). The resulting eigen-\nvectorsarethen arrangedin suchawaythat the first four\ndiagonal elements of the diagonalized L=U†KgK†U5\nmatrix are positive and the last four elements are nega-\ntive. The first four positivediagonalelements correspond\nto the magnon dispersions.\nTo calculate the sublattice magnetization miwe first\nconstruct the diagonal matrix, E=gLand then find the\ntransformation matrix T, which relates the boson modes\nXkwith the Bogoliubov modes αkviaXk=Tαk. The\nmatrixTis calculated using36:T=K−1UE1/2. mi=1,2,4\nof spins S1,S2,S4are positive but m3for spin S3is\nnegative. So we calculate the magnitude of mi=1−4for\neach of the four sublattices using\n|mi|= 0.5−|τi|. (12)\nwhereτiare the reduction caused by QSFs:\n|τi|=1\nN/summationdisplay\nk∈BZ/braceleftBig\nTkDT†\nk/bracerightBig\ni+4,i+4. (13)\nDis a diagonal matrix with [0 ,0,0,0,1,1,1,1] as the di-\nagonal elements. We again reiterate that the parameters\nη2,η3,η4are chosen such a way that the condition for the\nCholesky decomposition is satisfied, i.e. η4≤η4c.\nV. MAGNON DISPERSION AND SUBLATTICE\nMAGNETIZATION\nA. Magnon Dispersion\nEffects of inter-chain interaction on the magnon dis-\npersion is displayed in Fig. 2(a-e) where for illustration\nwe have chosen η2= 3,η3= 0.4 and the frustration pa-\nrameter η4is increased from 0.05 to 0.315. The disper-\nsion along ky(along the chains) is given for two values\nofkx:kx= 0 (top two panels) and kx=π/a(bottom\ntwo panels). Also for comparison we give the dispersions\nfor the non-interacting chains ( η3=η4= 0). Later we\nwill discuss the kxdependence for some special modes.\nAs expected, there are four magnon modes for each k.\nFor the non-interacting chains, there are two F-magnon\nmodes which are split (the lower mode ∼k2\nyfor small\nky) and two AF-magnons which are degenerate ( ∼kyfor\nsmallky). In the presence of couplings (discussed below)\nwe will (loosely) refer to these four modes as two F and\ntwo AF modes.\nFirst we consider the case kx=π/a(bottom two\npanels) where the hybridization between the F and AF\nmodes is absent (as γx= 0) - so the F and AF chains\ninteract only through effective fields. In this limit, we\nfind from Eq. (9a) and Eq. (9b) that the F-modes get\nrigidly shifted upwards by 2 J1S(η3−η4), the two degen-\nerate AF-modes are split by 4 JS(η3+η4), and both the\nmodes∼k2\ny. Atky= 0 the lower F-mode and the lower\nAF-mode are gapped, ∆ F(π/a,0) = 2J1S(η3−η4) and\n∆AF(π/a,0) = 2J1S[/radicalbig\n(η2+η3−η4)2−η2\n2−(η3+η4)].\nWhen the frustration parameter η4is increased towards\nη3, there is a critical value η4c=η2η3/(η2+ 2η3)< η3,11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 3\nky (π/b)01234ωk\n11.5 22.5 3\nky (π/b)01234(a) η4=0.05 (b) η4=0.1\n(c) η4=0.2A. kx=0, η2=3.0, η3=0.4\n(d) η4=0.3\n(f) η2=3.0, η3=η4=0 (e) η4=0.315F\nFAF\nAFAF AFAF\nAFAF\nAF\nAF\nAFAFF\nFF\nF\nF\nF\nFF\nF\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 3\nky (π/b)01234ωk\n11.5 22.5\nky (π/b)01234(a) η4=0.05 (b) η4=0.1\n(c) η4=0.2B. kx=π/a, η2=3.0, η3=0.4\n(d) η4=0.3\n(f) η2=3.0, η3=η4=0 (e) η4=0.315FFAFAF\nF\nFAFFFAFAF\nAF\nAFAF\nAF AF\nF\nFF\nF\nFAF\nFAF\nFIG. 2. Magnon dispersion of the ferrimagnetic state for A.\nkx= 0 and B. kx=π/a[Figs. (a-e)] with η2= 3.0,η3= 0.4.\nThe frustration parameter η4is varied from 0.05 (small frus-\ntration) to η4= 0.315. (f) Limiting case with no inter-\nchain coupling: the two AF-branches are degenerate, the F-\nbranches are gapped, and the lower F-branch vanishes at\nky= 2π/b(= 0). Note that due to 2 πperiodicity ky=\n[−π/b,π/b] = [π/b,3π/b].\nwhere ∆ AF(π/a,0) = 0 but ∆ F(π/a,0)>0. The fer-\nrimagnetic GS becomes locally unstable and the system\ntransits to a new ground state (For the parameter values\nwe have chosen η4c= 0.316 - this is also the place where\nCholesky decomposition fails because the matrix hkis\nnot positive definite). This is similar to the field induced\nquantum phase transition as a function of the external\nmagnetic field for the 1D quantum S 1= 1,S2= 1/2\nmodel discussed in the introduction.17,21Here the optic\nmode gap goes to zero at a critical field and the system\nundergoes a quantum phase transition from a ferrimag-\nnetic state to some other state. This phase transition oc-\ncurs in the range η3> η4> η4c. Fig. 3 shows a schematic\nphase diagram in the ( η4/η2,η3/η2) space. We also note\nthat for given η3andη4≤η3, the strength of the ex-\nchange in the AF chains η2should be greater than a\ncritical value η2c= 2η3η4/(η3−η4) for the ferrimagnetic\nstate to be stable.\nForkx= 0, the picture is qualitatively similar, but\nwith two fundamental differences resulting from hy-6\n0 0.2 0.4 0.60.8 1\nη3/η200.20.40.60.81η4/η2\nCollinear F-AF Ground StateCollinear F-AF Ground State\nFIG. 3. Phase diagram of H[Eq (2)]: normalized ˜ η4=η4/η2\nis plotted against normalized ˜ η3=η3/η2. The dashed lines\nare given by the equations ˜ η4= ˜η3/(1+2˜η3) (lower one) and\n˜η3= ˜η4/(1 + 2˜η4) (upper one). They are the boundaries of\nthe stability of the ferrimagnetic state. The solid thick li ne\n˜η4= ˜η3is most likely a critical line.\nbridization between ferro and antiferro chain excitations.\nFirst, the lower F-mode goes to zero when ky→0 as it\nshould for the Goldstone mode. However the dispersion\nfor large kydiffers qualitatively from the non-interacting\nchains. Second, hybridizationbetween the upper F-mode\nand the lower AF-mode opens up a hybridization gap\nat a finite kyand the size of the gap increases with\nη4. However, as for the ( kx,ky) = (π/a,0) the gap\n∆AF(π/a,0)→0 asη4→η4c. In fact ∆ AF(kx,0)→0\nfor all values of 0 ≤kx≤π/aforη4→η4c. In Fig. 4B we\nshow the kxdependence of ∆ AF(kx,0) for three different\nvalues of the frustration parameter η4. Also we show in\nFig. 4A the kxdependence of ∆ F(kx,0). This suggests\nthat the chains become dynamically decoupled and since\nthe decoupled AF chains are spin liquids without any\nlong range order, the system goes from an ordered state\nto a spin disordered state when η4> η4c. Exact calcula-\ntions will tell us about the precise nature of the ground\nstate for η4c< η4< η3.\n0 0.2 0.4 0.6 0.8 1\nkx (π/a)00.10.20.30.4∆F (kx,0)η4=0.05\nη4=0.2\nη4=0.315\n0 0.2 0.4 0.6 0.8 1\nkx (π/a)00.30.60.91.2∆AF(kx,0)η4=0.05\nη4=0.2\nη4=0.315A. B.\nFIG. 4. Gaps for F-mode (∆ F) and AF-mode (∆ AF) with\nincrease in kxforky= 0 with η2= 3.0,η3= 0.3 and three\ndifferent values of η4= 0.05,0.2,and 0.315.B. Sublattice Magnetization\nFollowing Colpa’s method we have calculated the sub-\nlattice magnetizations mifor the four sites. We have\nchecked that the sum of the reduction in the four sublat-\ntice moments due to quantum fluctuations,/summationtext4\ni=1τi= 0,\nwhich results in the total magnetic moment equal to\none as expected. This is equivalent to the results ob-\ntained for S 1= 1, S 2= 1/2 1D quantum ferrimag-\nnetic state for which the total magnetization/unit cell\nis equal to 0.5. Next we discuss the effect of frustra-\ntion on the quantum fluctuation induced reduction of\nthe long-range ordered moments for the four different\nspins of the unit cell. In the absence of interchain cou-\npling [Fig. 1], m1=/angbracketleftS1z/angbracketright=m2=/angbracketleftS2z/angbracketright= 0.5 and\nm3=/angbracketleftS3z/angbracketright=−m4=/angbracketleftS4z/angbracketright= 0 (due to quantum spin\nfluctuation in 1D AF). When we turn on η3, its effect\nis to produce an ordering field at the S3sites and or-\nder them in the direction opposite to the F-chain spins.\nThe intra AF chain interaction orders the S4spins par-\nallel to the F-chain spins, resulting in a 2D ferrimagnetic\nground state. If η2≪η3then the system will be more\n2D,m1=m2∼=0.5, andm3,m4will be non-zero with\nthe magnitude of m3larger than m4. On the other hand\nifη2≫η3, then intra-chain AF bonds will dominate,\nmaking the AF chains nearly decoupled and the LRO in\nthe AF chains will be small, m4≈ −m3≪0.5.\n0 0.1 0.2 0.3\nη400.10.20.30.40.5Sublattice Magnetization mi η2=3.0, η3=0.4m1=m2\n|m3|\nm4\nFIG. 5. Magnitude of sublattice magnetizations, mi, for\nη2= 3.0,η3= 0.4 as function of η4. Magnetizations for the\ntwo degenerate ( m1=m2) ferro-modes (solid) corresponding\nto spins 1 and 2 slowly increase as η4is increased. Magne-\ntizations m3,m4for spins 3 (dashed) and 4 (solid) decrease\ndue to the increase in quantum fluctuations with increase in\nη4. The ferrimagnetic ground state is stable in the parame-\nter space ( η2,η3,η4) as long as η3> η4andη4≤η4c, where\nη4c= 0.316 forη2= 3.0,η3= 0.4.\nIn Fig. 5, we show how the ordered moments change\nwith the increasing strength of the frustrated bond η4for\nspecificvaluesof η2= 3.0andη3= 0.4. Asη4approaches\nthe criticalvalue 0.316the magnetizationofthe AF chain\ndecreases but remains finite ( |m3| ∼0.07,|m4| ∼0.06)\njust before quantum phase transition to other ground\nstate within LSWT. This is in contrast to what happens7\nin the (J1,J2) model whereas J2approaches Jc(Jc1from\ntheNe´ elstateand Jc2fromtheCAFstate),thesublattice\nmagnetization goes to zero.\nFinally, in Fig. 6(a-b), we show the η2dependence of\nthemagnitudesofthefourorderparameters mi(i= 1..4)\nforη3= 0.4 for two fixed values of the frustrated inter-\nchain bond η4. For our assumed collinear ferrimagnetic\nground state η3> η4andη2> ηc\n2. Forη3= 0.4,η4= 0.1,\nthe critical value of ηc\n2is =0.27 and for η4= 0.2,η2c=\n0.80. For small η2i.eη2≪η3,m1=m2=|m3| ∼0.46\n1 2 3 4 56\nη200.10.20.30.40.5Sublattice Magnetization mi(a) η3=0.4, η4=0.1\nη2c=0.27m1=m2\n|m3|\nm4\n1 2 3 4 5 6\nη200.10.20.30.40.5Sublattice Magnetization mi(b) η3=0.4, η4=0.2m1=m2\nη2c=0.8|m3|\nm4\nFIG. 6. (a-b) Magnitude of sublattice magnetizations, mifor\nη3= 0.4,η4= 0.1 (Fig. a) and 0 .2 (Fig. b) as function of η2.\nThe ferrimagnetic ground state is stable for η2≥η2cwhere\nη2c= 0.27 forη4= 0.1 andηc\n2= 0.80 forη4= 0.2. Ferro\nmodes (solid) corresponding to spins 1 and 2 are degenerate\n(m1=m2). Magnetizations m3,m4for spins 3 (dashed) and\n4 (solid) decrease due to the increase in quantum fluctuation s\nwith increase in η2.\nandm4∼0.38, a reduction from 0.5 by 8% and 24%\nrespectively. The small antiferromagnetic coupling be-\ntween spins of the AF-chain induces a relatively large\nvalue of the moment at the site 4. When η2increases theQSF in the AF-chain reduces the moments at sites 3 and\n4. Notice that site 3 still has a larger moment (in magni-\ntude) than at site 4. For large η2values, say η2∼6, ferro\nchain spins have moments ∼0.495, whereas AF chain\nspins have moments of magnitude ∼0.14(>0) due to\nsmall stabilizing interchain coupling η3= 0.4 [Fig. 6(a)].\nIncreasing the strength of the frustrated bond η4essen-\ntially decouples the chains. For example with η4= 0.2,\natη2= 6.0 ferro chains have moments close to 0.5 and\nAF-chains have moments of magnitude 0 .08 [Fig. 6(b)].\nForη2< η2c, the system is most likely a spin liquid state\nwithout LRO.\nVI. CONCLUSIONS\nIn summary, we have proposed a 2D frustrated Heisen-\nberg model consisting of alternating 1D ferro ( J1) and\nantiferro( J2) chainswhich interact with alternatingfrus-\ntrated (J4) and unfrustrated( J3) bonds (strengths). The\nground state is a long range ordered ferrimagnetic state\nin certain region of the parameter space. Analysis using\nlinear spin wave theory suggests that the system under-\ngoesaquantumphasetransitiontoaquantumdisordered\nphasewith increasingstrengthof η4, similartothe classic\n2D (J1,J2) model. However in contrast to the ( J1,J2)\nmodel, the sublattice magnetizations of the AF chains\ndo not vanish at the critical value η4c, similar to the 1D\nS1= 1,S2= 1/2 model of a quantum ferrimagnet. The\nexact nature of the phase transition, the nature of the\nGS above η4c, and whether the order parameter vanishes\nat the transition should be explored by other theoretical\nand numerical techniques.\nVII. ACKNOWLEDGMENT\nSDM would like to thank Dr. Xianglin Ke for stimu-\nlating discussions.\nAppendix A: Equation of Motion Method\nWith inter-chain coupling (i.e. η3,η4>0), we are\nunable to find the the Bogoliubov transformations that\ndiagonalizes the Hamiltonian in Eq. (6). Thus we opt for\nanother way - the canonical equation of motion method\nto obtain the magnon dispersion.1The variouscommuta-\ntors that are needed for the canonical equation of motion\nmethod are:8\n/bracketleftbig\na(1)\nk,H0/2J1S/bracketrightbig\n=A1a(1)\nk−B1a(2)\nk+C1b(3)†\nk+C∗\n2a(4)\nk, (A1a)\n/bracketleftbig\na(2)\nk,H0/2J1S/bracketrightbig\n=A1a(2)\nk−B1a(1)\nk+C∗\n1b(3)†\nk+C2a(4)\nk, (A1b)\n/bracketleftbig\na(4)\nk,H0/2J1s/bracketrightbig\n=A2a(4)\nk+B2b(3)†\nk+C2a(1)\nk+C∗\n2a(2)\nk, (A1c)\n/bracketleftbig\nb(3)†\nk,H0/2J1s/bracketrightbig\n=−A3b(3)†\nk−B2a(4)\nk−C∗\n1a(1)\nk−C1a(2)\nk, (A1d)\n/bracketleftbig\na(1)†\nk,H0/2J1S/bracketrightbig\n=−A1a(1)†\nk+B1a(2)†\nk−C∗\n1b(3)\nk−C2a(4)†\nk, (A1e)\n/bracketleftbig\na(2)†\nk,H0/2J1S/bracketrightbig\n=−A1a(2)†\nk+B1a(1)†\nk−C1b(3)\nk−C∗\n2a(4)†\nk, (A1f)\n/bracketleftbig\na(4)†\nk,H0/2J1s/bracketrightbig\n=−A2a(4)†\nk−B2b(3)\nk−C∗\n2a(1)†\nk−C2a(2)†\nk, (A1g)\n/bracketleftbig\nb(3)\nk,H0/2J1s/bracketrightbig\n=A3b(3)\nk+B2a(4)†\nk+C1a(1)†\nk+C∗\n1a(2)†\nk, (A1h)\nwhere\nA1= (1+η3−η4), A2= (η2−2η4),(A2a)\nA3= (η2+2η3), B1=γy, B2=η2γy,(A2b)\nC1=η3γxeikyb/4, C2=η4γxeikyb/4.(A2c)\nWe notice that the first four commutators [Eqs. (A1a)\n- (A1d)] are decoupled from the second four commu-\ntators [Eqs. (A1e) - (A1h)]. With the basis vectors\nXk= (a(1)\nk,a(2)\nk,a(4)\nk,b(3)†\nk), thecanonicalequationofmo-\ntion can be deduced from the Hamiltonian in Eq. (6) in\nthe following way:\n/bracketleftBig\nX(i)\nk,H0\n2J1S/bracketrightBig\n=idX(i)\nk\ndt=gω(i)\nkX(i)\nk.(A3)In Eq. (A3) gis a 4×4 diagonal matrix with gii=\n(1,1,1,−1) in the diagonal elements. The eigenvalues,\nω(i)\nkare obtained by solving the determinant:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A1−ω(i)\nk)−B1 C2 C∗\n1\n−B1(A1−ω(i)\nk)C∗\n2 C1\nC∗\n2 C2(A2−ω(i)\nk)B2\nC1 C∗\n1 B2(A3+ω(i)\nk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(A4)\nThe above determinant leads to a fourth-order polyno-\nmial:\nω4\nk+aω3\nk+bω2\nk+cωk+d= 0, (A5)\nwhere the coefficients are:\na=−2(1−2η4), (A6a)\nb= (1+η3−η4)2−4(1+η3−η4)(η3+η4)−(η2+2η3)(η2−2η4)−(1−η2\n2)γ2\ny+2(η2\n3−η2\n4)γ2\nx, (A6b)\nc= 2(1+η3−η4)2(η3+η4)+2(1+ η3−η4)(η2−2η4)(η2+2η3)−2η2\n2(1+η3−η4)γ2\ny−2(η3+η4)γ2\ny\n−2(1+η2+η3−3η4)η2\n3γ2\nx+2(1−η2−η3−η4)η2\n4γ2\nx−2(η2\n3−η2\n4)γ2\nxγ2\ny+4η2η3η4γ2\nxγ2\ny, (A6c)\nd=−(1+η3−η4)/bracketleftBig\n(1+η3−η4)(η2−2η4)(η2+2η3)−(1+η3−η4)η2\n2γ2\ny−2(η2−2η4)η2\n3γ2\nx\n−2(η2+2η3)η2\n4γ2\nx+4η2η3η4γ2\nxγ2\ny/bracketrightBig\n+(η2−2η4)(η2+2η3)γ2\ny+2(η2−2η4)η2\n3γ2\nxγ2\ny+2(η2+2η3)η2\n4γ2\nxγ2\ny\n−η2\n2γ4\ny−4η2\n3η2\n4γ2\nx−4η3η4γ2\nxγ2\ny(η2−η3η4). (A6d)\nThe other set of four boson operators\n(a(1)†\nk,a(2)†\nk,a(4)†\nk,b(3)\nk) lead to a similar fourth or-\nder polynomial equation, but the signs before the linear\nand cubic terms are negative. There is thus a ωk↔ −ωk\nsymmetry between the two sets of solutions. This fourth\norder polynomial [Eq. (A5)] has to be solved numerically.\nThe four real eigen-values can be positive or negative. If\nwe solve the fourth order polynomial associated with the\nother four boson operators we will get again four real\nsolutions which are negative of the solutions of Eq. (A5).\nFor the magnon frequencies we will consider only thefour positive solutions. The diagonalized quadratic\nHamiltonian in terms of new basis vectors α(i=1..4)\nk\nbecomes:\nH0=E0+/summationdisplay\nk∈BZ4/summationdisplay\ni=1ǫ(i)\nkα(i)†\nkα(i)\nk.(A7)\nE0contributes to the LSWT correction to the classical\nground state energy.\nFor the special case kx=π/ai.e.γx= 0 the solutions\nofthepolynomial[Eq.(A5)]canbeobtainedanalytically.9\nThey are (we only consider the positive solutions):\nω(1,2)\nk= (1+η3−η4)±γy, (A8a)\nω(3,4)\nk= (η3+η4)±/radicalBig\n(η3−η4+η2)2−η2\n2γ2y,\n(A8b)and thus the energies of the magnon spectrum are:\nǫ(1,2)\nk= 2J1S/bracketleftbig\n(1+η3−η4)±γy/bracketrightbig\n, (A9a)\nǫ(3,4)\nk= 2J1S/vextendsingle/vextendsingle(η3+η4)±/radicalBig\n(η3−η4+η2)2−η2\n2γ2y/vextendsingle/vextendsingle.\n(A9b)\n∗majumdak@gvsu.edu\n†mahanti@msu.edu\n1H. T. Diep, Frustrated Spin Systems , 1st ed. (World Scien-\ntific, Singapore, 2004).\n2C. Lacroix, P. Mendels, and F. 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Kolezhuk, in Quantum Mag-\nnetism, Lecture Notes in Physics , Vol. 645, edited by\nU. Schollw¨ ock, J. Richter, D. J. Farnell, and R. F. Bishop\n(Springer, Berlin, Heidelberg, 2008) pp. 1 – 83.\n15A. V. Chubukov, K. I. Ivanova, P. C. Ivanov, and E. R.\nKorutcheva, J. Phys.: Condens. Matter 3, 2665 (1991).\n16A. K. Kolezhuk, H.-J. Mikeska, and S. Yamamoto, Phys.\nRev. B55, R3336 (1997).17S. Brehmer, H.-J. Mikeska, and S. Yamamoto, J. Phys.:\nCondens. Matter 9, 3921 (1997).\n18S. K. Pati, S. Ramasesha, and D. Sen, Phys. Rev. B 55,\n8894 (1997).\n19S. K. Pati, S. Ramasesha, and D. Sen, J. Phys.: Condens.\nMatter9, 8707 (1997).\n20N. B. Ivanov, Phys. Rev. B 57, R14 024 (1998).\n21A. K. Kolezhuk, H.-J. Mikeska, K. Maisinger, and\nU. Schollw¨ ock, Phys. Rev. B 59, 13 565 (1999).\n22S. Yamamoto, T. Fukui, K. Maisinger, and U. Schollw¨ ock,\nJ. Phys.: Condens. Matter 10, 11 033 (1998).\n23K. Maisinger, U. Schollw¨ ock, S. Brehmer, H.-J. Mikeska,\nand S. Yamamoto, Phys. Rev. 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Matter 27, 166002\n(2015)."    },    {        "title": "1206.6671v2.Frequency_and_power_dependence_of_spin_current_emission_by_spin_pumping_in_a_thin_film_YIG_Pt_system.pdf",        "content": "arXiv:1206.6671v2  [cond-mat.mtrl-sci]  29 Jun 2012Frequency and power dependence of spin-current emission by spin pumping in a thin\nfilm YIG/Pt system\nV. Castel,∗N. Vlietstra, and B. J. van Wees\nUniversity of Groningen, Physics of nanodevices, Zernike In stitute for Advanced Materials,\nNijenborgh 4, 9747 AG Groningen, The Netherlands.\nJ. Ben Youssef\nUniversite de Bretagne Occidentale, Laboratoire de Magnet isme de Bretagne CNRS, 29285 Brest, France.\n(Dated: October 31, 2018)\nThis paper presents the frequency dependence of the spin cur rent emission in a hybrid ferrimag-\nnetic insulator/normal metal system. The system is based on a ferrimagnetic insulating thin film of\nYttrium Iron Garnet (YIG, 200 nm) grown by liquid-phase-epi taxy (LPE) coupled with a normal\nmetal with a strong spin-orbit coupling (Pt, 15 nm). The YIG l ayer presents an isotropic behaviour\nof the magnetization in the plane, a small linewidth, and a ro ughness lower than 0.4 nm. Here\nwe discuss how the voltage signal from the spin current detec tor depends on the frequency [0.6 -\n7 GHz], the microwave power, Pin, [1 - 70 mW], and the in-plane static magnetic field. A strong\nenhancement of the spin current emission is observed at low f requencies, showing the appearance\nof non-linear phenomena.\nPACS numbers: 72.25.Ba, 72.25.Pn, 75.78.-n, 76.50.+g\nI. INTRODUCTION\nThe actuation, detection and control of the magneti-\nzation dynamics and spin currents in hybrid structures\n(magnetic material/normal metal) by using the (inverse)\nspin hall effect (ISHE and SHE), spin transfer torque\n(STT) and spin pumping, has attracted much attention\nin the last few years. The observation of these phenom-\nena in ferromagnetic (FM)/normal metal (NM) systems\nhas been reported by several groups1–4.\nSpin pumping is the generation of spin currents from\nmagnetization precession, which can be excited by mi-\ncrowave radiation (microstrip5, resonant cavity6, wave-\nguide7). In a FM/NM system, this spin current is in-\njected into the NM layer, where it is converted into a\ndc electric voltage using the ISHE. In 2010, Y. Kaji-\nwara et al.6opened new interest in this research field\nby the demonstration of the spin pumping/ISHE and\nSHE/STT processes in a hybrid system using the mag-\nnetic insulating material Yttrium Iron Garnet (YIG, 1.3\nµm), coupled with a thin layer of platinum (Pt, 10 nm).\nIt has been shown experimentally that the combination\nof these materials and the mentioned phenomena can\nbe used to transmit electrical information over several\nmillimeters6,8,9. The insulator/normal metal (YIG/Pt)\nsystem presents an important role for future electronic\ndevices related to non-linear dynamics effects10–14, such\nas active magnetostatic wave delay lines and signal to\nnoise enhancers, and bistable phenomena15.\nIn this paper, spin current emission in a hybrid struc-\nture YIG [200 nm]/Pt [15 nm] as a function of microwave\nfrequency f, microwave power Pinand applied magnetic\nfieldB(in-plane) is presented. The actuation of the spin\ncurrent emission is provided by a non-resonant 50 Ω mi-\ncrostrip reflection line16within a range of fbetween 0.6and 7 GHz. To our best knowledge, in all previous exper-\niments, the thickness of the single-crystal of YIG, grown\nby liquid-phase-epitaxy (LPE), is within a rangeof 1.3 to\n28µm, which is alwayshigher than the exchange correla-\ntion length defined in pure YIG11. In contrast, the thick-\nness of the YIG used for the experiments presented here\nis only 200nm. Experiments with lowerthickness of YIG\nhave been reported17,18, however these layers are grown\nby different methods than LPE. The different growing\nprocesses result in an enhancement of the linewidth and\nthese layers do not reach the high quality as when grown\nby LPE. Besides its thickness, two other points should\nbe made concerning our YIG sample. First the magnetic\nfield (in-plane) dependence of the magnetizationpresents\nisotropic behaviour and second, no stripe domains have\nbeen observed by Magnetic Force Microscopy (MFM).\nII. EXPERIMENTAL DETAILS\nA. Sample description\nSpin pumping experiments in FM/NM systems for dif-\nferent NM materials have been performed in order to\nstudy the magnitude of the dc voltage induced by the\nISHE19. It has been shown that the mechanism for spin-\ncharge conversion is effective in metals with strong spin-\norbit interaction. Therefore, for the experiments pre-\nsented in this paper, Pt is used as normalmetal layer. As\nmagnetic layer, the insulating material Y 3Fe5O12(YIG)\nis used. The sample is based on a layer of single-crystal\nY3Fe5O12(YIG) (111), grown on a (111) Gd 3Ga5O12\n(GGG) single-crystal substrate by liquid-phase-epitaxy\n(LPE). The thickness of the YIG is only 200 nm, which\nis very low compared to other studies6,11,12,20,21. The2\nYIG layer has a roughness of 0.4 nm. X-ray diffraction\nwas used in orderto estimate the quality ofthe thin layer\nof YIG. The spectrum (not shown) shows epitaxial grow-\ning of YIG oriented along the (111) direction with zero\nlattice mismatch.\n800 µm \n1750 \nµm \nGGG [500 µm] (111)  Pt [15 nm]  \nYIG [200 nm]   (111) \nz, hrf y, B b) \nc) \na) \nFIG. 1. a) and b) schematics of the experimental setup for\nspin pumping measurements. The ferromagnetic resonance\nin the YIG is excited by using a microstrip line in reflection\nbetween 0.6 and 7 GHz. The thickness of the YIG and the\nGGG substrate is 200 nm and 500 µm, respectively. Ti/Au\nelectrodes are attached on top of the Pt layer in order to\ndetect the ISHE voltage. The magnetic field Bis applied\nin the plane of the sample along the ydirection and B⊥\nhrf, where hrfis the microwave field. c) magnetic field (in-\nplane) dependence of the magnetization M(normalized by\nMs, the saturation magnetization) of the pure single-crystal\nof YIGperformed byVibratingSample Magnetometer (VSM)\nat room temperature.\nFor the realization of the hybrid structure, two steps\nof lithography have been used. First, to create the Pt\nlayer (15 nm thick), an area of 800 ×1750µm2has been\npatterned on top of a YIG sample (1500 ×3000µm2), by\nelectron beam lithography (EBL). Before deposition of\nthePtlayerbydcsputtering,argonetchinghasbeenused\nto clean the surface. Etching was done during 5 seconds\nat a beam voltage (intensity) of 500 V (14 mA) with an\nacceleration voltage of 200 V. The second lithography\nstep realizes the Ti/Au electrodes of 30 µm width and\n100 nm thick. For both lithography steps, PMMA with a\nthickness of 270 nm has been used as resist. A schematic\nof the final device is shown in Fig.1 b).\nB. Static and dynamic magnetization\ncharacterizations\nBy using specific growing conditions, the anisotropic\ncontributions (growth, and magneto-elastic) in the YIG\nfilm can be optimized in order to keep the magnetization\nin-plane. Fig.1 c) shows the dependence of the longitudi-\nnal component of the magnetization as a function of themagnetic field applied in the plane of the YIG sample,\nas measured by using a Vibrating Sample Magnetometer\n(VSM) at room temperature. The saturation magneti-\nzation is µ0M=0.176 T, corresponding to the value ob-\ntained for YIG in bulk6,11. The low coercive field ( ≃0.06\nmT) and the shape of the hysteresis loop provide an easy\nproofofthe magnetizationbeingin theplane, with avery\nlow dissipation of the energy. VSM measurements along\nthe two crystallographic axis, [1, ¯1,0] and [1,1, ¯2], show\nsimilar responses indicating isotropic behaviour of the\nmagnetization in the film plane. In addition, no stripe\ndomains have been observed by MFM.\nIn order to well characterize the pure single-crystal of\nYIG, before realizing the YIG/Pt structure, broadband\nferromagneticresonance(FMR)measurementshavebeen\nperformed using a highly sensitive wideband resonance\nspectrometer in the perpendicular configuration (the ap-\nplied magnetic field, B, is normal to the film plane). The\nmicrowave excitation is provided with a non-resonant 50\nΩ microstrip reflection line within a range of microwave\nfrequencies between 2 and 25 GHz. The FMR is mea-\nsured via the first derivative of the power absorption\ndP/dHby using a lock-in measurement technique. The\nvalueofthe modulation field (lock-inreference)used dur-\ning the field sweeping is much smaller than the FMR\nlinewidth. The dependence of the frequency resonance,\nωres, as a function of the resonant magnetic field is\nused to determine the gyromagnetic ratio γ=1.80 1011\nradT−1s−1(and the Lande factor, g=2.046). The intrin-\nsic Gilbert damping parameter is extracted from the de-\npendence of the linewidth as a function of the microwave\nfrequency ( α≈2 10−4)22.\nC. Spin pumping measurement\nFor the actuation of the magnetization resonance in\nthe YIG layer, a different FMR setup has been used.\nTo connect the device, the YIG/Pt system is placed as\nshown in Fig.1 a). In this configuration, the microwave\nfieldhrfis perpendicular to the static magnetic field,\nB. To optimize the electric voltage recording, a lock-in\nmeasurement technique was used. The frequency refer-\nence, generated by the lock-in, is send to the network\nanalyser trigger. This command (with a frequency of 17\nHz) controlsthe microwavefield by the networkanalyser.\nThe microwave field is periodically switched on and off\nbetween PHigh\nrfandPLow\nrf, respectively. PLow\nrfis equal to\n0.001 mW and PHigh\nrfcorresponds to the input microwave\npower, so-called in the following, Pin. The dc voltages\ngenerated between the edges of the Pt layer are ampli-\nfied and detected as a difference of V(PHigh\nrf)−V(PLow\nrf).\nUsing this measurement setup, the dependence of the\nelectric voltage signal as a function of the microwave\npower [1-70 mW] and the frequency [0.6-7 GHz] is anal-\nysed, while sweeping the applied static magnetic field, B.\nBis large enough in order to saturate the magnetization\nalong the plane film. All measurements were performed3\nat room temperature.\nIII. RESULTS AND DISCUSSION\nConversion of spin currents into electric voltage via\nthe ISHE is given by the relation6:EISHE∝Js×σ, where\nEISHE, andσare the electric field induced by the ISHE\nand the spin polarization, respectively. In YIG/Pt, the\n/s45/s53/s48 /s45/s50/s53 /s48 /s50/s53 /s53/s48/s45/s48/s46/s55/s53/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53\n/s52/s50 /s52/s52 /s52/s54 /s52/s56 /s53/s48 /s53/s50 /s53/s52/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s32/s32/s86\n/s73/s83/s72/s69/s32/s91 /s86/s93\n/s66/s32/s91/s109/s84/s93/s82/s101/s115/s111/s110/s97/s110/s116/s32\n/s99/s111/s110/s100/s105/s116/s105/s111/s110/s86/s32/s32/s66/s32/s124 /s124 /s32/s43/s121\n/s32/s66/s32/s124 /s124 /s32/s45/s32/s121/s32\n/s32\n/s32/s32\nFIG. 2. Dependence of the electric voltage signal, VISHE, as a\nfunction of the magnetic field, B, for the YIG [200 nm]/Pt [15\nnm] sample. Bis applied in-plane and the microwave param-\neters are fixed at 3 GHz and 20 mW. The inset shows VISHE\nat resonant condition for the positive and negative configur a-\ntion of the magnetic field (along + yand−y, respectively, see\nFig.1b)).\norigin of the spin current, Js, injected through the Pt\nlayer differs from the conventional spin current in con-\nducting systems like Py/Pt. The spin pumping origi-\nnates from the spin exchange interaction between a lo-\ncalized moment in YIG at the interface and a conduction\nelectron in the Pt layer.\nThe magnetic field dependence of the voltage signal in\nYIG [200 nm]/Pt [15nm] at 3 GHz is shown in Fig. 2.\nThe rf microwave power is fixed at 20 mW. The sign of\nthe electric voltage signal is changed6by reversing the\nmagnetic field along yand no sizeable voltage is mea-\nsured when Bis parallel to z, as expected. The reversing\nof the sign of V(by reversing the magnetic field) shows\nthat the measured signal is not produced by a possible\nthermoelectric effect, induced by the microwave absorp-\ntion. A direct measurement of the electric voltage signal\n(without lock-in amplifier) has been performed in order\nto define the sign of VISHEas a function of the magnetic\nand electric configuration. The voltage detected between\nthe edges of the Pt layer shows resonance-like behaviour,\nwith a maximum value (∆ V) at the resonant condition\nof the system as defined in the inset of Fig.2.In Fig.3 the in-plane magnetic field dependence of the\nelectric voltage signal for a large range of microwave fre-\nquencies between 0.6 and 7 GHz is shown. For each value\nof microwave power ( Pin=1, 10, and 20 mW) and fre-\nquency,f, thevoltagesignal, VISHE=f(Pin,f), atresonant\nconditions has been extracted. To our best knowledge,\nonly two groups6,11have studied the electric voltage sig-\nnalinahybridYIG/Ptsystemasafunctionofmicrowave\nfrequency, but only one11in a large frequency range of\n[2-6.8 GHz]. The difference between our structure and\nRef.11lies in the thickness of the YIG, which is 5.1 µm\nin their case and only 200nm in this work. The thickness\nof the Pt is the same (15 nm). As can be seen from Fig.3,\nthe frequency dependence of ∆ Vpresents a complicated\nevolution, partly resulting from the S11dependence of\nthe microstrip in reflection itself, as a function of fre-\nquency. Nevertheless, note that ∆ Vpresents high values\nat low frequency.\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53/s49/s46/s56\n/s54/s32/s71/s72/s122/s53/s32/s71/s72/s122/s51/s32/s71/s72/s122\n/s52/s32/s71/s72/s122/s50/s32/s71/s72/s122\n/s32/s49/s32/s109/s87\n/s32/s49/s48/s32/s109/s87\n/s32/s50/s48/s32/s109/s87/s32/s86\n/s73/s83/s72/s69/s32/s91 /s86 /s93\n/s66/s32/s91/s109/s84/s93/s49/s32/s71/s72/s122\nFIG. 3. Dependence of the electric voltage signal, VISHE, for\ntheYIG[200nm]/Pt[15nm]sample asafunctionofthestatic\nmagnetic field (in-plane) within a microwave frequency rang e\nof [0.6-7 GHz] at 20 mW. Symbols correspond to the value of\n∆Vfor different microwave power: 1, 10, and 20 mW.\nFig.4 a), b), and c), present the dependence of the\nelectric voltage VISHEas a function of the magnetic field\nand the microwave power for different frequencies (1, 3,\nand 6 GHz, respectively). Two points should be made\nregarding these graphs. First, one can see in those spec-\ntra multiple resonance signals, which are attributed to\nthe Magnetostatic Surface Spin waves (MSSW, when the\nmagnetic field is lower than the resonant condition) and4\nBackward Volume Magnetostatic Spin Waves (BVMSW,\nwhen the magnetic field is higher than the resonant\ncondition)23,24. Second, the strong non-linear depen-\ndence observed at low frequency is well represented by\nthe resonance magnetic field shift and the asymmetric\ndistortion of the resonance line as observed in Fig.4 a).\nThese observations are correlated with the pioneering\nworks of Suhl25and Weiss26related to non-linear phe-\nnomena occurring at large precession angles. The simple\nexpression27of the magnetization precession cone angle\nat resonance is given by Θ = hrf/∆H, wherehrfand\n∆Hcorrespond to the microwave magnetic field and the\nlinewidth of the absorption line of the uniform mode, re-\nspectively. This expression shows that by decreasing the\nexcitation frequency, an enhancement ofthe cone angle is\ninduced. Therefore, the system becomes more sensitive\nto the rf microwave power, Pin.\nIn addition, the non-linear behaviour measured at 1\nGHz (also at 3 GHz, but less) is well represented by\nFig.4 e). This figure represents evolutions of ∆ Vas a\nfunction of the microwave power, Pin, performed at 1, 3\nand 6 GHz between 1 and 70 mW. Y. Kajiwara et al.6\nhave proposed an equation to represent the dependence\nof the electric voltage signal as a function of B,f,hac\n(microwave magnetic field), and the parameters of the\nbilayer system. They showed that VISHEat resonant con-\nditions depends linearly on the microwave power. This\ndependence is well reproduced only at 6 GHz. Fig.4 d)\nrepresents the ratio of ∆ Vextracted from measurements\nat 1 and 6 GHz, ∆ V1GHz/∆V6GHz, as a function of the\nmicrowave power Pin, to emphasize the non-linearity ob-\nserved at 1 GHz. Note that, for a very low microwave\npower of 1 mW, ∆ Vat 1 GHz is 14 times greater than\n∆Vat 6 GHz, whereas by increasing Pin, this difference\nis drastically reduced11and reached a factor of 5 at 60\nmW.\nTo investigate the frequency dependence of ∆ V, the\nresponse of the microstrip line should be taken into ac-\ncount. Between 30 and 40 mT and between 70 and 100\nmT, the microstrip line induces an artificial increase of\nVISHE, as can be observed in Fig.3. The correction fac-\ntor for this artificial increase is determined by measuring\nthe reflection parameter S11, for the system being out of\nresonance.\nFig.5 a) represents the frequency dependence of\n∆˜V/Pin, where ∆ ˜Vcorresponds to the the dc voltage\ncorrected by the response of the microstrip line itself.\nThisfigurepermitstodefinethefrequencyrangeinwhich\nthis evolution presents non-linear behaviour. Note that\nbetween 3.4 and 7 GHz, values of ∆ ˜V/Pinpresent a slow\ndecrease as a function of the microwave frequency. In\nthis regime, ∆ ˜V/Pinvalues are similar for the different\nrf microwave powers of 1, 10, and 20 mW due to the fact\nthat in this frequency range, the rf power dependence of\n∆Vis linear6,13. The interesting feature of the frequency\ndependence of ∆ ˜V/Pinis observed at frequencies below\n3.4 GHz. At those frequencies, the frequency dependence\ndoes not follow the trend observed at higher frequencies/s52 /s53 /s54 /s55/s48/s49/s50/s51/s52\n/s52/s54 /s52/s56 /s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s49/s51/s50 /s49/s51/s53 /s49/s51/s56 /s49/s52/s49/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s49/s50/s51/s52/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s53/s49/s48/s49/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s32/s49/s48/s32/s109/s87\n/s32/s50/s48\n/s32/s53/s48\n/s32/s56/s48\n/s32/s32/s86\n/s73/s83/s72/s69/s32 /s91 /s86 /s93/s66/s32/s91/s109/s84/s93\n/s86\n/s49/s71/s72/s122/s47 /s86\n/s54/s71/s72/s122/s32/s86\n/s73/s83/s72/s69/s32 /s91 /s86 /s93/s32\n/s66/s32/s91/s109/s84/s93\n/s80\n/s105/s110/s32/s91/s109/s87/s93/s32/s86\n/s73/s83/s72/s69/s32 /s91 /s86 /s93\n/s66/s32/s91/s109/s84/s93/s54/s32/s71/s72/s122/s51/s32/s71/s72/s122 /s49/s32/s71/s72/s122\n/s100/s41\n/s101/s41/s99/s41/s98/s41/s32 /s86/s32 /s91 /s86 /s93/s32/s49/s32/s71/s72/s122\n/s32/s51/s32/s71/s72/s122\n/s32/s54/s32/s71/s72/s122/s32\n/s80\n/s105/s110/s32/s91/s109/s87/s93/s97/s41\n/s32/s32 \n/s32 /s32 /s32 \nFIG. 4. a), b), and c) present the dependence of the electric\nvoltage signal, VISHE, as a function of the static magnetic field,\nB, for different microwave power, Pin, at 1, 3, and 6 GHz,\nrespectively. d) microwave power dependence of the ratio of\nthe values of ∆ Vmeasured at 1 and 6 GHz. e) representation\nof ∆Vas a function of the microwave power between 1 and\n70 mW at 1, 3, and 6 GHz. The inset corresponds to the\ndependence of ∆ Vfor low rf power.\n(>3.4 GHz). In the frequency range [0.6-3.4 GHz], the\npreviously observed non-linear behaviour affects the val-\nues of ∆ ˜V/Pinas a function of the input rf microwave\npower. The enhancement ∆ ˜V/Pinis more efficient at\nlow powers and gradually reduces with increasing the\nmicrowave power. The discrepancy is especially strong\naround the maximum at 1 GHz. H. Kurebayashi et al.11\nobtained 17.1 and 62.8 nV/mm at 2 and 6 GHz, respec-\ntively, whereas in our system, for the same frequencies,\n∆Vreaches 542.85 and 108.6 nV/mm.\nThe question arising now: what is the origin of the\nstrong enhancement of ∆ Vat low frequency. Is it only\ndue to the frequency dependence of the cone angle? The\nassumption of a single magnetization precession angle is\nnot warranted, due to the fact that several spin wave5\nFIG. 5. a) dependence of ∆ ˜V/Pinas a function of the mi-\ncrowave frequency with a microwave power of 1, 10, and 20\nmW. The red dashed line corresponds to the analytical ex-\npression of the frequency dependence of ∆ Vextracted from\nRef.6. b) dependence of the resonant frequency, f, as a func-\ntion of the applied magnetic field. Open circles indicate the\nexperimental data when k⊥B(in-plane magnetic field) and\nthe solid black curve is calculated from the Kittel’s formul a28\ngiven by: f=/radicalbig\nfH(fH+fM). Note that fH=γµ0Hand\nfM=γµ0M. c) dispersion relation of spin waves29: depen-\ndence of the frequency as function of the wavevector, k, when\nk/bardblBfor different thickness of YIG. The magnetic field is\nfixed at 40 mT.\nmodes contribute to the dynamic response of the sys-\ntem. Therefore, assuming that no spin waves are created\nin the YIG, the normalization of ∆ Vby Θ (defined by\nα) andPcannot explain the enhancement of ∆ ˜V/Pin\nat low frequency. Here, Pcorresponds to the correction\nfactor related to the ellipticity trajectory of the magneti-\nzation precession of the uniform mode30due to the mag-\nnetic field configuration (in-plane). The analytical ex-\npression (red dash line in Fig.5a)) extracted from Ref.6,\nin which the spin current at the YIG/Pt interface is de-\nfined by the uniform mode, cannot reproducethe dc volt-\nagebehaviouratlowfrequency. AsreportedpreviouslyinRef.11,12, this behaviour has been attributed to the pres-\nence of non-linear phenomena. H. Kurebayashi et al.11\nhavedemonstratedthe possibilityto controlthe spin cur-\nrent at the YIG/Pt interface by three-magnon splitting.\nThis non-linear phenomenon can be easily actuated for\nvery low rf power31. Kurebayashi et al.11have observed\nthat the threshold power of the splitting in their system\nwas around 18 µW, which is very low with respect to the\nrf power used for FMR and dc voltage measurements.\nFig.5 b) introduces the frequency limit of the three-\nmagnon splitting boundaries calculated for our sample\n(200 nm) and for a thick sample of YIG. The split-\nting induces the creation of two magnons (with short-\nwavelength) from the uniform mode (long-wavelength),\nfollowing the equations: f=f1+f2andk=k1+k2,\nwherefandkare the frequency and wave vector with\nf1=f2=1\n2fandk1=−k223,31. In agreement with\nKurebayashietal.11,astrongenhancementofthedcvolt-\nage at low frequency has been observed, but this depen-\ndencedoes notnecessarilymeanthat three-magnonsplit-\ntingisinvolvedinoursystem. Byfollowingthe schemaof\nthe three-magnon, one can easily see that this phenom-\nena is allowed for a specific frequency range. The upper\nfrequency limit ( fcutoff) for the splitting is defined by the\nminimum of the BVSWM dispersion curve ( fmin) result-\ning from the competition between the dipole interaction\nand the exchange interaction. This minimum depends of\nthe thickness of the YIG sample.\nFor a thick sample of YIG, fmin≈fH, wherefHis the\nLarmor frequency. The FMR frequency cannot be lower\nthanfH32, and thus, the excitation frequency should be\nhigher than 2 fHin order that the process described by\nthe above equation can take place. Consequently, the\nupper frequency limit, fcutoff, for a thick sample system\nof YIG is fcutoff=2\n3fM, where fM=γµ0M. In the\nexperimentofKurebayashiet al.11, they haveused aYIG\nsample with a thickness of 5.1 µm, which is higher than\nthe exchange correlationlength, and therefore fmin≈fH.\nNevertheless, by taking into account a YIG thickness\nof 200 nm, the dependence of fmin(from Ref.29) shows a\nstrong difference with fH(see Fig.5 b)). The model of\nthe three-magnon splitting ( f >2fmin) suggested that in\nour case this process is not allowed. Fig.5 c) represents\nthe spin wave spectrum in YIG when the magnetic field\nis parallel to the wavevector, k, for different thickness\nof YIG29. The calculation has been performed with a\nmagnetic field of 40 mT inducing a microwave frequency\nf=2.66 GHz with γ=1.80 1011rad T−1s−1. A crossing\nof the dispersion curve with the black dotted line ( f/2)\nshows that the splitting is permitted. By reducing the\nthickness, the minimum frequency increases. For thin\nlayers of YIG, the dispersion curve does not cross the\nblackdottedlineanymore,suggestingthatherethethree-\nmagnon splitting is no longer allowed.\nThe role of the three-magnon splitting process for the\nspin pumping is not fully clear and there are many non-\nlinear phenomena which can induce the creation of spin6\nwaves with short-wavelength (multi-magnon processes\nsuch as four-magnon and two-magnon scattering). It\nhas been shown by Jungfleisch et al.14that the two-\nmagnon process (due to the scattering of magnons on\nimpurities and surfaces of the film) contributes to en-\nhance the spin current at the YIG/Pt interface. The\nstrong enhancement of ∆ Vobserved at low frequency\nis due to the fact that the dc voltage induced by spin\npumping at the YIG/Pt interface is insensitive to the\nspin waves wavelength11,14. In other words, ∆ Vis not\nonly defined by the uniform mode but from secondary\nspin wave modes, which present short-wavelength. It is\nnot obvious to identify the contributions of the different\nmulti-magnon processes, involved in our system, to the\nenhancement of the dc voltage at low frequency.\nIV. CONCLUSION\nIn summary, we have shown spin current emission in\na hybrid structure YIG [200 nm]/Pt [15 nm] as a func-\ntion of microwavefrequency f, microwave power Pinand\napplied magnetic field B(in-plane). We have observed a\nstrong enhancement of the voltage signal emission across\na spin current detector of Pt at low frequency. This be-\nhaviour can be understood if we assume that the mea-\nsured signal is not only driven by the FMR mode (which\ncontributestothespin-pumpingattheYIG/Ptinterface)\nbut also from a spectrum of secondary spin-wave modes,\npresenting short wavelengths.In YIG-based electronic devices, the creation of short-\nwavelength spin waves is considered as a parasitic effect.\nHowever, in this case it can be used as a spin current am-\nplifier. Before to integrate this system in a device, many\nquestions related to the contribution for the spin pump-\ning of the spin waves with short-wavelength should be\nsolved. To date, no systematic studies of the spin current\nemission on a YIG/Pt system havebeen done as function\nof the YIG thickness. By choosing a specific thickness\nrange, it should be possible to follow the contribution\nof the three-magnon splitting ( fcutoff) by a combination\nof Brillouin Light Scattering (BLS)11and spin pumping\nmeasurements. More details of other multi-magnon pro-\ncesses should be given by temperature dependence mea-\nsurements. Nevertheless, the enhancement of ∆ V, which\nwe have observed in the frequency range [0.6 - 3.2 GHz],\ncouldbeusedtodownscaleahybridstructureofYIG/Pt.\nThe isotropic behaviour of the in-plane magnetization,\nthe absence of stripe domains, and the high quality thin\nlayerofYIG(200nm)grownbyliquid-phase-epitaxygive\nkeys to success in this way.\nWe would like to acknowledgeN. Vukadinovic for valu-\nable discussions and B. Wolfs, M. de Roosz and J. G.\nHolstein for technical assistance. This work is part of the\nresearchprogram(Magnetic Insulator Spintronics) of the\nFoundation for Fundamental Research on Matter (FOM)\nand supported by NanoNextNL, a micro and nanotech-\nnology consortium of the Government of the Netherlands\nand 130 partners (06A.06), NanoLab NL and the Zernike\nInstitute for Advanced Materials.\n∗v.m.castel@rug.nl\n1A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-\nSu´ arez, A. B. Oliveira, and S. M. Rezende,\nJournal of Applied Physics 97, 10C715 (2005).\n2E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,\nApplied Physics Letters 88, 182509 (2006).\n3K. Ando, S. Takahashi, K. Harii, K. Sasage,\nJ. Ieda, S. Maekawa, and E. Saitoh,\nPhys. Rev. Lett. 101, 036601 (2008).\n4A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-\nSu´ arez, A. F. Lacerda Santos, and S. M. Rezende,\nPhys. Rev. B 83, 144402 (2011).\n5K. Harii, T. An, Y. Kajiwara, K. Ando,\nH. Nakayama, T. Yoshino, and E. Saitoh,\nJournal of Applied Physics 109, 116105 (2011).\n6Y. 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State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of \nSciences, Beijing 100083, China  \n2.Cornell University, Ithaca, New York 14850, USA  \n3. Kavli Institute at Cornell, Ithaca, New York 14850, USA  \n \n+These authors contributed equally to this work.  \n*ljzhu@semi.ac.cn  \n \nAbstract:  Efficient  manipulation of antiferromagnetically coupled materials  that are integration -friendly and have \nstrong perpendicular magnetic anisotropy (PMA) is of great interest for low -power, fast, dense magnetic storage \nand computing. Here, we report a distinct, giant bulk damping -like spin -orbit torque in strong -PMA ferri magnetic \nFe100-xTbx single layers that are integration -friendly ( composition -uniform, amorphous, sputter -deposited ). For \nsufficiently -thick layers, this bulk torque is constant in the efficiency per unit layer thickness, 𝜉DL𝑗/t, with a record -\nhigh value of 0.036 ±  0.008 nm-1, and  the dampinglike torque efficiency 𝜉DL𝑗 achieves very large values for thick \nlayers, up to 300% for 90 nm layers. This giant bulk torque by itself switches tens of nm thick Fe 100-xTbx layers \nthat have very strong PMA and high coercivity at current densities as low as a few MA/cm2. Surprisingly, f or a \ngiven layer thickness, 𝜉DL𝑗 shows strong composition dependence and becomes negative for composition where \nthe total angular momentum i s oriented parallel to the magnetization rather than antiparallel. Our findings of giant \nbulk spin torque efficiency and intriguing torque -compensation correlation will stimulate study of such unique \nspin-orbit phenomena in a variety of ferrimagnetic hosts . This work  paves a promising avenue for developing \nultralow -power, fast, dense ferrimagnetic storage and computing devices.  \nKeywords : spin -orbit torque, spin current, inversion symmetry breaking, ferromagnetic resonance, \nferrimagnetism  \n \nFerrimagnetic materials  can host a variety of  exotic  properties that are promising for technolog y, e.g.  \nmagnetization/angular momentum compensation, giant perpendicular magnetic anisotropy  (PMA)[1-4], ultrafast \nmagnetic domain wall velocit ies[5-7], reduced sensitivity to stray magnetic fields than ferromagnets, and easier and \nfast detection than antiferromagnets . Therefore , ferrimagnets ( FIMs ) are potentially advantageous for dense and \nfast magnetic recording, memory, and computing applications  [8,9]. However, successful i ntegration of FIMs  into \nhigh-performance functional devices  requires efficient manipulation of strong -PMA  and integration -friendly FIMs , \nwhich has remained a big challenge. So far, electrical switching of FIMs by interfacial spin -orbit torque (SOT)[11] \nis only reported in samples  with poor PMA, small coercivities (e.g. Hc < 0.2 kOe for GdFeCo[3,10,12], Hc <0.25 kOe \nfor CoTb[13,14]), and low SOT efficienc ies (𝜉DL𝑗 = 0.017 for Pt/CoTb[15]). Moreover, the yet-known bulk SOT s [16-\n22] in magnetic single layers ha ve low efficiency  per unit film thickness [20-27] and/or usually require a  single -crystal \nstructure  [17] and a composition gradient[16,18,19,22], which limit  practical application .  2 \n Here , we report a record -strong bulk SOT  within single layers of  Fe100-xTbx alloys (FeTb for short) that have \nstrong  PMA , spatially -uniform, amorphous structure, and are grown simply by sputtering on to oxidized silicon \nsubstrate s, which are highly desirable for large -scale and high -density integration with CMOS circuits . \nSurprisingly, t he efficiency of the  bulk torque varies rapidly as a function of stoichiometry and reverses sign in \nbetween the compensation points for magnetization and angular momentum . We also show that t his bulk SOT can \nswitch  tens of nm thick FeTb layers  with strong PMA and high Hc at extremely  low current densit ies. \n \nSample characterizations  \nFor this work,  we deposit  a series of Fe 100-xTbx single layers with different thicknesses and Tb volume \npercentages ( x = 15-78) by co -sputtering at room temperature. Each sample is capped by a MgO (1.6 nm)/Ta (1.6 \nnm) bilayer that is fully oxidized upon exposure to the atmosphere [see the electron energy loss spectrum (EELS) \nresults in Ref. 26].  For measurements of SOT and SOT -induced switching, the layers are patterned into Hall bars \nthat are 60 μm long and 5 μm wide.  More details  about sample preparation and characterization can be found in \nSupplementary Sec. 1. These FeTb alloys have an amorphous and homogeneous texture and reasonably sharp \ninterfaces, as indicated by c ross-sectional scanning transmission electron microscopy (STEM) measurements  [see \nFig. 1(a)  for the example of a Fe57Tb43 sample ( 66 nm) ]. Electron energy loss spectroscopy  (EELS) measurements \ndemonstrate that there is no obvious variation in the ratio of Fe to Tb concentrations through  the thickness of the \nfilms  (Fig. 1(b) ). There is an  overall  variation in the absolute EELS intensities of Fe and Tb  due to variations in \nthickness along the sample wedge created during the ion -beam thinning process (see Supplementa l Sec. 2 and ref. \n[27]), but this should not be misinterpreted as a composition gradient . There is also no obvious indication of \noxidization of the FeTb from the EELS imaging, which is consist with the previous observation[16] that the MgO \n(1.6 nm)/Ta (1.6 nm) bilayer well protect s the magnetic layers from oxidization . \nThe FeTb samples have strong PMA and square perpendicular magnetization hysteresis loops over a  wide \ncomposition range , x = 29-61, when the thickness is greater than 7 nm [ Fig. 1(c) and Supplementary Fig. S3]. The \nperpendicular magnetic anisotropy field ( Hk) is very high and reaches values as large as  100 kOe for 20 nm films \nnear x = 47 [Fig. 1(d) ]. In Fig. 1(e)-1(g) we summarize the saturation magnetization ( Ms), the coercivity ( Hc), and \nthe anomalous Hall resistance ( RAHE) for 20 nm Fe 100-xTbx samples as a function of x. As expected, Ms shows a \nstrong, “V -shaped” variation with x, which is a characteristic of the competing magnetic moment contributions of \nthe antiparallel Fe and Tb sub -lattices. From the fit of the data to the relation Ms = (1-0.01x) MFe – 0.01x M Tb (see \nthe orange  solid line in Fig. 1( e)), we obtain MFe = 824 ± 40 emu/cm3 (≈ 1.05 µB) and MTb = 940 ± 80 emu/cm3 (≈ \n3.14 µB) of our samples. The values of MFe and MTb of our disordered thin films are smaller than typical bulk \nvalues ( i.e., 2.20 µB/Fe or 9.72  µB/Tb[8]). The magnetization compensation point  (xM ≈ 47  for our 20 nm FeTb \nsamples at room temperature ), at which Ms vanishes  and Hc appears to diverge , also diffe rs from previously \nreported  values for 20 nm FeTb  grown on Pt ( xM ≈ 25)[28-30].  However, it is common that the magnetic properties \nof thin films of ferrimagnetic alloy s can vary  depending on growth protocols and substrate choices . For example, \nxM of CoTb  at room temperature  has been reported to be ≈ 22[13], ≈ 35[31], ≈ 44[32] when the CoTb is grown on Ta, \nPt and SiN, respectively. We also observe that the magnetic properties of our FeTb samples are sensitive to the \nlayer thickness  [Supplementary Fig . S3(c)-(e)], in line with previous reports[30]. As expected , RAHE for the 20 nm 3 \n FeTb is negative for x < xM but positive for x > xM (see Supplementary Fig . S4(a) for more details ) because, when \nx < xM (x > xM), the Fe moment is parallel (antiparallel) to the total magnetization and to a strong  applied \nperpendicular magnetic field ( Hz). The 3d states of Fe govern the anomalous Hall effect because the 4 f states of \nTb is expected to be located well below the Fermi level and are less involved in transport phenomena of FeTb.  \n \nGiant bulk spin -orbit torque  \nWe measure the efficiencies of SOTs in the perpendicularly magnetized FeTb samples using  the polar -angle \ndependent  harmonic Hall voltage response (HHVR) technique[33,34], after carefully taking into account current -\ninduced heating and  thermoelectric effect s (Supplementary Sec. 5 and 6 ). This HHVR  technique is accurate when  \nthe magnetization rotate s coherently at small polar angles  (𝜃M). This condition is fulfilled in the FeTb samples , as \nindicated by a well-defined parabolic scaling of the first harmonic Hall signal versus 𝜃M (Supplementary Fig . S8). \nTo determine the dampinglike SOT, we rotate the magnetization by  scanning  a fixed magnitude of  magnetic field \n(𝐻𝑥𝑧) relative to the sample at small values of θM in the x-z plane ( Fig. 2a ), and collect the first and the second \nHHVRs, Vω and  V2ω, as a function of θM under the excitation of  a low-frequency sinusoidal electric field E in the x \ndirection . As we discuss in detail in the Supplementary Sec. 6, the HHVR signal s are given by  \nVω = VAHE cos 𝜃M,                                               (1) \nV2ω ≈ (1\n2𝑉AHE𝐻DL\n𝐻k+𝐻𝑥𝑧 + 𝑉ANE ,𝑧) sin  𝜃M+𝑉ANE ,𝑥,   (2) \nwhere 𝑉AHE is the anomalous Hall voltage, 𝑉ANE ,𝑧(𝑥) is the anomalous Nernst voltage induced by an out-of-plane  \n(in-plane)  temperature gradient,  and 𝐻DL is damping -like effective SOT field.  As shown in Fig. 2(b), the measured \nV2ω varies linearly with sin𝜃M for each fixed magnitude of Hxz. The value of HDL can be obtained from the fits of \ndata to Eq. ( 2) as shown in Fig. 2(c). In this determination we ignore the so -called “planar Hall correction”[35] \nbecause the planar Hall resistance ( RPHE) samples is negligibly small compared to RAHE (|RPHE/RAHE| ≤0.04, \nSupplementa ry Fig. S9), and even if this were not the case the  planar Hall correction is  generally  found to give \nincorrect values when it is not negligible[36-38]. As shown in Fig. 2(d) , HDL for the FeTb single layers with different \nx increase much more slowly than 1/Ms scaling upon approaching the  magnetization  compensa tion point  (xM ≈ \n47), which is in sharp contrast to the behavior observed for HM/FM bilayers in which HDL is proportional to 1/ Ms. \nHDL reverses sign betwee n xM and the angular momentum compensation point ( xA ≈ 38, see below ).  \nUsing the obtained HDL values,  we calculate 𝜉DL𝑗 of these FeTb single layers following : \n𝜉DL𝑗 ≡ js /j = (2𝑒/ℏ)𝐻DL𝑀s𝑡/𝑗  (3) \nwhere e is the elementary charge, ћ the reduced Plank’s constant, Ms the saturation magnetization of the spin \ncurrent detector,  t the thickness of the spin current detector, js the spin current density absorbed by the spin current \ndetector,  and j = E/ρ xx the current density in the spin current generator  with electrical resistivity  ρxx (Supplementary \nFig. S10).  As we  justify in the Supplementary Sec. 10, Eq. (3) hold s for FIMs  regardless of the sign of  effective \ngyromagnetic ratio ( γeff). As plotted in Fig. 2(e) , 𝜉DL𝑗 of the 20 nm FeTb first increases rapidly from +0.11 at x = \n29 to +0.41 at x = 37, then  (like 𝐻DL) suddenly  becomes negative for 38 < x < 47, and finally becomes positive 4 \n again and starts to decrease  from  the value  + 0.16 at x = 49 upon further increase of x (see Supplementary Sec. 11 \nfor m ore details of the torque determination ).  This sign reversal of the dampinglike spin -orbit torque is re affirmed  \nby the opposite polarity of the current -induced magnetization switching of the Fe 57Tb43 and the Fe 67Tb33 \n(Supplementary Sec. 16). We find that the sign reversal appears to be correlated to that of the angular momentum. \nIn Fig. 2(f), we show the effective gyromagnetic ratio  for the 20 nm FeTb with different composition as calculated \nusing the relation [39-41] 𝛾eff=(𝑚Fe−𝑚Tb)/(𝑚Fe/|𝛾Fe|−𝑚Tb/|𝛾Tb|), the magnetic moments of the two \nsublattices mFe = (1-0.01x)MFe and mTb = 0.01 xM Tb, and the individual gyromagnetic ratios γFe = -2.1μB/ℏ[42] and \nγTb = -1.5μB/ℏ[43]. The composition  of the angular momentum compensation point , where  the total angular \nmomentum S = 𝑚Fe/|𝛾Fe|−𝑚Tb/|𝛾Tb| is zero, is estimated to be xA ≈ 38.5 for the 20 nm FeTb at the room \ntemperature . We note that 𝛾eff, xA, and  xM in Figs. 2(d) -2(f) are only the 20 nm FeTb samples and different from \nthat for the thicker films (e.g. for the films in Fig. 2(g), xA < xM < 43). The FeTb also shows a  field-like torque that \nis relatively small compared to the damping -like torque (Supplementary Fig . S12). \n \nBulk characteristics and m icroscopic origin  \nTo analyze these data , we first show that  the strong damping -like torque we observe within FeTb is a bulk \neffect. Qualitatively similar to previous measurements  of CoPt  single layers[26], 𝜉DL𝑗 of the FeTb  layers  increases \nlinearly with layer thickness when the thickness is greater than  about  40 nm  as shown in Fig. 2( g) for Fe 57Tb43, \nyielding in the bulk limit  a SOT efficiency per thickness of 𝜉DL𝑗/t = 0.036 ±  0.008 nm-1. This behavior is not \nconsistent with an interfacial torque , for which  𝜉DL𝑗 should be approximately independent of the magnet ic-layer  \nthickness[24,44]. We also find that this torque is insensitive to the details of the sample interfaces because we measure \nessentially the same value of 𝜉DL𝑗 from symmetric MgO/Fe 61Tb39 20 nm/MgO samples and asymmetric \nSiO 2/Fe 61Tb39 20 nm/MgO samples. We thus conclude from these characteristics that the damping -like spin torque \nin FeTb single layers is a bulk effect . This bulk torque is microscopically distinct from the previously reported \n“interface -engineered” self -torque concluded from a study of GdFeCo [22].  \n We suggest that the source of the strong damping -like SOT in the perpendicularly magnetized  FeTb is most \nlikely a strong conventional  bulk spin Hall effect  (SHE). We have considered the possibility of origins associated \nwith the anomalous Hall effect or planar Hall effect , but these  can only generate spin polarization  collinear with \nthe magnetization[45-47]. Magnetic and antiferromagnetic spin Hall effects[48,49] are also not relevant because they \nare odd under time reversal , while we find the damping -like torque  efficiencies  generated  in FeTb for a given \napplied electric field does not reverse  orientation  when the magnetization reverses. We have  further verif ied the \nexistence of  a strong SHE in FeTb by measuring the spin current emi tted by FeTb layers . We perform ed thickness -\ndependent spin -torque ferromagnetic resonance (ST -FMR) experiments[50,51] on a control sample of Fe 50Tb50 (20 \nnm)/Ti  (1 nm)/Fe ( 3.8-10.5 nm)  and used the in -plane magnetized Fe layer to detect the spin current emitted from \nthe FeTb ( Fig. 2( h)). Here, the 1 nm Ti spacer layer was used to suppress the exchange coupling between the FeTb \nand the Fe layers  (Supplementary Fig. S13(b) ). The PMA FeTb produces no measurable  FMR  excitation  under  \nthe condition of  small in-plane magnetic field,  so the ST -FMR  signal  we measure from  the FeTb /Ti/Fe trilayers \ncorresponds only to magnetic dynamics from the Fe  layer . If we define the apparent FMR spin -torque efficiency 5 \n (𝜉FMR) from the ratio of the symmetric and anti -symmetric components of the magnetoresistance response of the \nST-FMR  (Supplementary Sec. 13), the actual  efficiency of the damping -like torque acting on the Fe layer due to \nthe spin current emitted by the  Fe50Tb50 (𝜉DL,ext𝑗\n ) can be  determined by the method of ref. [ 51] based on the y -axis \nintercept in a linear fit of 1/ ξFMR versus 1/ tFe. As shown in Fig. 2( i), we measure  𝜉DL,ext𝑗\n = 0.16 ±  0.02  for \nFe50Tb50/Ti/Fe, which  is 3 times stronger than that of Pt/Fe bilayers (0.051 ±  0.002 , also  shown  in Fig. 2(i) ) for Pt \nwith resistivity 38 μΩ cm .  We have also measured 𝜉DL,ext𝑗\n for x = 43 (where 𝜉DL𝑗 is negative) and for x = 29 (on \nthe other side of the  angular momentum  compensation point where 𝜉DL𝑗 is positive again), and we find that the sign \nof 𝜉DL,ext𝑗 is unambiguously positive at all three concentrations. The value of  𝜉DL,ext𝑗\n that we quote  for Fe 50Tb50 \nonly represents a lower bound for the internal value of spin Hall  ratio because the torque applied to the Fe is \nreduced by  spin attenuation in  the Ti spacer[26,52], interfacial spin backflow[11], and spin memory loss[53].  Spin \nmemory loss, in particular, should be significant at the Ti/Fe interface because it possesses  strong interfacial  spin-\norbit coupling[11] as indicated by the large interfacial magnetic anisotropy energy  density  of 1.43 ± 0.05 erg/cm2 \n(Supplementary Sec. 14). \nA non -zero SOT in a single magnetic layer requires that the sample structure is not symmetric  relative to a \nmirror parallel to the  sample plane[16,26]. The required broken symmetry within the FeTb layers seems unrelated  to \nany vertical composition gradient  because  there is  no evidence  of a composition gradient  in the EELS studies of  \nour films. We also find that a deliberately introduced  vertical composition gradient does not enhance the damping -\nlike torque in FeTb. A control sample of 20 nm thick Fe 100-xTbx in which x varied from 27 to 41 with thickness  \ngave 𝜉DL𝑗 of 0.05± 0.01 (Supplementary Fig. S15), which is similar to the average d value of whole  film over the \nthickness using the composition -dependent values in Fig. 2(e) , but significantly smaller  in magnitude  than -0.46 \nfor 20 nm Fe 61Tb39. In addition,  the source of the symmetry breaking is not a  vertical thermal gradient because the \nmagnitude of HDL scales in proportion to the applied electric field ( Supplementary Fig. S7) and thus 𝜉DL𝑗 is \nindependent of the applied electric field  (symmetry breaking due to Joule heating would give 𝜉DL𝑗∝𝐸2).  \n  \nStrong composition dependence and sign change  \nWe now turn to analyz e the dependence on x of the bulk anti -damping spin torque efficiency 𝜉DL𝑗 for Fe 100-\nxTbx (Fig. 2(e) ). We observe that  |𝜉DL𝑗| for the 20 nm Fe 100-xTbx samples shows a broad peak around x = 43, \nsuggesting an enhanced SHE in the intermediate composition range , near and between the two compensation \npoints for magnetization and angular momentum . |𝜉DL𝑗| reaches 0.5 for 20 nm  Fe57Tb43 films  and 3 for 90 nm  \nFe57Tb43 films .  Our result differs  from the case of GdFeCo, which was reported to have zero self -torque at the \ncompensation point of angular momentum in a previous temperature -dependen ce study  [22].   \nThe sign change of 𝜉DL𝑗 that we observe  in the 20 nm Fe 100-xTbx samples  between the compensation points for \nmagnetization and angular momentum appears to be correlated with the relative orientation  of the magnetization \nand angular momentum  vector .  Outside the region between the two compensation points  the magnetization \n(𝑚𝐹𝑒−𝑚𝑇𝑏) and angular momentum  (𝑠Fe−𝑠Tb) are antiparallel  (𝛾eff<0), but between the compensation points 6 \n the total magnetization  becomes parallel to the total angular momentum  (𝛾eff>0). A change in the sign of 𝜉DL𝑗 \nindicates a change in the sign of the spin angular momentum being transferred to the magnet.  However, our ST -\nFMR measurements on Fe 100-xTbx/Ti/Fe indicate no sign change in the polarization of the spin current emitted from \nFeTb  regardless of composition.  The microscopic origin of the sign change of 𝜉DL𝑗 remains a puzzle  and worth  \nstudy in the future . \n \nPractical impact and self -torque -driven magnetization switching  \nFrom technological point of view, a strong bulk torque can be advantageous by itself or in combination with \ninterface -applied torques for applications, such as perpendicular magnetic recording and chiral domain \nwall/skyrmion devices,  that require relatively large thickness for high thermal stability. The damping -like SOT \nefficiency per  unit thickness  that we measure in the bulk limit for Fe 57Tb43, 𝜉DL𝑗/t ≈ 0.036 nm-1, is much  greater \nthan previous reports  for other magnetic  single layers, e.g ~ 0.0017 nm-1  for in-plane NiFe[25], ~ -0.008 nm-1 for \nin-plane  CoPt[26], ~ 0.005 nm-1 for in -plane FePt[16], and ~ 0.016 nm-1 for perpendicular GdFeCo[22]. Here we do \nnot compare our 𝜉DL𝑗/t result with those out -of-plane HHVR results  obtained by applying a large “planar Hall \ncorrection” (e.g. 0.045 nm-1 for L10-FePt single crystals in ref. [19]), because , as we noted above,  the planar Hall \ncorrection is generally foun d to give incorrect values when it is not negligible  [36-38]. \nThe bulk SOT of FeTb  is sufficiently strong to drive SOT switching of layers with very large thicknesses and  \nstrong PMA . In Fig. 3(a)-3(d) we compare magnetic -field-driven switching and SOT switching  for both a 20 nm \nFe67Tb33 device (x < xA, Fe-dominated, Ms = 250 emu/cm3, Hk=33.5 kOe, Hc=1.59 kOe) and a 20 nm Fe 42Tb58 \ndevice (x > xM, Tb-dominated, Ms= 16 4 emu/cm3, Hk=17.5 kOe, Hc=1.72 kOe) as two representative examples. \nFigs. 3(a) and 3(b) show the Hall resistance ( RH) of the samples as a function of Hz, which indicate sharp full \nswitching for both samples with ΔRH = 2RAHE = +11 Ω (-12.2 Ω) for the Fe- (Tb-) dominated sample . In Figs. 3(c) \nand 3(d), we show RH of the two samples  measured following the application of sequences of current pulses of \ndifferent amplitudes  (0.2 seconds in duration) under the application of a  constant  symmetry -breaking  in-plane bias \nfield Hx along the current direction ( x direction). We measure a  switching current density of only (8.2± 1.2) × 106 \nA/cm2 for the 20 nm Fe 67Tb33 and (5.5± 0.2) × 106 A/cm for the  20 nm  Fe39Tb61. The current -driven switching is \nonly partial ( ∼16% of the full value of Δ RH for magnetic -field-driven switching) , likely because the non-uniform \npinning impedes free motion of domain walls in this domain -wall-mediated switching regime.  Full current -driven \nreversal is likely still possible  in nanodot device s with improved magnetic homogeneity  as recently demonstrated \nin CuPt/CoPt bilayers[54]. Here,  the switching chirality is opposite for the Fe -dominated  Fe67Tb33 and Tb -dominated  \nFe42Tb58, i.e., clockwise (anti -clockwise) for the former but anti -clockwise (clockwise) for the latter when Hx > 0 \n(Fig. 3(c) -3(d)). This is because the SOT fields are of the same sign for the two samples  (𝜉DL𝑗> 0), but the \nanomalous Hall resistances are of opposite sign s (ΔRH > 0 for Fe 67Tb33, but <0 for Fe 42Tb58). We also note that the \n20 nm Fe 57Tb43 (xA < x < xM, ΔRH > 0, 𝜉DL𝑗< 0) can be  also switched at a low current density of (5.5±0.1)× 106 A/cm  \n(Supplementary Fig. S16(c) ), but the switching polarity is opposite to that of the Fe 67Tb33 (x < xA, ΔRH > 0, 𝜉DL𝑗> \n0) due to the negative sign of the bulk spin-orbit torque  in the Fe 57Tb43. 7 \n   \nConclusion  \nWe have demonstrated a giant damping -like SOT arising from the SHE in composition -uniform, amorphous,  \nsputter -deposited  ferrimagnetic Fe100-xTbx single layers with giant PMA . This bulk torque exhibits no apparent \ncorrelation to the interfaces or the absence/presence of a composition gradient. The torque reaches a constant value \nof efficiency per  unit layer thickness in the bulk limit , 𝜉DL𝑗/t ≈ 0.036 nm-1. This is more than twice greater  any \nprevious report for other magnetic single layers.  The torque varies strongly with composition and achieves giant \nefficiencies  |𝜉DL𝑗| of 0.5 for 20 nm  Fe61Tb39 and 3 for 90 nm  Fe57Tb43. Interestingly , the torque become s negative \nin sign in the intermediate composition range where  total angular momentum becomes parallel to the  \nmagnetization rather than antiparallel . We also show that the bulk SOT can  drive  switch ing in tens of nm thick \nFeTb layers with strong PMA and high coercivity . For example, the bulk SOT can switch a 20 nm  FeTb  at very \nlow current densit ies of a few M A/cm2. Our findings of giant bulk SOT efficiency and intriguing torque -\ncompensation correlation will stimulate study of such unique spin -orbit phenomena in a variety of ferrimagnetic \nhosts. Our work suggest s a promising strategy  for self -driven -switching  perpendicular ferrimagnetic devices with \nlow power , high density,  and straightforward  integration with CMOS circuit s because there is  no requirement for \nepitaxy  or composition gradient . \n  \nData availability  \nThe data that support this study are available from the corresponding author upon reasonable request.  \nAcknowledgement s \nThe authors thank Robert A. Buhrman for support. This work was funded  in part by the Office of Naval \nResearch (N00014 -19-1-2143 ), in part by the Defense Advanced Research Projects Agency \n(USDI  D18AC00009),  and in part by the NSF MRSEC program (DMR -1719875) through the Cornell Center for \nMaterials Research . Device fabrication was performed at the Cornell Nanofabrication Facility, in part by the NSF \n(NNCI -2025233 ) as part of the National Nanotechnology Coordinated Infrastructure , and in part by the Strategic \nPriority Research Program of the Chinese Academy of Sciences (XDB44000000). Q. Liu acknowledges the  \nfinancial  support by  the China Scholarship Council (File No. 201906460052).  \n \nConflict of Interest  \nThe authors declare no conflict of interest.  \n \nAdditional Information  \nSupplementary Information is available for this paper at xxxxx . \n \nReferences   \n[1] C. Kaiser, A. F. Panchula, S. S. P. Parkin, Phys. Rev. Lett. 95, 047202  (2005 ).  \n[2] H. Awano, J. Magn. Magn. Mater. 383, 50  (2015) .  \n[3] N. Roschewsky, T. Matsumura, S. Cheema, F. Hellman, T. Kato, S. Iwata, S. Salahuddin,  Appl. Phys. Lett. 8 \n 109, 112403  (2016) . \n[4] J. Finley, C. Lee, P. Y . Huang,  L. Liu, Adv. Mater. 31,  1805361  (2019) . \n[5] K. Kim, S. K. Kim, Y . Hirata, S. Oh, T. 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(d ) Perpendicular magnetic anisotropy field ( Hk), (e) \nSaturation magnetization ( Ms), (f) Coercivity (Hc), and (g) Anomalous Hall resistance ( RAHE) of Fe 100-xTbx films \nwith varying Tb concentration  (x). Here, the values of Hk, H c, and RAHE are determined from transport \nmeasurements.  \n \n \n \n11 \n  \nFig. 2 Spin -orbit torque s. (a) Geometry of the HHVR measurement.  (b) Second HHVR  V2ω vs sin θM for a 20 nm \nFe71Tb29 sample under  for constant magnitudes of  applied  magnetic field Hzx = 20 and 80 kOe . (c) dV2ω/dsinθM vs \nVAHE/2(Hk+Hzx) for 20 nm Fe 71Tb29. Dependence on the Tb concentration x for (d) the d amping -like effective SOT \nfield HDL, (e) the d amping -like torque efficiencies per current  density  𝜉DL𝑗 and |𝜉DL𝑗|, and (f) the calculated value of \n𝛾eff for the 20 nm Fe 100-xTbx. In (d) -(f), the dashed  lines indicate  the angular momentum compensation point xA \n(blue dashed line) and  the magnetization compensation point xM (red dashed line). ( g) 𝜉DL𝑗vs. the thickness (t) of \nFe47Tb43 samples  (𝜉DL𝑗> 0 because the composition  x = 43 is located in the Tb -dominated regime when the thickness \nis greater than ≈30 nm, see the Supplementary Fig. S4(e) ). (h) Schematic of ST-FMR measurements on Fe50Tb50 \n(20 nm)/Ti (1 nm)/Fe ( tFe) samples.  (i) Inverse FMR efficiency (1/ ξFMR) vs. inverse Fe thickness (1/ tFe) for the \nFe50Tb50 (20 nm)/Ti (1 nm )/Fe (tFe nm) sample  and a control sample Pt  (4 nm)/Fe ( tFe). \n \n12 \n  \nFig. 3 . Anomalous Hall resistance  hysteresis  of (a) a 20 nm thick Fe67Tb33 single layer (x<xA, Fe-dominated,  Ms = \n250 emu/cm3, Hk=33.5 kOe, Hc=1.59 kOe ) and (b) a 20 nm thick Fe42Tb58 single layer (x>xM, Tb-dominated, Ms= \n164 emu/cm3, Hk=17.5 kOe, Hc=1.72 kOe ). Current induced magneti zation  switching of (c) the Fe67Tb33 and (d) \nthe Fe42Tb58, under  a constant  in-plane bias field  Hx = ±3 kOe  that overcomes the DMI field within the domain \nwalls .  \n \n1 \n Supplementary Information  for  \n \nGiant bulk spin-orbit torque  and efficient electrical switching in  single  ferrimagnetic FeTb  layers  \nwith strong perpendicular magnetic anisotropy  \n \nQianbiao Liu1,2+, Lijun Zhu1,2+*, Xiyue S. Zhang2, David A. Muller2, Daniel C. Ralph2,3 \n \n1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of \nSciences, Beijing 100083, China  \n2.Cornell University, Ithaca, New York 14850, USA  \n3. Kavli  Institute at Cornell, Ithaca, New York 14850, USA  \n \n+These authors contributed equally to this work.  \n*ljzhu@semi.ac.cn  \n \n \nTable of Contents:  \nSection 1. Sample fabrication and characterizations  \nSection 2. Thickness gradient of FIB -thinned STEM samples  \nSection 3. Magnetic properties of Fe 100-xTbx single layer s \nSection 4. Anomalous Hall resistance and coercivity of Fe 100-xTbx single layer  \nSection 5. Estimation of current -induced temperature increase  \nSection 6. Subtraction of effects of anomalous Nernst vo ltage from harmonic Hall voltage response   \nSection 7. Coherent rotation at small polar angles  \nSection 8. Validation of Eq. (3) in the main text  \nSection 9. Planer Hall resistance   \nSection 10. Resistivity of the 20 nm Fe 100-xTbx \nSection 11. Raw data for representative samples exhibiting different torque signs  \nSection 1 2. Field -like spin -orbit torque in Fe 100-xTbx single layer  \nSection 1 3. Spin -torque ferromagnetic resonance measurements  \nSection 1 4. Interfacial  perpendicular magnetic anisotropy energy density  \nSection 1 5. HHVR measurement  of a 20 nm thick composition -gradient Fe100-xTbx (x = 27→41)  \nSection 1 6. More examples of magneti zation  switching  by the bulk spin -orbit torque  \n \n \n \n \n \n \n \n \n 2 \n Section 1. Sample fabrication and characterization s \nA series of Fe 100-xTbx (FeTb for short) single layers with different fixed Tb volume percentages ( x = 15-\n78) were  deposited on oxidized Si substrate s by co -sputtering at room temperature , and then protected  by capping \nwith MgO (1.6 nm)/Ta (1.6 nm)  without breaking vacuum . The Tb volume percentages were  calibrated from the \nfraction of the Tb deposition rates over the total deposition rate during the co -sputtering deposition. The argon \npressure was 2 mTorr during the sputteri ng process, and the base pressure was ~10-9 Torr. To make devices for \ndetermin ing the efficiencies of the spin-orbit torques by  harmonic Hall  voltage  response  (HHVR) measurements \nand spin -torque ferromagnetic resonance (ST-FMR) measurements , the layers  were patterned by photolithography \nand ion milling into Hall bars (5×60 μm2, Fig. S1a ) and simple  microstrips (10×20 μm2, Fig. S1b ) followed by \ndeposition of 5 nm Ti and 150 nm Pt as electrical contacts .  \n \n \nFigure S1 . Optical images of (a) a Hall bar  device  and (b) a ST-FMR device . \n \nThe magnetic properties  of the films  were characterized using a superconducting quantum interference \ndevice -vibrating sample magnetometer (SQUID -VSM). The sample  for scanning transmission electron \nmicroscopy (STEM) imaging and electron energy loss spectrum (EELS) measurements was thinned using a \nfocused ion beam (FIB) system using a FEI/Thermo Fisher Titan Themis STEM system at 300 kV. Anomalous \nHall resistance and planar Hall resistance were measured using  a physical property measurement system (PPMS). \nDuring HHVR  measurements, a Signal Recovery DSP Lock -in Amplifier Model 7625 was used to source a \nsinusoidal current  onto the Hall bars and to detect the first and second harmonic Hall voltage responses. During \nthe switching measurements, a pulsed write current with a duration of 0.2 ms was source d to the Hall bar devices \nusing a Keithley 2400 . The anomalous Hall voltage was detected either by a Keithley 2182A after each pulse d \nwrite current (with a reading current of 0.1 mA ) or by a  Signal Recovery DSP Lock -in Amplifier Model 7625  \n(with a reading excitation  of 0.1 V ). For S T-FMR measurements, the rf current was sourced by signal generator  \n(E8257D ) and the mixing voltage was detected using a Signal Recovery DSP Lock -in Amplifier Model 7625. All \nthe measurements were performed at room temperature . \n \n \n3 \n Section 2. Thickness gradient of the FIB-thinned STEM sample  \n \nOur standard FIB thinning process typically results in a thickness gradient in  the cross -sectional view  of \nSTEM sample s, with the top side of film thinner than the substrate side. This thickness gradient may cause \nsignificant variation of thickness -sensitive signals (e.g. the absolute intensity EELS), but it usually does not affect \nthe relative strengths of the signals  from  different elements . Therefore, the quasi -linear variation of the absolute \nEELS intensity of the 66 nm  FeTb sample in Fig. 1(b) of the main text is simply due to the thickness gradient \nformed during the FIB thinning process and should not be misinterpreted as a composition gradient.  \nFor a detailed understanding of the formation of the thickness gradient, we now briefly describe the  standard \nFIB thinning process we employ in this work. We first protect the initial thin -film sample surface by depositing \n20-30 nm carbon and 0.8 -1 µ m Pt and prepare a lamella on the specimen using  a Ga ion beam. The lamella is \ntransferred to a TEM grid u sing a needle and then fixed by sputtering the “ paste ” Pt ( Fig. S2(a)), followed by \nrepeatedly cleaning the cross -section by 30 keV Ga ion milling from both sides until the thickness is down to 200 -\n500 nm. The lamella is then  thinned further from both side s using 5 keV Ga ions and smaller milling box (see Fig. \nS2(b) until the protective Pt layer on the top is reduced to  0-20 nm ). Because the  lamella is tilted 2 -3 degrees \nduring the Ga ion beam thinning process, the end result includes  a thickness gradient with the top of the lamella  \nthe thinnest.  \n \n \nFigure S2 FIB thinning process . (a) Carton of a lamella attached to the TEM grid before thinning. (b) Scanning \nelectron microscopy of a lamella after thinning from both sides until the top protective Pt layer reaches 0 -20 nm. \nThe thinning box is moved gradually toward the top and the inside of the lamella . A thickness gradient with the \ntop side thinnest is finally formed because of the tilted thinning from both sides . \n \n  \n4 \n Section  3. Magnetic properties of  Fe100-xTb x single layers  \n \nWe obtained FeTb  single layers with square perpendicular magnetization hysteresis loops over a wide \ncomposition range (the Tb composition  x = 29-61, Fig. S3(a)) and over a wide thickness range (even down to 7 \nnm, Fig. S3(b)). The saturation magnetization and the perpendicular magnetic anisotropy field of the FeTb can be \ntuned strongly by the composition ( Figs. S3(c) and S3(d)) or by the layer thickness  (Figs. S3(e)). The thickness \ndependence of the magnetic properties was interesting and was attributed to a varying alignment of the Tb \nmagnetic moments  with the layer thickness  in a previous report  [S1].   \n \n \nFigure S3. Magnetic properties of Fe 100-xTbx single layer s as measured by SQUID -VSM . The out -of-plane \nmagnetization hysteresis loops ( M-H curves ) of (a) 20 nm -thick Fe100-xTbx single layer s with different Tb \nconcentration s (x = 31 -61) and (b)  a 7 nm thick Fe57Tb43. (c) The s aturation magnetization ( Ms) and (d) the \nperpendicular magnetic field (Hk) of the 20 nm Fe100-xTbx thin films  plotted as a function of the Tb concentration . \nThe values of Hk around x = 47 are not measureable by SQUID . (e) Ms and the perpendicular coercivity  of the \nFe57Tb43 films plotted as a function of the layer thickness.  \n  \n5 \n Section 4. Anomalous Hall resistance and coercivity of Fe100-xTb x single layer s \n \nFigure S4. (a) Anomalous Hall resistance ( RH) hysteresis  of 20 nm Fe100-xTbx with different composition s. RH is \nnegative for x < xM and positive for x > xM (xM = 47 for the  20 nm films ).  (b) The coercivity  of the 20 nm Fe100-\nxTbx as measured from the electrical measurement (red points) and magnetic hysteresis measurement (black points) . \nThe coercivity  difference around the magnetic compensation point is attributed to  enhanced domain pinning  during \ndevice fabrication . Anomalous Hall resistance hysteresis of (c) 7 nm , (d) 20 nm and ( e) 66 nm Fe 57Tb43. (f) \nAnomalous Hall resistance and coercivity of the Fe57Tb43 films with different thicknesses . Thickness -induced sign \nreversal of the anomalous Hall resistance  indicat es a compensation thickness of ≈30 nm for the Fe57Tb43. The bias \ncurrent is 0.1 mA during the measurement so that Joule heating  is negligible.  \n \nSection 5. Estimation of current -induced temperature increase  \nWe can estimate the temperature  increase of our  FeTb samples induced by Joule heating  during  \nmeasurements of  spin-orbit torque and  current -driven  switching by simultaneously measuring the overall change \nin the sample resistance . We first measure the resist ance of the Hall bar devices as a function of temperature using \na small dc bias of 0.1 mA (see Fig. S5(a) for example of 20 nm Fe 59Tb41) so that the result is not affected by \ncurrent -induced heating . Then, the temperature  rise during a harmonic Hall voltage response measurement can be \ncalibrated according to the resistance. As summarized in Fig. 5(b), the current -induced temperature increase  (∆T) \nrelative to 300 K varies  from  11 K to 22 K, depending on the Tb concentration ( thus resistivity) of the  FeTb. This  \ntemperature increase is sufficient  to affect the saturation magnetization of FeTb during harmonic Hall voltage \nresponse measurements ( Fig. S5(c)). Therefore, in the calculation of spin orbit torque efficiency in our samples  \nwe have used the magnetization at the calibrated temperature rather than at 300 K .   \n6 \n  \nFigure S5. Current -induced temperature increase.  (a) Temperature  dependence  of the longitudinal resistance (Rxx) \nof a 20 nm thick Fe59Tb41 sample measured with a  small  DC current (0.1 mA in a Hall bar that is 5 µm wide and \n60 nm long).  (b) The calibrated  temperature  increase ( ∆𝑇) relative to 300 K of the FeTb with different Tb \nconcentration during the h armonic  Hall voltage response measurements . (c) Variation of the magnetization of a \n20 nm thick Fe59Tb41 layer as a function of  temperature.  \n \nSection 6. Subtraction of effects of anomalous Nernst voltage  from  harmonic Hall voltage response   \n \nFigure S6. Schematic of measurement coordinates.  \n \nA charge current flow in a resistive perpendicularly -magnetized Hall bar device can induced thermal \ngradients in  both the out-of-plane  (∇Tz) and in -plane direction s (∇Tx) [S2]. Sizable anomalous Nernst voltages  can \narise from the se thermal gradients and contribute to measurements of the  harmonic Hall voltage response s \n(HHVR s). In a general case,  the first and the second HHVR s, Vω and V2ω, are given by  [S2]: \nVω =VAHE cosθM +VPHEsin2θM sin2φ,                         (S1) \nV2ω = 𝑉2ω,SOT+ VANE,x cos𝜃M + VANE,z  sin 𝜃Mcos 𝜑.    (S2) \nwhere θM is the polar angle of the magnetization  (with θM = arccos( Vω/VAHE) if the angle of applied magnetic field \nis chosen so that sin2φ = 0), φ is the azimuthal angle of the magnetization and is the same as that of the magnetic \nfield for a uniaxial anisotropy system, 𝑉2ω,SOT is the  second HHVR  due to the spin-orbit torque s, VAHE is the \nanomalous Hall voltage  coefficient , VPHE is the planar Hall voltage  coefficient , VANE,x = cANE ∇Tx and VANE,z = cANE \n∇Tz are the anomalous Nernst voltage s associated with  the in -plane and perpendicular thermal gradient s, and cANE \nis the anomalous Nernst coe fficient. VAHE can be determined from the dependence of Vω on a swept out -of-plane \nmagnetic field ( Hz) or on a  small  sweep in -plane field Hx if θM is determined.  VANE, z is equal to the value of V2ω \nwhen  θM = 90◦ and φ = 0◦ (current direction), while  VANE, x is equal to the value of V2ω when θM=θH = 0◦ (film normal \ndirection ).  In Eq. (S2) we have ignored possible contributions from the ordinary Nernst effect that tend s to be \nmuch smaller than the anomalous Nernst effect in samples with metallic ferromagnets  [S3]. \n7 \n  (i) W hen a magnetic field applied in the xz plane ( Hxz) tilts the magnetization by  a small  θM, Eqs. (S1) and \n(S2) can be simplified as : \nVω =VAHE cosθM，                                                      (S3) \nV2ω = (1\n2𝑉AHE\n𝐻𝑘+𝐻xz𝐻DL + VANE,z) sinθM + VANE,x.              (S4) \nIn this case,  𝐻DL can be  determined f rom the dependence of 𝜕V2ω/𝜕sinθM on the magnitude of Hxz. \n(ii) When a  small in -plane field Hx is applied along the current  direction ( with Hx/Hk <<1 so that sinθM≈ \nHx/Hk), Eqs. (S1) and (S2) can be simplified as:  \nVω = VAHE  (1- 𝐻𝑥2\n2𝐻k2),                                               (S5) \nV2ω = 1\n2𝑉AHE\n𝐻k2𝐻DLHx + 𝑉ANE ,z\n𝐻kHx + VANE,x  .               (S6) \nNow, since  both the first and the second terms  in Eq. (S6) are proportional to  Hx, the anomalous Nernst contribution \nto the second HHVR cannot be separated from the dampinglike torque contribution simply from the  dependences \nof V2ω on Hx. Generally, one can define an  apparent longitudinal effective field  \nHL = -2𝜕𝑉2ω\n𝜕𝐻𝑥 /𝜕2𝑉ω\n𝜕𝐻𝑥2  = 𝐻DL + 2𝐻k𝑉ANE ,z\n𝑉AHE  .                  (S7) \nBoth HDL and the second term of Eq. ( S7) are proportional  to the magnitude of the applied ac electric field  E. This \nsuggests that the anomalous Nernst effect should be carefully taken into account in the analysis of  resistive \nmagnetic systems with a small  SOT  field and a large value of 𝐻k.  This can be done by measuring each of the \nparameters contributing to the second term in Eq. (S7)  separately, as described above.  We find that the anomalous \nNernst term is significant for some FeTb single layers  (x≤51). As an example, we show the results of VANE,z, VAHE, \nHL, and HDL as a function of E for a 20 nm thick Fe71Tb29 in Fig. S7 (a)-(c). However, we find that for the Pt  5 \nnm/FeTb 20 nm , HL is the same as  𝐻DLwithin the experimental uncertainty , indicating minimal influence of  \nthermoelectric effect .  \n(iii) When a small in -plane field Hy is applied transverse to the current  (Hy/Hk <<1 so that s inθM≈ Hy/Hk), Eq. \n(S2) can be  simplified  as: \nVω = VAHE  (1- 𝐻𝑦2\n2𝐻k2).                                            (S8) \nV2ω = 1\n2𝑉AHE\n𝐻k2𝐻FLHy + VANE,x .                                   (S9) \nFrom this equation, the field-like spin -orbit  torque field 𝐻FL can be determined f rom the slope of the linear fit of \nV2ω vs Hy.  \nHT = -2𝜕𝑉2ω\n𝜕𝐻𝑦 /𝜕2𝑉ω\n𝜕𝐻𝑦2  = 𝐻FL.                                                    (S10) 8 \n  \nFigure S7. Electric field (E) dependence  of (a) VANE,z, (b) VAHE,  (c) HL and HDL for a 20 nm Fe71Tb29  layer.  \n \nSection 7. Coherent rotation at small polar angles  \n \nFigure S8. Polar -angle dependence of the f irst harmonic Hall voltage response ( Vω) of a 20 nm Fe71Tb29 Hall bar \ndevice under an external magnetic field  with a constant magnitude  Hzx= 80 kOe  in the x -z plane . The s olid parabolic \nline represents  the best fit to the expression  Vω= VAHEcosθM ≈ VAHE(1-𝜃M2/2), indicating  a coherent magnetization \nrotation at small polar angles.  \n \nSection 8. Plan ar Hall resistance  \n \nFigure S9. Planar Hall resistance of a 20 nm thick Fe55Tb 45 layer . (a) Hall resistance ( RH) for a Fe55Tb45 single \nlayer plotted as a function of the in -plane angle ( 𝜑) of an in-plane magnetic field (90 kOe) relative to the current \ndirection . The planar Hall resistance ( RPHE) is determined from the best fit (red line) of the data to  the relation \n𝑅H=𝑅PHEsin2 𝜑+𝐶sin𝜑 (the 𝐶sin𝜑 term comes from the small misalignment of the magnetic field with respect \nto the sample plane). (b) RPHE and (c) the ratio of the planar to anomalous Hall resistance s (RPHE/RAHE) plotted as \na function of the Tb concentration. The |RPHE/RAHE| ratios are ≤ 0.04 for Fe 100-xTbx. \n9 \n Section 9. Resistivit ies of 20 nm  thick  Fe100-xTb x layers  \n \nFigure S10. Resistivity of the 20 nm  thick  Fe100-xTbx layers with different  Tb concentration s. \n \nSection  10. Validation of Eq. (3) in the main text  \nWe compare the effect of an external magnetic field to the effect a  dampinglike  spin-orbit torque on a \nferrimagnet composed of two  oppositely oriented magnetic sublattices (1 and 2) with different gyromagnetic ratios \n𝛾𝑖 (i=1,2) . We assume the individual gyromagnetic ratios are negative, so that the angular momentum ( 𝑆⃗𝑖) and \nmagnetization ( 𝑀⃗⃗⃗𝑖) vectors are related as 𝑆⃗𝑖=−𝑀⃗⃗⃗𝑖/|𝛾𝑖| with magnitudes 𝑆𝑖=𝑀𝑖/|𝛾𝑖|. All calculations below \nwill assume a unit area of a sample thin film.  \nThe LLG equation for an individual sublattice subject only to a magnetic field has the form  \n𝑑\n𝑑𝑡𝑀⃗⃗⃗𝑖=−|𝛾𝑖|𝑀𝑖𝑚̂𝑖×𝐻⃗⃗⃗  +  𝛼𝑖𝑀𝑖𝑚̂𝑖×𝑑𝑚̂𝑖\n𝑑𝑡 .            (S11) \nwhere 𝛼𝑖 is the Gilbert damping and 𝑀𝑖 is the magnetic moment.  It will be more convenient to write the combined \nequation of motion for the two sublattices in terms of the total angular momentum rather than the total \nmagnetization, because the exchange inter action between the two sublattices will conserve total angular \nmomentum, while it does not conserve the total magnetization.   \n \nThe LLG equation for sublattice 1 can be re -written  \n𝑑\n𝑑𝑡𝑆⃗1=−𝑀1𝑠̂1×𝐻⃗⃗⃗ − 𝛼1\n|𝛾1|𝑀1𝑠̂1×𝑑ŝ1\n𝑑𝑡 .               (S12) \nand for sublattice 2  \n𝑑\n𝑑𝑡𝑆⃗2=−𝑀2𝑠̂2×𝐻⃗⃗⃗  − 𝛼2\n|𝛾2|𝑀2𝑠̂2×𝑑ŝ2\n𝑑𝑡=+𝑀2𝑠̂1×𝐵⃗⃗ − 𝛼2\n|𝛾2|𝑀2𝑠̂1×𝑑ŝ1\n𝑑𝑡. (S13) \nAdding these equations, the equation of motion for the total angular momentum of the ferrimagnet subject only to \na magnetic field is   \n𝑑\n𝑑𝑡(𝑆⃗total)  =−(𝑀1− 𝑀2) 𝑠̂1×𝐻⃗⃗⃗ −(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2) 𝑠̂1×𝑑ŝ1\n𝑑𝑡.       (S14) \nIf the ferrimagnet is also subject to an antidamping spin -orbit torque, with spin angular momentum in the 𝜎̂ \ndirection, we can add this to the equation of motion  \n𝑑\n𝑑𝑡(𝑆⃗total) =−(𝑀1− 𝑀2) 𝑠̂1×𝐻⃗⃗⃗ −(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2) 𝑠̂1×𝑑ŝ1\n𝑑𝑡+ℏ\n2𝑒𝜉DL𝑗𝑗𝑒(𝑠̂1×𝜎̂×𝑠̂1).     (S15) \nSince 𝑆⃗total = (𝑆1−𝑆2)𝑠̂1=(𝑀1\n|𝛾1|−𝑀2\n|𝛾2|)𝑠̂1=(𝑀1−𝑀2)𝑠̂1/𝛾eff, where we define 𝛾eff≡ (𝑀1−𝑀2)/(𝑀1\n|𝛾1|−\n𝑀2\n|𝛾2|), in terms of unit vectors Eq. (S15)   can be rewritten as  \n10 \n 𝑑\n𝑑𝑡(𝑠̂1)  =−𝛾eff 𝑠̂1×𝐻⃗⃗⃗  −𝛼eff𝑠̂1×𝑑ŝ1\n𝑑𝑡+𝛾effℏ\n2𝑒(1\n𝑀1−𝑀2)𝜉DL𝑗𝑗𝑒(𝑠̂1×𝜎̂×𝑠̂1).  (S16) \nHere 𝛼eff=𝛾eff(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2)/(𝑀1−𝑀2). \n \nFrom Eq. (S16) , we can read off that the spin -transfer torque is equivalent to an effective magnetic field  \n𝐻⃗⃗⃗DL=−ℏ\n2𝑒(1\n𝑀1−𝑀2)𝜉DL𝑗𝑗𝑒(𝜎̂×𝑠̂1)=ℏ\n2𝑒1\n|𝑀1−𝑀2|𝜉DL𝑗𝑗𝑒(𝜎̂×𝑚̂).                 (S17) \nIf the effective magnetic field is measured relative to the direction 𝜎̂×𝑚̂, we therefore have that  \n \n             𝜉DL𝑗=2𝑒\nℏ𝐻𝐷𝐿|𝑀1−𝑀2|\n𝑗𝑒 ，                                                                                 (S18) \nWith the identification that the total magnetization per unit area |𝑀1−𝑀2| is equal to 𝑀s𝑡, the measured \nmagnetization per unit area , one obtain                              \n                      𝜉DL𝑗=2𝑒\nℏ𝐻DL𝑀s𝑡\n𝑗𝑒 .                                                 (S19) \nThis is the same as  Eq. (3) in the main text , and indicates that  𝜉DL𝑗 has no dependence on the sign of 𝛾eff. \n \nSection 11. Raw data for representative samples exhibiting different torque signs  \n \nFigure S1 1. Raw data for three representative samples: 20 nm Fe 71Tb29 (VAHE  > 0, 𝜉DL𝑗>0), 20 nm Fe 61Tb39 \n(VAHE  > 0, 𝜉DL𝑗< 0), and 20 nm Fe 47Tb53 (VAHE< 0, 𝜉DL𝑗>0). (a) Frist (V1ω) vs sin θM, (b) Second HHVR (V2ω) vs \nsinθM, and (c) dV2ω/dsinθM vs VAHE/2(Hk+Hzx) for under different applied magnetic field Hzx. The definition of \nsymbols are the same as those  in the Supplementary  Section 6.  \n11 \n Section 12. Field-like spin-orbit  torque in Fe 100-xTb x single layer s \n \nFigure S12. (a) Field -like effective torque field for 20 nm  thick  Fe100-xTbx layers with different Tb concentration s. \n(b) Field -like effective torque field for Fe57Tb43 layers with different thickness es. The field -like torque is relatively \nsmall compared to the damping -like torque ( Fig. 2(d) of the main text).  \n \nSection 13. Spin -torque ferromagnetic resonance measurements  \nTo confirm the spin Hall effect of the FeTb, we performed spin-torque ferromagnetic resonance (ST -FMR) \nto measure emission  of spin current from FeTb  layers that is absorbed by  an Fe detector  layer. During the ST -\nFMR measurements, an in -plane magnetic field ( H) was swept at a fixed angle of 45º  with respect to the magnetic \nmicrostrip. As shown in Fig. S13(a), the amplitudes of the symmetric and anti -symmetric components of a ST -\nFMR spectrum, S and A, can be  determined by fitting the data to [S4]: \n𝑉mix=𝑆∆𝐻2\n∆𝐻2+(𝐻−𝐻r)2+𝐴∆𝐻(𝐻−𝐻r)\n∆𝐻2+(𝐻−𝐻r)2 ,   (S11) \nwhere  ∆H is the FMR linewidth  and Hr the resonance field.  From the S/A ratio, we can define  as an intermediate \nparameter an effective  spin-orbit torque efficiency [S5, S6]: \n𝜉FMR =𝑆\n𝐴𝑒𝑀s𝑡𝑑\nℏ√1+4π𝑀eff\n𝐻r,       (S12) \nwhere  e is the electron charge, ℏ is the reduced Planck constant, 𝜇0 is the vacuum permeability, Ms is the saturation \nmagnetization. 𝑡 is the layer thickness of the magnetic detector, 𝑑 is the layer thickness of the spin -current -\ngenerating layer , and 4𝜋𝑀eff is the effective demagnetization field of the magnetic detector.  The true damping -like \nspin-torque efficiency is determined from plots of 1/ 𝜉FMR versus 1/ t, as described in the main text.  Here, there is \nonly negligible Oersted field due to the in -plane current flow in the extremely resistive thin Ti layer [S7]. As \nshown in Fig. S13(b), 4𝜋𝑀eff can be determined from the resonance frequency  (f) dependence of Hr following the \nKittel’s equation  [S6] \n𝑓=𝛾\n2𝜋√(𝐻𝑟+𝐻ex)(𝐻𝑟+𝐻𝑒𝑥+4𝜋𝑀eff) ,              ( S13) \nwhere 𝐻ex is the exchange field from the other layer . \n \n12 \n  \nFigure S13. (a) ST-FMR spectrum at 15 GHz for Fe 50Tb50 (20 nm)/Ti (1 nm)/Fe (tFe), in which a clea r symmetric \ncomponent (solid blue line) is from  the damping -like torque, and the asymmetric component  (solid pink line)  is \nfrom  the field -like torque  and Oersted field torque . (b) Frequency dependence of f erromagnetic resonance field Hr \nof the Fe layer in the representative samples Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (9.8 nm ) and Fe50Tb50 (20 nm)/Ti (1 \nnm)/Fe (3.8 nm ). The solid red line represent s fit of the data to Kittel’s equation  (Eq. (13) ). (c) A and (d) S of the \nrepresentative samples Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (9.8 nm ) (12 GHz) and Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (3.8 \nnm) (14 GHz) plotted as a function of in -plane angle ( φ) of the magnetic field relative to  the rf current . The blue \nand red solid curves represent the best fits of the data to a sin2 φ cosφ dependence.  \nSection 14. Interfacial  perpendicular magnetic anisotropy energy density  \n \nFigure S14. Dependence of the effective demagnetization field (4π Meff) on the inverse thickness of the Fe layer \nin the Fe50Tb50 (20 nm)/Ti(1  nm)/Fe( tFe) sample.  The linear fit of the data to the relation  4𝜋𝑀eff ≈ 4πM s -2Ks/MstFM \nyields a total interfacial magnetic  anisotropy energy density of Ks=2.07 ±  0.05 erg/cm2 and a saturation \nmagnetization Ms = 1500 emu/cm3. After subtraction of the contribution from  the top Fe/MgO interface (≈ 0.64 \nerg/cm2) [S8], we obtain the Ks ≈ 1.43 ±  0.05  erg/cm2 for the Ti/Fe interface . \n13 \n Section 15. HHVR measurement of  a 20 nm thick composition -gradient Fe100-xTb x (x = 27→41)  \n \nFigure S15 𝑑V2ω/𝑑sinθM versus  VAHE/2(Hk+Hzx) for the 20 nm  thick composition -gradient  Fe100-xTbx (x = 27→41) \nsample , where x is varied continuously from 27 at the bottom  of the layer  to 41 at the top . Using the extracted  \nvalue of  HDL obtained from a linear fit to Eq. (3) in the main text,  we obtain 𝜉DL𝑗 = 0.05 ±  0.01  for this sample . \nSection 16. More examples o f magnetization  switching by the bulk spin -orbit torque  \n \nFigure S1 6. Current -induced  switching  of the 20  nm Fe 67Tb33, Fe61Tb39, and Fe57Tb43 films  under the same applied \nbias magnetic field of +3 kOe. The arrows  indicate the switching polarity . \n \n14 \n In Fig. S16(a) -(c), we show the c urrent -induced  switching  of the 20  nm-thick  Fe67Tb33, Fe61Tb39, and \nFe57Tb43, under the same applied bias magnetic field of 3 kOe. Since all the three films are Fe-dominated  (RAHE > \n0), their anomalous Hall voltages have the same sign. As indicated by the arrows, the switching polarity of the \nFe67Tb33 is opposite to that of Fe57Tb43, which is consistent with the opposite sign of the bulk spin -orbit torques in \nthe Fe67Tb33 (𝜉DL𝑗 >0) and the Fe57Tb43 (𝜉DL𝑗 < 0). Interesting, the Fe61Tb39 shows the same switching polarity as \nthe Fe67Tb33, which is because the current -induced heating during the switching has driven the Fe61Tb39 deep into \nthe Fe -dominated regime where the Fe67Tb33 is also located and the bulk spin -orbit torque efficiency is positive \n(𝜉DL𝑗 >0). It has been reported that the ferrimagnetic CoTb near the compensation point can become Co -dominated \nat high er temperatures and Tb dominated at lower temperatures [ S9]. \n \nReferences  \n[S1] B. Hebler, A. Hassdenteufel, P. Reinhardt, H. Karl and M. Albrecht , Front. Mater. 3, 8 (2016).  \n[S2] A. Ghosh, K. Garello, C. O. Avci, M. Gabureac, and P. Gambardella, Phys. Rev. Appl. 7, 014004 (2017).  \n[S3] M. Weiler, M. Althammer, F.  D. Czeschka, H. Huebl, M.  S. Wagner, M. Opel, I. -M. Imort, G. Reiss, A. \nThomas, R. Gross,  and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602  (2012).  \n[S4] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,  Phys. R ev. Lett. 106, 036601 (20 11). \n[S5] C.-F. Pai, Y . Ou, L. H . Vilela -Leao, D. C. Ralph, R. A. Buhrman, Phys. Rev. B 92, 064426 (2015).  \n[S6] C. Kittel,  Phys. Rev. 73, 155 (1948).  \n[S7] L. Zhu and R. A. Buhrman, Phys. Rev. Applied 15, L031001(2021).  \n[S8] Y. Iida, J. Okabayashi, and S. Mitani, Appl. Phys. Lett. 113, 252401 (2018) . \n[S9] K. Ueda,  M. Mann, P. W. P. de Brouwer, D. Bono, and G. S. D. Beach,  Phys. Rev. B 96, 064410 (2017).  "    },    {        "title": "1209.3965v1.Nontrivial_ferrimagnetism_of_the_Heisenberg_model_on_the_Union_Jack_strip_lattice.pdf",        "content": "arXiv:1209.3965v1  [cond-mat.str-el]  18 Sep 2012Nontrivial ferrimagnetism of the Heisenberg model on the Un ion\nJack strip lattice\nTokuro Shimokawa∗and Hiroki Nakano\nGraduate School of Material Science,\nUniversity of Hyogo, Kamigori, Hyogo 678-1297, Japan\n(Received 31 May 2012)\nAbstract\nWe study the ground-state properties of the S= 1/2 antiferromagnetic Heisenberg model on\nthe Union Jack strip lattice by using the exact-diagonaliza tion and density matrix renormaliza-\ntion group methods. We confirm a region of the intermediate-m agnetization state between the\nN´ eel-like spin liquid state and the conventional ferrimag netic state of Lieb-Mattis type. In the\nintermediate-state, we find that the spontaneous magnetiza tion changes gradually with respect to\nthe strength of the inner interaction. In addition, the loca l magnetization clearly shows an in-\ncommensurate modulation with long-distance periodicity i n the intermediate-magnetization state.\nThese characteristic behaviors lead to the conclusion that the intermediate-magnetization state is\nthe non-Lieb-Mattis ferrimagnetic one. We also discuss the relationship between the ground-state\nproperties of the S= 1/2 antiferromagnetic Heisenberg model on the original Union Jack lattice\nand those on our strip lattice.\nPACS numbers: 75.10.Jm, 75.30.Kz\nKeywords: quantum spin system, frustration, ferrimagnetism, D MRG, exact diagonalization\n∗Electronic address: t.shimokaw@gmail.com\n0I. INTRODUCTION\nFerrimagnetism is a fundamental phenomenon in the field of magnetis m. The most\nfamous type of ferrimagnetism is called Lieb-Mattis (LM) one[1–6]. Fo r example, this fer-\nrimagnetism appears in the ground state of the ( s,S)=(1/2, 1) mixed spin chain with\nnearest-neighbor antiferromagnetic interaction. In this system , the occurrence of the LM\nferrimagnetism originates from the situation that two different spin s are arranged alter-\nnately in a line owing to the AF interaction. In the LM ferrimagnetic sta te, the spontaneous\nmagnetization occurs and the magnitude is fixed to a simple fraction o f the saturated mag-\nnetization. As in the case of this mixed spin chain, not only the magnet ic properties but\nalso the occurrence mechanism of the LM ferrimagnetism are well kn own since this type of\nferrimagnetism has been studied extensively. Especially, the ferrim agnetism in the quantum\nHeisenberg spin model on the bipartite lattice without frustration is well understood within\nthe Marshall-Lieb-Mattis (MLM) theorem[1, 2].\nOn the other hand, a new type of ferrimagnetism that is clearly differ ent from the LM\nferrimagnetism has been found in the ground state of several one -dimensional frustrated\nHeisenberg spin systems[7–13]. The spontaneous magnetization in t his new type of ferri-\nmagnetism changes gradually with respect to the strength of frus tration. In addition, the\nincommensurate modulation with long-distance periodicity in local mag netizations is ob-\nserved as a characteristic quantum behavior of the new type of fe rrimagnetism. Hereafter,\nwe call the new type of ferrimagnetism non-Lieb-Mattis (NLM) type . The mechanism of the\noccurrence of the NLM ferrimagnetism have not yet been clarified in contrast to the case of\nthe LM ferrimagnetism.\nHistorically, some candidates of the NLM ferrimagnetism among the 2 D systems were\nalready reported. For examples, there are the mixed-spin J1-J2Heisenberg model on the\nsquare lattice[14] and the S= 1/2 Heisenberg model on the Union Jack lattice of Fig. 1(a)\n[15–18]. These 2D frustrated systems have the intermediate grou nd-state, namely “canted-\nferrimagnetic state” as described in Fig. 2, in which the spontaneou s magnetization is\nchanged when the inner interaction of the system is varied. It has n ot been, however, in-\nvestigated whether the incommensurate modulation with long-dista nce periodicity exists or\nnot in the local magnetization of the intermediate-magnetization st ate owing to the diffi-\nculty of treating these 2D frustrated systems numerically and the oretically. Therefore, the\n1'JH\u000f\u0001\u0012\tB\n 'JH\u000f\u0001\u0012\tC\n \n\"\n\"` $\n$` # %+\u0012\n+\u0013 +\u0012\n+\u0013\nFIG. 1: (Color online) Structures of the lattices: the Union Jack lattice (a), the Union Jack strip\nlattice (b). An S= 1/2 spin is located at each site denoted by a black circle. Antif erromagnetic\nbondsJ1(bold straight line) and J2(dashed line) are represented. Sublattices in a unit cell of\nlattice (b) are represented by A, A′, B, C, C′, and D.\nrelationships between the intermediate-magnetization states of t hese 2D frustrated systems\nand the NLM ferrimagnetic state are still unclear.\nUnder such circumstances, quite recently, the S= 1/2 antiferromagnetic Heisenberg\nmodel on the spatially anisotropic kagome lattice was studied[19]. In t his model, the\nintermediate-magnetization states exist between the LM ferrimag netic state and the non-\nmagnetic one[20–24]. It was reported that the local magnetization in these intermediate-\nstate shows large dependence on the position of the sites although it is difficult to judge\nclearly whether the incommensurate modulation with long-distance p eriodicity is present or\nabsent. In addition, the S= 1/2 Heisenberg models on the quasi-one-dimensional kagome\nstrip lattices were studied[25, 26]. These strip lattices share the sa me lattice structure in\ntheir inner part with the spatially anisotropic kagome lattice. The loca l magnetizations in\nthe intermediate-state clearly show incommensurate modulations w ith long-distance peri-\nodicity irrespective of the strip width. Therefore, these results s trongly suggest that the\nintermediate-magnetization states not only of the kagome strip lat tices but also of the orig-\ninal kagome lattice are the NLM ferrimagnetism.\nThese kagome results motivate us to investigate the ground-stat e properties of the quasi-\none-dimensional strip model whose lattice structure is common to t he part of the 2D lattice\n2/FFMʅ $BOUFE\u000eGFSSJ 4FNJTUSJQFE\u000eGFSSJ\n+\u0013\u0010+ \u0012П\nП\nЋ\u0012Ћ\u0013\nFIG. 2: (Color online) Ground-state phase diagram of the S= 1/2 Heisenberg model on the\nUnion Jack lattice depicted Fig. 1(a). Here, black and white circles represent up-spin and down-\nspin respectively.\nknown as the other candidates of the NLM ferrimagnetism. In this s tudy, we treat the\nS= 1/2 Heisenberg model on the Union Jack strip lattice depicted in Fig. 1(b ). This strip\nlattice share the same lattice structure in the inner part with the or iginal Union Jack lattice\ndepicted in Fig. 1(a). Our numerical calculations lead to the conclusio n that the NLM\nferrimagnetic phase appears in the ground-state of the Union Jac k strip model of Fig. 1(b).\nWe also discuss the relationship between the ground-state proper ties of the present strip\nmodel and those of the original 2D model.\nII. MODEL\nThe Hamiltonian of the S= 1/2 antiferromagnetic Heisenberg model on the Union Jack\nstrip lattice depicted in Fig. 1(b) is given by\nH=J1/summationdisplay\ni[Si,A·Si,C+Si,B·Si,D+Si,A′·Si,C′+Si,A·Si,B+Si,B·Si,A′\n+Si,C·Si,D+Si,D·Si,C′+Si,C·Si+1,A+Si,D·Si+1,B+Si,C′·Si+1,A′]\n+J2/summationdisplay\ni[Si,A·Si,D+Si,A′·Si,D+Si,D·Si+1,A+Si,D·Si+1,A′], (1)\nwhereSi,ξis anS= 1/2 spin operator at ξ-sublattice site in i-th unit cell. Positions\nof the six sublattices in a unit cell are denoted by A, A′, B, C, C′and D in Fig. 1(b).\nWe fixed J1= 1 hereafter as a energy scale. In what follows, we examine the reg ion of\n30≤J2/J1≤3.5 in the present study. Note that the number of total spin sites is d enoted\nbyN; thus, the number of unit cells is N/6.\nLet us introduce here the ground-state phase diagram of the S= 1/2 Heisenberg model\non the original Union Jack lattice of Fig. 1(a). At small J2/J1, one can see immediately that\nthe antiferromagnetic N´ eel order is observed since this model co rresponds to the S= 1/2\nantiferromagnetic Heisenberg one on the simple square lattice in the limit ofJ2/J1= 0.\nWhen the J2/J1is increased, the intermediate-magnetization state appears. A va riational\nanalysis for the classical model revealed that the spin configuratio ns in this intermediate\nregion are defined by the angle of cant ϕas illustrated in Fig. 2[18]. Therefore, this\nintermediate-state is called canted-ferrimagnetic one. The phase transition between the N´ eel\nandcanted-ferrimagneticphasesoccursat α1≡J2/J1∼0.84fromtheviewpointofthespin-\nwave theory[15, 16]. On the other hand, it was reported that this p hase transition occurs\natα1∼0.65 by using the series expansion (SE)[17] and coupled cluster metho d (CCM)[18]\ntechniques. It was also discussed the possibility that the semistripe d-ferrimagnetic state as\nillustrated in Fig. 2 appears at very large values of J2/J1(α2≡J2/J1≈125)[18].\nIn what follows, we examine the ground-state phase diagram of the S= 1/2 Heisenberg\nmodel on the Union Jack strip lattice depicted in Fig. 1(b) and compar e the results of the\noriginal 2D lattice with those of the our strip lattice.\nIII. NUMERICAL METHODS\nWe employ two reliable numerical methods: the exact diagonalization ( ED) method and\nthe density matrix renormalization group (DMRG) method[27, 28]. Th e ED method can\nbe used to obtain precise physical quantities for finite-size cluster s. This method does not\nsuffer from the limitation of the shape of the clusters. It is applicable even to systems with\nfrustration, in contrast to the quantum Monte Carlo (QMC) metho d coming across the\nso-called negative-sign problem for systems with frustration. The disadvantage of the ED\nmethod is the limitation that available system sizes are very small. Thus , we should pay\ncareful attention to finite-size effects in quantities obtained from this method.\nOn the other hand, the DMRG method is very powerful when a syste m is (quasi-)one-\ndimensional under the open-boundarycondition. The methodcan t reat much larger systems\n40 1 2 3\nJ2/J 100.20.4M/M sN=24, periodic\nN=48,  open\nN=96,  open\nN=144, open'JH\u000f\u0001\u0014\tB\n 'JH\u000f\u0001\u0014\tC\n \n0 10 20\nStotz–50050 E nergy J2/J 1=0.1 \nJ2/J 1=1.5 \nJ2/J 1=3.5 \nFIG. 3: (Color online) (a) Dependences of the lowest energy o nSz\ntot. Results of J2/J1= 0.1, 1.5\nand 3.5 for the system size of N= 48 are presented. Arrows indicate the values of the spontan eous\nmagnetization Min eachJ2/J1. (b)J2/J1-dependence of M/Msobtained from ED calculations for\nN= 24 (black cross) under the periodic-boundary condition an d DMRG calculations for N= 48\n(red triangle), 96 (blue square) and 144 (green pentagon) un der the open-boundary condition.\nthan the ED method. Note that the applicability of the DMRG method is irrespective of\nwhether or not systems include frustrations. In the present res earch, we use the “finite-\nsystem” DMRG method. Note that we carefully choose the maximum n umber of retained\nstates (MS) and the number of sweeps ( SW) in our DMRG calculations.\nIV. RESULTS\nIn this section, we present our numerical results in the ground-st ate of the S= 1/2\nHeisenberg model on the Union Jack strip lattice of Fig. 1(b). First, let us explain the way\nto obtain the spontaneous magnetization Min the ground state of the quantum system with\nisotropic interactions. We calculate the lowest energy E(J2/J1,Sz\ntot,N), where Sz\ntotis the\nz-component of the total spin. For example, the energies for each Sz\ntotin the three cases of\nJ2/J1are shown in Fig. 3(a). In this figure, the results of the DMRG calcula tions with the\nMS= 700 and SW= 15 are presented when the systems size is N= 48 for J2/J1=0.1, 1.5,\n3.5. The spontaneous magnetization M(J2/J1,N) is determined as the highest Sz\ntotamong\n5'JH\u000f\u0001\u0015\tB\n 'JH\u000f\u0001\u0015\tC\n \nFIG. 4: (Color online) (a) Spin configuration in the N´ eel-li ke spin liquid phase. (b) Spin configu-\nration in the LM ferrimagnetic phase of M/Ms= 1/3, where this configuration is obtained from\nthe numerical results of the local magnetization shown in Fi g. 5(b).\nthose at the lowest common energy [see arrows in Fig. 3(a)].\nOur results of the J2/J1-dependence of the M/Msare shown in Fig. 3(b), where Ms\nmeans saturated magnetization value, namely, Ms=N/2. In the limit of J2/J1= 0, this\nUnion Jack strip model depicted in Fig. 1(b) is reduced to the S= 1/2 antiferromagnetic\nHeisenberg model on the three-leg ladder as known the typical sys tem of the gappless spin-\nliquid ground states[29, 30]. According to the study of the S= 1/2 frustrated three-leg\nspin ladder[31], it is expected that the N´ eel-like spin liquid phase occur s in the ground-state\nwhen the strength of the J2/J1is small but finite, where the schematic spin configuration in\nthe N´ eel-like spin liquid state is illustrated in Fig. 4(a). Indeed, our nu merical calculations\nlead to the same conclusions that the nonmagnetic phase of M/Ms= 0 appears in the region\nwhereJ2/J1is relatively small.\nFor larger J2/J1, on the other hand, the magnetic phases with M/Ms/negationslash= 0 appears in\nthe ground state. Careful observation enables us to find that th ere are two magnetic phases\nin the thermodynamic limit; one is the intermediate magnetic phase of 0 < M/M s<1/3\nand the other is the phase of M/Ms= 1/3. It should be noted here that the phase of\nM/Ms= (1\n3−2\nN) which is found only under the open-boundary condition merges with the\nphase of M/Ms= 1/3 in the thermodynamic limit of N→ ∞since the value of M/Ms\nbecomes gradually larger and approaches the value of M/Ms= 1/3. This change due to the\nincrease of the system size comes from finite-size effect. It is impor tant that we successfully\nobserve the intermediate-magnetization phase where the sponta neous magnetization M/Ms\nchanges continuously with respect to the strength of J2/J1.\nNext, we calculate the local magnetization /angbracketleftSz\ni,ξ/angbracketrightto investigate the spin configurations in\n6'JH\u000f\u0001\u0016\tB\n 'JH\u000f\u0001\u0016\tC\n \n10 20 \ni–0.2 00.20.4<S i, ξz>N=144 J 2/J 1=3.5 M=24\nC’ D C A ’ B A\n10 20 \ni–0.2 00.20.4<S i, ξz>N=144 J 2/J 1=1.8 M=17\nC’ D C A ’ B A\nFIG. 5: (Color online) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice ξ. Panels (a) and(b) areresults\nforJ2/J1=1.8 and 3.5 respectively. These results are obtained from o ur DMRG calculations for\nN= 144 (i=1,2,···, 24).\nthese two magnetic states, where /angbracketleftA/angbracketrightdenotes the expectation value of the physical quantity\nAandSz\ni,ξis thez-component of Si,ξ. Figure 5 depicts our results for a system size N= 144\non the lattice depicted in Fig. 1(b) under the open-boundary condit ion; Fig. 5(a) and Fig.\n5(b) correspond tothe case of J2/J1=1.8and3.5 respectively; we use greeninverted triangle\nforξ=A, blue pentagon for ξ= A′, red circle for ξ= B, black cross for ξ=C, aqua triangle\nforξ= C′, and purple square for ξ=D. In Fig. 5(a), we find clearly incommensurate\nmodulations with long-distance periodicity in the behavior of the local magnetization at\ntheB-sublattice sites. Therefore, we conclude that the intermediate- magnetization phase\nof 0< M/M s<1/3 is the NLM ferrimagnetic one. On the other hand, in Fig. 5(b),\nwe observe the uniform behavior of upward-direction spins at subla ttice-sites B, C, C′and\nD and downward-direction spins at sublattice A and A′as illustrated in Fig. 4(b). The\nspin configuration in the phase of M/Ms= 1/3 can be understood from the viewpoint of the\nMLMtheorembecausethepresent stripmodelcorrespondstoth eS= 1/2antiferromagnetic\nHeisenberg model on the diamond chain in the limit of J2/J1=∞. Therefore, it is naturally\nlead to the conclusion that the phase of M/Ms= 1/3 is the LM ferrimagnetic one.\nFinally, we discuss the relationship between the ground-state prop erties of the original\nUnion Jack model depicted Fig. 1(a) and those of the strip model de picted in Fig. 1(b).\n7It is confirmed that the schematic spin configuration in the N´ eel-like spin liquid state de-\npicted in Fig. 4(a) is consistent to that in the N´ eel state depicted in Fig. 2. We also\nconfirm that the schematic spin configuration depicted in Fig. 4(b) a grees completely with\nthat in the semistriped-ferrimagnetic state of the original Union Ja ck model as shown in\nFig. 2. In addition, there exists the intermediate-magnetization st ate where the sponta-\nneous magnetization is gradually changed in the ground-state of th e both models although\nthe incommensurate modulation with long-distance periodicity has no t been confirmed in\nthe case of 2D Union Jack model. Therefore, one finds that the the ground-state phase\ndiagram of the S= 1/2 antiferromagnetic Heisenberg model on the Union Jack strip lat-\ntice is qualitatively consistent to that of the S= 1/2 antiferromagnetic Heisenberg model\non the original Union Jack lattice. The intermediate-magnetization s tate of the original\nUnion Jack lattice have been understood from the view point of the c lassical configuration,\nnamely ”canted-state”. However, our numerical results of the U nion Jack strip model leads\nto the possibility that the intermediate-state of the original Union J ack lattice is also the\nNLM ferrimagnetic one whose characteristic behavior in the local ma gnetization originates\nfrom pure quantum effects. The future studies are desirable to co nfirm the presence of the\nincommensurate modulation in the canted-state of the 2D Union Jac k model.\nV. CONCLUSIONS\nWe have studied the ground-stateproperties of the S= 1/2 antiferromagnetic Heisenberg\nmodel on the Union Jack strip lattice depicted in Fig. 1(b) by the ED an d DMRG meth-\nods. Our numerical calculations have revealed that the intermediat e-magnetization state\noccurs between the N´ eel-like spin liquid state corresponding to the N´ eel state of the original\n2D model and the LM ferrimagnetic state which agrees with the semis triped-ferrimagnetic\nstate of the original 2D model. In this intermediate-magnetization s tate of this strip model,\nthe spontaneous magnetization changes gradually with respect to the strength of the inner\ninteraction. We have also found the existence of the incommensura te modulation with long-\ndistance periodicity of the local magnetization. From the finds of th ese characteristic be-\nhavior, it has concluded that the intermediate state of this strip mo del is the NLM ferrimag-\nnetism. These results naturally lead to the expectation that the int ermediate-magnetization\nstate of the original model is also the NLM ferrimagnetic one.\n8Acknowledgments\nWe wish to thank Professor T. Sakai for fruitful discussions. One of the authors (T. S.)\nacknowledgesthefinancialsupportfromtheMotizukiFundofYuk awaMemorialFoundation.\nThis work was partly supported by Grants-in-Aid (Nos. 20340096, 23340109, 23540388,\nand 24540348) from the Ministry of Education, Culture, Sports, S cience and Technology of\nJapan. 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Gull, S. Gurtler,\nA. Honecker, R. Igarashi, M. Korner, A. Kozhevnikov, A. Lauc hli, S. R. Manmana, M. Mat-\nsumoto, I. P. McCulloch, F. Michel, R. M. Noack, G. Pawlowski , L. Pollet, T. Pruschke, U.\nSchollwock, S. Todo, S. Trebst, M. Troyer, P. Werner, S. Wess el, J. Magn. Magn. Mater. 310,\n1187 (2007) (see also http://alps.comp-phys.org).\n10"    },    {        "title": "2109.00169v1.Epitaxial_Integration_of_a_Perpendicularly_Magnetized_Ferrimagnetic_Metal_on_a_Ferroelectric_oxide_for_Electric_Field_Control.pdf",        "content": "Epitaxial Integration of a Perpendicularly  Magnetized Ferrimagnet ic \nMetal  on a Ferroelectric  oxide  for Electric -Field Control  \nXin Zhang, Pei-Xin Qin, Ze -Xin Feng, Han Yan, Xiao -Ning Wang, Xiao -Rong Zhou,  Hao-\nJiang Wu,  Hong -Yu Chen, Zi -Ang Meng, Zhi-Qi Liu*  \nSchool of Materials Science and Engineering, Beihang University, Beijing 100191, China  \nEmail: zhiqi@buaa.edu.cn  \n \n \nAbstract : Ferrimagnets , which contain the advantages of both ferromagnets  (detectable \nmoments)  and antiferromagnets  (ultrafast spin dynamics), have recently attracted great \nattention. Here we report the optimization of epitaxial growth of a tetragonal \nperpendicularly magnetized ferrimagnet  Mn 2Ga on MgO . Electrical transport, magnetic \nproperties and the anomalous Hall effect (AHE) were systematically studied. \nFurthermore,  we successfully integrated high-quality epitaxial ferrimagnetic Mn 2Ga \nthin films onto ferroelectric PMN -PT single crystals with a MgO bu ffer layer. It was \nfound that the AHE  of such a ferrimagnet can be effectively modulated by a small \nelectric field over a large temperature range in a nonvolatile manner. This work thus \ndemonstrates the great potential of ferrimagnets for developing high -density and low -\npower spintronic device s. \n \nKeywords : ferroelectric oxides; ferrimagnetic metals; PMN -PT; Mn 2Ga; anomalous \nHall effect  \n  1. Introduction  \nContemporary mass storage for data centers predominantly relies on hard disk drives that are \nbased on perpendicularly magnetized ferromagnetic granular films , the spin states of which can \nbe easily manipulated by external magnetic fields generated by current coils  due to remarkable \nmacroscopic moments. However, as limited by the characteristic GHz spin dynamics, \nferromagnetic materials could hardly be utilized for the static random -access  memory \ntechnology  and the in -memory computing, which needs a sub -ns response speed. Similar to \nferromagnets, ferrimagnets possess large magnetic moments. Besides, they exhibit ultrahigh \nspin dynamics of THz as a result of antiferromagnetic exchange coupling  [1-6] akin to \nantiferromagnets. Hence, ferrimagnets are rising  star materials for new -generation  sub-ns \ninformation devices  [7-12]. \nTetragonal D0 22 Mn 2Ga with a = 3.905 Å and c = 7.193 Å , a classical ferrimagnet with a high \nCurie temperature of ~710 K, exhibits large magnetic anisotropy along its [001] \ncrystallographic orientation. (001) -oriented single -crystalline or textured films are thus \nperpendicularly magnetized. In addition, D0 22 Mn2Ga is of high spin polarization at the Fermi \nlevel and low Gilbert damping constant  [13], both of which are useful for spin valves, \nperpendicular magnetic tunnel junctions  [14], narrow -band terahertz emission from coherently \nexcited spin precession  [15], and high -density spin -transfer -torque magnetoresistive random \naccess memories  (STT -MRAM) [16]. \nHowever,  the information writing (corresponding to the spin state manipulation) of Mn 2Ga-\nbased spintronic devices such as spin val ves and STT -MRAM mostly depend on electric al-\ncurrent -generated magnetic fields or electric al currents, which generates significant Joule \nheating and hence results in high power consumption.  Alternatively, if one can effectively \ncontrol the spin state of Mn 2Ga with a non -electrical -current manner, the energy for writing a \nbit could be substantially lowered.  \nAn electric field applied onto a conductor yields an electrical current. In contrast, for highly \ninsulating ferroelectric oxide materials, the application of an electric field generates a negligible \ncurrent. Instead, it induces piezoelectric strain  [1-7,17-23]. Assuming (001) -oriented D0 22 \nMn 2Ga could be epitaxially integrated onto ferroele ctric oxides, electric -field-generated strain could harness  the spin state of Mn 2Ga as the electronic states of solid -state materials are all \nsensitive to the periodic lattice.   \nNevertheless, epitaxial growth of intermetallic Mn 2Ga on a ferroelectric oxide is quite \nchallenging. The key reasons are: (1) highly -ordered epitaxial films need high thermal energy \nto let the atoms diffuse into their equilibrium sites during thin film growth and therefore high \ngrowth temperatures are requ ired; (2) intermetallic alloys are of strong chemical activity, which \ncan easily be oxidized  or form secondary alloys at high  temperatures while landing on \nferroelectric oxide substrates ; (3) lattice mismatch between intermetallic alloys and \nferroelectric oxides could break epitaxial growth. Accordingly , a proper oxide substrate or an \noxide buffer layer with robust high -temperature stability and low oxygen diffusion coefficient \ncould be crucial for realizing epitaxial growth of Mn 2Ga on ferroelectrics.  In this letter, we \nreport the optimal  epitaxial growth of D0 22 Mn 2Ga on ferroelectric PMN -PT. Furthermore, \nelectric -field control of its AHE  has been achieved over a large temperature range, which paves \nthe way for low -power Mn 2Ga spintroni c device applications.  \n2. Experimental  \nMgO, a transparent  insulating oxide with a large bandgap of 7.8 eV, shows excellent high -\ntemperature stability with a melting point of 2800°C. It has a cubic lattice with a = 4.212 Å , \nwhich is close to the in -plane lattice constant of D0 22 Mn 2Ga. Therefore, Mn 2Ga films were \nfirstly grown on single -crystalline (001) -oriented  MgO substrates using the magnetron \nsputtering technique  at different temperatures ranging from 30 to 550°C . The base pressure and \nthe D.C.  sputtering power were  7.5×10-9 Torr and 60 W, respectively. During the deposition, \nthe Ar pressure was kept at 3mTorr. With X -ray reflectometry, t he deposition rate was \ndetermined  to be 35 Å/ min and the total thickness was first kept at 100 nm , which was later \nchanged to 50 and 30 nm f or thickness dependence studies . The crystal structure of Mn 2Ga thin \nfilms was measured by X -ray diffraction (XRD). Magnetic characterization and electrical \nmeasurements were performed by a Quantum Design VersaLab with a vibrating sample \nmagnetometer option . The standard linear four -probe method and the Hall geometr y were used \nfor longitudinal and Hall resistance measurements, respectively.  \n3. Results and discussion  Figure 1a shows single -crystal XRD spectra of Mn 2Ga films grown on MgO substrates \nfabricated at different growth  temperatures  TG. For TG below 450°C, no thin -film peaks are seen, \nimplying disordered polycrystalline films. At 450°C, the (002) peak and a weak (001) peak of \nD0 22 Mn 2Ga show up. With increasing the growth temperature to 550°C, both the (001) and \n(002) peaks of D0 22 Mn 2Ga become sharpe r, indicating enhanced crystallinity and chemical \nordering. The growth temperature could not be further raised. That is because at higher \ntemperatures the surface energies of intermetallic Mn 2Ga and ferroelectric oxide PMN -PT are \nof large different, due to  the non -wetting issue, Mn 2Ga could not form continuous thin films \nbut only separate islands.   \nThe metallicity, which could be examined by the normalized resistivity relative to room -\ntemperature values, is demonstrated in Fig. 1b. When the substrate temperature is lower than \n450°C, Mn 2Ga/MgO  films are semiconducting, which is contrary to its bulk met allic behavior. \nThis suggests the existence of a large degree of chemical disorder  including grain boundaries . \nThe films fabricated at 450 and 550°C  are metallic and the metallicity is improved with \nenhancing substrate temperature.  All these electrical transport results are consistent with the \nXRD spectra shown in Fig. 1a . \nThe out -of-plane and in -plane magnetic moments versus magnetic fields ( M-H) are mea sured \nat 50 and 300 K  for single -crystalline Mn 2Ga/MgO film s (Fig. 1c , f). Overall, both films exhibit \nthe feature of perpendicularly magnetized anisotropy. The room -temperature saturation \nmagnetization of Mn 2Ga/MgO films are ~290 and ~335 emu/cc for growth temperature of 450 \nand 550°C, respectively, which are comparable with the magnetization values of previously \nreported ferrimagnetic Mn 2Ga films  [24]. Similar to the XRD and electrical resistivity data, the \nincrease of the growth temperature improves the squareness of the out -of-plane M-H loop, \nwhich is favorable for perpendicular spintronic device applications. Therefore, 550°C is the \noptimized growth t emperature for high -quality epitaxial and perpendicularly magnetized \nferrimagnetic Mn 2Ga films.  \nIn addition, the room -temperature anisotropy field 0Hk is determined to be ~10 T for the \n550°C -fabricated Mn 2Ga film, which corresponds to a uniaxial magnetocrystalline anisotropy energy Ku of ~1.68 MJ/m3, which is of the same order with the previously reported highest Ku \n(~2.17 MJ/m3) [25] for molecular -beam -epitaxy -fabricated ferrimagnetic Mn-Ga films. For real \napplications, the duration of information storage needs to be more than 10 years, which requires \nthe ratio of magnetocrystalline energy of a bit KuV greater than 40 times room -tempe rature \nthermal energy  40kBT ~ 1.67×10-19 J. Considering a bit cell consisting of a single Mn 2Ga layer \nwith a cubic shape, the critical bit size could accordingly be reduced to ~ 4.6 nm. This thus \nimplies great potential of our optimized Mn 2Ga ferrimagnetic thin films with perpendicular \nmagnetized anisotropy for high -density data storage.  \nLongitudinal magnetotransport properties of Mn 2Ga thin films fabricated at various \ntemperatures were examined  for out -of-plane magnetic fields . It was found that for TG < 300 °C, \nthe magnetoresistance (MR) effect is rather weak  and almost not affected by the growth \ntemperature . Figure 2 shows  the MR curves collected at different temperatures ranging from 50 \nto 300 K for Mn 2Ga thin films with TG > 30°C. For TG = 150°C  (Fig. 2a ), the MR above 50 K \nis positive, which implie s that the orbital scattering due to the Lorent z force is dominant. \nHowever, the MR turns into negative for 50 K, suggesting the important role of magnetic \nmoments of Mn. For TG = 300°C (Fig. 2 b), the room -temperature MR is negligible while the \nlow-temperature MR curves are interestingly linear. It is worth noticing that the positive MR at \n150 K is the largest, reminiscent of the maximal magnetotransport properties at ~200 K for \nnoncollinear antiferromagnets Mn 3Sn [26] and Mn 3Ge [4]. For crystallized epitaxial Mn 2Ga \nfilms , the butterfly -shape hysteresis MR curves  (Fig. 2c , d) are clearly seen, characteristic of \nlong-range ferrimagnetic/ferromagnetic order. In addition, the positive orbital scattering is more \nsignificant at low temperatures, leading to suppressed negative MR.  \nSystematic transverse magnetotransport properties, i.e., the Hall effect, of the Mn 2Ga/MgO \nfilms deposited below 450°C are demonstrated in Fig. 3. Similarly, the Hall curves for TG = 30 \nand 150°C (Fig. 3a , b) are comparable  and are linear above 50 K . At 50 K, the Hall effect \nbecomes nonlinear , signature of the magnetic -moment -related AHE, which is consistent with \nthe negative MR at 50 K in Fig. 2a . For TG = 300°C,  the AHE is obvious below 200 K , indicating \nthe formation of magnetic order.  The Hall effect of epitaxial Mn 2Ga films is shown in Fig. 4a , b. The g eneral shape of the Hall \ncurves is in excellent agreement with that of the M-H loops. Therefore, the Hall effect could \nserve as a sensitive electrical probe to magnetic p roperties of perpendicularly magnetized \nMn 2Ga. Detailed scaling law analysis  [7] (Fig. 4c) on the optimized film reveals that for low \nlongitudinal resistivity range 280 < ρxx < 330 Ωcm, ρxy ∝ ρxx2, the Berry curvature is the \ndominant origin for the AHE , which is a pseudo magnetic field in momentum space  and \ndetermined by the topological bands interaction of Bloch electrons  [27,28] . While for the large \nresistivity region with ρxx > 330 Ωcm, skew scattering becomes more important for generating \na transverse Hall voltage , leading to  ρxy ∝ ρxx [29]. The excellent perpendicular magnetic \nanisotropy remains in thinner films such as 50 and 30 nm  (Fig. 5) , which, in turn, lead to \nsignificantly enhanced anomalous Hall resistance. The much larger anomalous Hall resistance \nin thinner Mn 2Ga films could facilitate the electrical read -out for memory devices.  \nBased on the experimental results mentioned above,  30-nm-thick Mn 2Ga films were further  \nepitaxially integrated onto (001) -oriented PMN -PT ferroelectric oxides with a 25-nm-thick  \nMgO buffer layer  so as  to manipulate its AHE  or magneti sm by electric -field-induced \npiezoelectric strain  [1-6,17-23,30] . The MgO  buffer layer s were grown by a pulsed laser \ndeposition system  at 400°C and post -annealed at 600°C for 1 h, which  was utilized to prevent  \nPb element  in PMN -PT substrate s from diffusing  into the chamber and Mn 2Ga film s to form \nsecondary alloy s at high temperature s [18,31] . As shown in  Fig. 6a, the XRD spectrum  \ncontain ing the (002) peak of  the MgO buffer layer and the (001) and (002) peaks of the Mn 2Ga \nthin film  indicates  the epitaxial growth of the Mn 2Ga on the MgO buffer layer. The f ield-\ndependent Hall signals of an epitaxial Mn 2Ga/MgO/PMN -PT heterostructure  (Fig. 6b) at \ndifferent temperatures are in consistence with that of  Mn 2Ga/MgO  in Fig . 4b, which suggest s \nthe excellent perpendicular magnetic anisotropy  of the epitaxially integrated Mn 2Ga films on \nferroelectric PMN -PT. \nTo explore  the effect of piezoelectric strain on the AHE in  the Mn 2Ga/MgO/PMN -PT \nheterostructure , an electric  field EG of -5 kV/cm was perpendicularly applied across the PMN -\nPT substrate  (Fig. 7a) to pole the ferroelectric substrate at room temperature. To examine any \npossible variation of the AHE, the Hall curves were re -measured after electric poling of the \nPMN -PT. It turns out that under such an electric -field excitation, the AHE is enhanced for all the temperatures  (Fig. 7b-g). The relative electric -field-induced nonvolatile modulation of the \nzero-field anomalous Hall resistance is extracted and plotted in Fig. 7h, which reaches ~ 16% \nbelow 200 K and ~ 14% at 300 K.  \nTo further confir m the nonvolatile nature of the electric -field-induced piezoelectric strain in \nPMN -PT, the room -temperature electric -field dependent longitudinal resistance of the Mn 2Ga \nfilm in the Mn 2Ga/MgO/PMN -PT heterostructure is measured with the linear four -probe \ngeometry  (Fig. 8a). As shown in Fig. 8b, the positive and negative peaks in perpendicular gating \ncurrent through the MgO buffer layer and PMN -PT clearly exhibit the reversible ferroelectric \npolarization switching feature.  Correspondingly, the electric -field-dependent longitudinal \nresistance  (Fig. 8c)  shows an asymmetric and nonvolatile butterfly loop, which is similar to \nwhat we obtained in  previous measur ements  [3,23] . \nEmpir ically , the AHE in ferromagnetic materials is closely related to magnetization. Motivated \nby this understanding, we examined the out -of-plane magnetization change of the Mn 2Ga film \nfor the Mn 2Ga/MgO/PMN -PT heterostructure  upon  electric -field poling (Fig. 9a) of the \nferroelectric substrate PMN -PT. As shown in Fig. 9b , c, the perpendicular magnetization has \nbeen changed substantially. At 50 K, the electric -field poling of the PMN -PT alter the saturation \nmagnetization of Mn 2Ga alters from ~360 to ~423 emu/cc  (Fig. 9b) , which corresponding to an \n~17.5% magnetization enhancement, similar to the anomalous Hall resistance increase in Fig. \n7b. For 300 K, the out -of-plane magnetization changes from ~335 to ~382 emu/cc  (Fig. 9c) , \nwell consistent with the ~14% anomal ous Hall resistance variation in Fig. 7g. Thus, these \nexperimental results clearly illustrate that the piezoelectric -strain -induced anomalous Hall \neffect modulation is predominantly caused by the strain -induced magnetization variation. For \nferrimagnetic ma terials with two opposite unequal sublattices, the enlargement of the net \nmagnetization would likely pertain to the weakening of the compensation of two sublattices  in \nterms of the spin rotation, reminiscent of the scenario of noncollinear antiferromagnetic spin \nstructure modulation by piezoelectric strain as theoretically described by  Lukashev  et al. [32].  \n4. Conclusions  \nIn conclusion, we have fabricated epitaxial ferrimagnetic Mn 2Ga thin films with perpendicular \nmagnetic anisotropy on MgO substrates. The mechanisms of the AHE  were unveiled for different longitudinal resistivit y ranges . When MgO is used as buffer layer, Mn 2Ga thin films \nwith perpendicular anisotropy have been successfully integra ted onto ferroelectric  PMN -PT \nsubstrates  which is useful  for utilizing ferrimagnetic material s in high-density spintronic \ndevices  and could enable the fabrication of other exotic epitaxial heterostructures with \nferrimagnets interfacing with some novel mate rials [33-45]. Via the defects engineering in thin \nfilms  [46], the spin structure and the AHE of ferrimagnetic Mn 2Ga could further be modulated \nto realize the topological Hall effect . More  importantly , the AHE  of ferrimagnetic Mn 2Ga films \nis largely modulated by the electric -field-induced piezoelectric strain , which paves the way for \nmagnetic -field-free low -power ferrimagnetic spintronic device applications.  \n \nAcknowledgements:  Zhi-Qi Liu acknowledges financial support from National Natur al \nScience Foundation of China  (NSFC Grant Nos: 52121001, 51822101, 51861135104 & \n51771009).  \n  References  \n[1] Feng Z, Yan H, Wang X, Guo H, Qin P, Zhou X, Chen Z, Wang H, Jiao Z, Leng  Z, Hu Z, Zhang \nX, Wu H, Chen H, Wang J, Zhang T, Jiang C, Liu Z. Nonvolatile electric control of the anomalous \nHall effect in an ultrathin magnetic metal. Adv Electron Mater. 2019;6(2):1901084.  \n[2] Wang X, Feng Z, Qin P, Yan H, Zhou X, Guo H, Leng Z, Chen W, Jia Q, Hu Z, Wu H, Zhang X, \nJiang C, Liu Z. Integration of the noncollinear antiferromagnetic metal Mn 3Sn onto ferroelectric \noxides for electric -field control. 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Phys Rev B. 2021;104(6):064428.  \n  Figure 1  \n \nFig. 1 a X-ray diffraction pattern s of 100-nm-thick Mn 2Ga/MgO heterostructure s grow n at different \ntemperatures ; b Temperature -dependent normalized resistivity of Mn 2Ga thin films deposited at different \ntemperatures ; c-f The magnetization curves of Mn 2Ga samples deposited at 450 and 550 °C are measured \nat 50 and 300 K, respectively  \n  \nFigure 2  \n \nFig. 2  Magnetoresistance of 100-nm-thick Mn 2Ga/MgO  films fabricated at different growth \ntemperatures TG. a TG = 150 °C; b TG = 300 °C; c TG = 450 °C; f TG = 550 °C \n  \nFigure 3  \n \nFig. 3 Hall effect for 100-nm-thick Mn 2Ga/MgO  films fabricated below 450 °C. a TG = 30°C; b TG \n= 150 °C; c TG = 300 °C  \n  \nFigure 4  \n \nFig. 4 a Hall effect for a 100-nm-thick Mn 2Ga/MgO film fabricated at TG = 450 °C; b Hall effect for a \n100-nm-thick Mn 2Ga/MgO  film fabricated at TG = 550 °C; c Scaling law analysis for the Mn 2Ga film \nfabricated at TG = 550 °C  \n  \nFigure 5  \n \nFig. 5  Hall effect of Mn 2Ga/MgO film s with smaller thickness  fabricated at TG = 550 °C. a 50 nm ; \nb 30 nm   \nFigure 6 \n \nFig. 6 a X-ray diffraction pattern of a Mn 2Ga/MgO/PMN -PT heterostructure ; b Hall effect \nmeasurement s of the Mn 2Ga/MgO/PMN -PT heterostructure at different temperatures ranging from 50 to \n400 K   \n  \nFigure 7 \n \n \nFig. 7 a Schematic of the perpendicular electric -field modulation of the AHE for the Mn 2Ga(30 \nnm)/MgO(25 nm)/PMN -PT heterostructure ; b-g AHE curves for the Mn 2Ga film before and after electric \npoling at various temperatures ; h Relative zero -field Hall resistance modulation as a function of \ntemperature  \n  \nFigure 8 \n \n \nFig. 8 a Schematic of the perpendicular electric -field modulation of the longitudinal resistance of Mn 2Ga \nin a Mn 2Ga/MgO/PMN -PT heterostructure ; b Room -temperature gating current as a function of \nperpendicular electric field ; c Room -temperature longitudinal resistance of the Mn 2Ga film versus \nelectric field.  \n  \nFigure 9  \n \n \nFig. 9  a Schematic of the  out-of-plane magnetization measurements upon electric -field poling of the \nferroelectric PMN -PT substrate for the Mn 2Ga(30 nm)/MgO(25 nm)/PMN -PT heterostructure ; b Out-of-\nplane magnetization of the Mn 2Ga film at 50 K before and after electric -field poling ; c Out-of-plane \nmagnetization of the Mn 2Ga film at 300 K before and after electric -field poling  \n"    },    {        "title": "2001.08701v2.Magnetization_plateaus_and_bipartite_entanglement_of_an_exactly_solved_spin_1_2_Ising_Heisenberg_orthogonal_dimer_chain.pdf",        "content": "Magnetization plateaus and bipartite entanglement of an exactly solved\nspin-1 /2 Ising-Heisenberg orthogonal-dimer chain\nLucia G ´alisov ´aa,\u0003, Jozef Stre ˇckab, Taras Verkholyakc, Samuel Havadejb\naInstitute of Manufacturing Management, Faculty of Manufacturing Technologies with the seat in Preˇ sov, Technical University of Koˇ sice,\nBayerova 1, 080 01 Preˇ sov, Slovakia\nbInstitute of Physics, Faculty of Science, P . J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slovakia\ncInstitute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Street 1, 790 11 L’viv, Ukraine\nAbstract\nSpin-1 /2 orthogonal-dimer chain composed of regularly alternating Ising and Heisenberg dimers is exactly solved in a presence\nof the magnetic field by the transfer-matrix method. It is shown that the ground-state phase diagram involves in total six di \u000berent\nphases. Besides the ferromagnetic phase with fully polarized spins one encounters the singlet antiferromagnetic and modulated\nantiferromagnetic phases manifested in zero-temperature magnetization curves as zero magnetization plateau, the frustrated fer-\nrimagnetic and singlet ferrimagnetic phases causing existence of an intermediate one-half magnetization plateau, and finally, the\nintriguing modulated ferrimagnetic phase with a translationally broken symmetry leading to an unconventional one-quarter mag-\nnetization plateau. The quantum character of individual ground states is quantified via the concurrence, which measures a strength\nof the bipartite entanglement within the pure and mixed states of the Heisenberg dimers at zero as well as nonzero temperatures.\nThe parameter region, where the bipartite entanglement may be in contrast to general expectations reinforced upon increasing of\ntemperature and /or magnetic field, is elucidated.\nKeywords: Ising-Heisenberg model, orthogonal-dimer chain, magnetization plateaus, bipartite entanglement\n1. Introduction\nThe gapped quantum ground states manifested in low-tem-\nperature magnetization curves as intermediate plateaus remain\nat a forefront of intense theoretical studies, because several in-\ntriguing fractional magnetization plateaus were experimentally\ndetected in high-field magnetization curves of the magnetic com-\npound SrCu 2(BO 3)2[1–5] providing a long-sought experimen-\ntal realization of the Shastry-Sutherland model [6, 7]. Despite\nsubstantial e \u000borts, the number and microscopic nature of inter-\nmediate magnetization plateaus of SrCu 2(BO 3)2still remain un-\nresolved issue due to extraordinary mathematical di \u000eculties re-\nlated to a theoretical modeling of the Shastry-Sutherland model\nat zero [8–13] as well as nonzero [14, 15] temperatures.\nBy contrast, the magnetization process of one-dimensional\ncounterpart of the Shastry-Sutherland model, which is com-\nmonly referred to either as the spin-1 /2 Heisenberg orthogonal-\ndimer or dimer-plaquette chain [16–18], is quite well estab-\nlished nowadays. Except three most pronounced magnetiza-\ntion plateaus at zero, one-quarter and one-half of the saturation\nmagnetization, one additionally encounters an infinite series of\nsmaller fractional magnetization plateaus at rational numbers\nn=(2n+2)=1=4;1=3;3=8;:::; 1=2 ranging in between one-\nquarter and one-half magnetization plateaus [19, 20]. Unfortu-\nnately, a respective magnetic compound that would enable an\n\u0003Corresponding author\nEmail address: galisova.lucia@gmail.com (Lucia G ´alisov ´a)experimental testing of this peculiar sequence of the fractional\nmagnetization plateaus is not available to date.\nRecent experimental discovery of the polymeric coordina-\ntion compound [Dy(hfac) 2(CH 3OH)] 2[Cu(dmg)(Hdmg)] 2[21,\n22], which will be further referred to as the polymeric chain\n[Dy 2Cu2]n, has a \u000borded a valuable experimental realization of\nthe spin-1 /2 Ising-Heisenberg orthogonal-dimer chain with a\nregular alternation of the highly anisotropic dimeric units of\nDy3+magnetic ions (Ising dimers) with the almost isotropic\ndimeric units of Cu2+magnetic ions. It is supposed that the\nmost dominant coupling in [Dy 2Cu2]nis by far an antiferromag-\nnetic interaction between the nearest-neighbouring Dy3+and\nCu2+magnetic ions, whereas the dinuclear entities of Dy3+ions\nand Cu2+ions are coupled presumably through much weaker\nferromagnetic interaction [22]. It is worthwhile to remark that\nthe spin-1 /2 Ising-Heisenberg orthogonal-dimer chain with a\nregularly alternating Ising and Heisenberg dimers arranged in\nan orthogonal fashion can be exactly solved by adapting the ap-\nproach elaborated in Refs. [23–25] for various versions of this\nintriguing one-dimensional quantum spin model.\nIn the present work we will introduce and rigorously solve\na spin-1 /2 Ising-Heisenberg orthogonal-dimer chain composed\nof regularly alternating Ising and Heisenberg dimers in a pres-\nence of the external magnetic field. Although the investigated\nquantum spin chain is somewhat oversimplified in that it does\nnot take into account two di \u000berent exchange pathways between\nDy3+and Cu2+magnetic ions existing within the polymeric\ncompound [Dy 2Cu2]n, we believe that this quantum spin chain\nPreprint submitted to Physica E June 2, 2020arXiv:2001.08701v2  [cond-mat.stat-mech]  31 May 2020may shed light on the nature of unconventional quantum ground\nstates invoked by the external magnetic field in a low-temperatu-\nre magnetization process of the polymeric complex [Dy 2Cu2]n.\nThe organization of this paper is as follows. In Section 2 we\nwill introduce and solve the spin-1 /2 Ising-Heisenberg orthogo-\nnal-dimer chain within the framework of the transfer-matrix\nmethod. Section 3 includes a comprehensive discussion of the\nmost interesting results obtained for the ground-state phase dia-\ngram, the magnetization process and the bipartite entanglement\nemergent within the Heisenberg dimers. The most important\nfindings and future outlooks are briefly mentioned in Section 4.\n2. Model and its exact solution\nIn the present paper, we will consider the quantum spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain schematically depict-\ned in Fig. 1 and defined through the total Hamiltonian:\nˆH=JHNX\ni=1\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001+J0\nINX\ni=1ˆ\u001bz\n1;iˆ\u001bz\n2;i\n+JINX\ni=1\u0002ˆSz\n1;i\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001+ˆSz\n2;i\u0000ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001\u0003\n\u0000hHNX\ni=1\u0000ˆSz\n1;i+ˆSz\n2;i\u0001\u0000hINX\ni=1\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001: (1)\nIn above,\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001= \u0001(ˆSx\n1;iˆSx\n2;i+ˆSy\n1;iˆSy\n2;i)+ˆSz\n1;iˆSz\n2;i,ˆS\u000b\n1(2);i\n(\u000b=x;y;z) label the spatial components of the standard spin-\n1/2 operators corresponding to the Heisenberg spins forming\nhorizontal dimers, ˆ \u001bz\n1(2);iare the spatial components of the stan-\ndard spin-1 /2 operators related to the Ising spins forming ver-\ntical dimers, the parameter JHdenotes the XXZ Heisenberg\nintra-dimer interaction on horizontal bonds with the parameter\nof exchange anisotropy \u0001,J0\nIrepresents the Ising intra-dimer\ninteraction between the Ising spins on vertical bonds, and JI\ndenotes the Ising inter-dimer interaction between the nearest-\nneighboring Ising and Heisenberg spins. The last two terms hH\nandhIare Zeeman terms, which account for the magnetostatic\nenergy of the Heisenberg and Ising spins in an applied longitu-\ndinal magnetic field, respectively. Finally, Ndenotes the total\nnumber of the Heisenberg and Ising spin dimers and the peri-\nodic boundary conditions ˆ \u001bz\n1(2);N+1\u0011ˆ\u001bz\n1(2);1are assumed for the\nsake of simplicity.\nFor further convenience, the total Hamiltonian (1) of the\nspin-1=2 Ising-Heisenberg orthogonal-dimer chain can be alter-\nnatively rewritten as a sum of the six-spin cluster Hamiltonians\nschematically delimited in Fig. 1 by a dotted rectangle:\nˆH=NX\ni=1ˆHi; (2)\nˆHi=JH\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001+J0\nI\n2\u0000ˆ\u001bz\n1;iˆ\u001bz\n2;i+ˆ\u001bz\n1;i+1ˆ\u001bz\n2;i+1\u0001\n+JI\u0002ˆSz\n1;i\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001+ˆSz\n2;i\u0000ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001\u0003\n\u0000hH\u0000ˆSz\n1;i+ˆSz\n2;i\u0001\u0000hI\n2\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i+ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001:(3)\n/s49/s44 /s105/s43/s49\n/s50/s44 /s105/s43/s49/s83\n/s49/s44 /s105/s43/s49/s83\n/s50/s44 /s105/s83\n/s49/s44 /s105/s49/s44 /s105\n/s50/s44 /s105/s83\n/s50/s44 /s105/s43/s49\n/s105/s116/s104/s32/s115/s105/s120/s45/s115/s112/s105/s110/s32/s99/s108/s117/s115/s116/s101/s114/s74\n/s72/s40 /s41\n/s74\n/s73/s39/s74\n/s73\n/s74\n/s73/s74\n/s73\n/s74\n/s73/s83\n/s50/s44 /s105/s45/s49Figure 1: The magnetic structure of the frustrated spin-1 /2 Ising-Heisenberg\northogonal-dimer chain. Green (red) balls denote lattice sites occupied by the\nIsing (Heisenberg) spins, thin green (thick red) lines correspond to the Ising\n(Heisenberg) intra-dimer bonds and dashed black lines illustrate the Ising inter-\ndimer bonds. The dotted rectangle represents the ith six-spin cluster described\nby the cluster Hamiltonian (3).\nIt is noteworthy that the ith six-spin cluster Hamiltonian (3) in-\nvolves the vertical Ising dimers from two adjacent unit cells,\nwhereas the factor 1 =2 emergent at the Ising coupling J0\nIand\nthe Zeeman term hIavoids a double counting of these two in-\nteraction terms being symmetrically split into two consecutive\ncluster Hamiltonians.\nIt is obvious from Eq. (3) that di \u000berent cluster Hamiltoni-\nans satisfy the standard commutation relation [ ˆHi;ˆHj]=0, and\nthus, the determination of eigenvalues of ˆHiis su\u000ecient to ex-\nactly resolve the considered spin-1 =2 Ising-Heisenberg orthogo-\nnal-dimer chain. The relevant calculation can be performed in\na matrix representation of the Hilbert subspace spanned over\nthe orthonormal basis of four available spin states correspond-\ning to the ith Heisenberg spin pair\bj\"\"i i=j\"i1;ij\"i2;i;j\"#i i=\nj\"i 1;ij#i 2;i;j#\"i i=j#i 1;ij\"i 2;i;j##i i=j#i 1;ij#i 2;i\t, where\nj\"i 1(2);iandj#i 1(2);idenote eigenvectors of the spin operator\nˆSz\n1(2);iwith the eigenvalues Sz\n1(2);i=1=2 and\u00001=2, respectively.\nAs a result, one obtains the following set of eigenvalues:\nEi;1=JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n+JI\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001\u0000hH;(4a)\nEi;2=JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n\u0000JI+hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001+hH;(4b)\nEi;3=\u0000JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n+1\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2\n\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001; (4c)\nEi;4=\u0000JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n\u00001\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2\n\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001; (4d)\n2and corresponding eigenvectors:\nj ii;1=j\"\"i i; (5a)\nj ii;2=j##i i; (5b)\nj ii;3=sin'ij\"#i i+cos'ij#\"i i; (5c)\nj ii;4=sin'ij\"#i i\u0000cos'ij#\"i i; (5d)\nwhere tan (2'i)=JH\u0001=\u0002JI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u0001\u0003.\nHaving the full set of eigenvalues of the cluster Hamilto-\nnian (3), the partition function Zof the investigated quantum\nspin chain can be derived by applying the standard transfer-\nmatrix approach [26, 27]:\nZ=X\nf\u001b1;i;\u001b2;igNY\ni=1TrS1;i;S2;iexp\u0000\u0000\fHi\u0001=X\nf\u001b1;i;\u001b2;igNY\ni=14X\nj=1exp\u0000\u0000\fEi;j\u0001\n=X\nf\u001b1;i;\u001b2;igNY\ni=1T\u0000\u001bz\n1;i;\u001bz\n2;i;\u001bz\n1;i+1;\u001bz\n2;i+1\u0001=TrTN=4X\nj=1\u0015N\nj:(6)\nIn above,\f=1=(kBT) is the inverse temperature ( kBis the\nBoltzmann’s constant and Tis the absolute temperature), the\nsymbolP\nf\u001b1;i;\u001b2;igdenotes a summation over all possible config-\nurations of the Ising spins from all vertical bonds, the prod-\nuctQN\ni=1runs over all six-spin clusters visualized in Fig. 1\nand Tr S1;i;S2;istands for a trace over degrees of freedom of the\nith Heisenberg spin dimer. Apparently, the applied formalism\nenables one to express the partition function Zof the spin-\n1/2 Ising-Heisenberg orthogonal-dimer chain in terms of four\neigenvalues \u00151,\u00152,\u00153,\u00154of the 4\u00024 transfer matrix T, whose\nelements are formed by 16 Boltzmann’s weights corresponding\nto all available states of two adjacent Ising spin dimers from the\nith six-spin cluster (see Fig. 1) as defined by the formula:\nT\u0000\u001bz\n1;i;\u001bz\n2;i;\u001bz\n1;i+1;\u001bz\n2;i+1\u0001=2exp\"\n\u0000\fJ0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001#\n\u0002exp\"\fJH\n4+\fhI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001#\n\u0002(\nexp\u0012\n\u0000\fJH\n2\u0013\ncosh\u0014\fJI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001\u0000\fhH\u0015\n+cosh\"\f\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2#)\n:(7)\nFrom the physical point of view, the transfer matrix (7) rep-\nresents the e \u000bective Boltzmann’s factor, which was obtained\nafter tracing out spin degrees of freedom of two Heisenberg\nspins from the ith horizontal dimer. Explicit expressions of the\ntransfer-matrix eigenvalues emerging in the final form of the\npartition function (6) are:\n\u00151=0; (8a)\n\u0015j=a\n3+2\n3sgn(q)ppcos26666641\n3tan\u000010BBBBB@p\np3\u0000q2\nq1CCCCCA+2\u0019(j\u00002)\n33777775\n(j=2;3;4);(8b)\nwhere:\na=A1+2A0+A\u00001;\np=a2\u00003(A1A\u00001+2A0A\u00001+2A0A1\u00002B2\n1\u00002B2\n\u00001\u0000B2\n0);q=a3\u00009a(A1A\u00001+2A0A\u00001+2A0A1\u00002B2\n1\u00002B2\n\u00001\u0000B2\n0)\n+27(A1A0A\u00001\u0000A\u00001B2\n1\u0000A1B2\n\u00001\u0000A0B2\n0+2B1B0B\u00001):\nThe coe \u000ecients AxandBx(x=\u00001;0;1) entering into the for-\nmula (8b) either directly, or through the parameters p,q, are\ngiven by:\nAx=2exp\"\fJH\n4+\fJ0\nI(\u00001)x\n4+\fhIx#\n\u0002\"\n2exp\u0012\n\u0000\fJH\n2\u0013\ncosh (\fJIx\u0000\fhH)+cosh \fJH\u0001\n2!#\n;\nBx=2exp266664\fJH\n4+\fJ0\nI(x2\u00001)\n4+\fhIx\n2377775\n\u0002(\nexp\u0012\n\u0000\fJH\n2\u0013\ncosh\u0012\fJIx\n2\u0000\fhH\u0013\n+cosh\u0014\f\n2q\nJ2\nI(x2\u00002)2+(JH\u0001)2\u0015)\n:\nAfter the explicit form of the transfer-matrix eigenvalues (8a)-\n(8b) is substituted into the final expression for the partition\nfunction (6) one obtains a crucial result of our calculations,\nfrom which the whole thermodynamics of the spin-1 /2 Ising-\nHeisenberg orthogonal-dimer chain directly follows. As a mat-\nter of fact, the Gibbs free energy Gper elementary unit cell can\nbe expressed in the thermodynamic limit N!1 in terms of\nthe largest transfer-matrix eigenvalue:\nG=\u0000kBTlim\nN!11\nNlnZ=\u0000kBTln(maxf\u00150;\u00151;\u00152;\u00153g):(9)\nOther important physical quantities, such as the local magne-\ntization mI=hˆ\u001bz\n1;i+ˆ\u001bz\n2;ii=2,mH=hˆSz\n1;i+ˆSz\n2;ii=2 per Ising\nand Heisenberg spin, respectively, the total magnetization m\nper lattice site, as well as the pair correlation functions cz\nII=\nhˆ\u001bz\n1;iˆ\u001bz\n2;ii,cxx(yy)\nHH=hˆSx\n1;iˆSx\n2;ii=hˆSy\n1;iˆSy\n2;ii,czz\nHH=hˆSz\n1;iˆSz\n2;iiand\nczz\nIH=hˆSz\n1;iˆ\u001bz\n1;ii=hˆSz\n1;iˆ\u001bz\n2;ii=hˆSz\n2;iˆ\u001bz\n1;i+1i=hˆSz\n2;iˆ\u001bz\n2;i+1i, which\nbring insight into the local order of the nearest-neighbour spins\ncan be subsequently obtained by means of the di \u000berential cal-\nculus:\nmI=\u00001\n2@G\n@hI;mH=\u00001\n2@G\n@hH;m=1\n2(mI+mH);(10)\nczz\nII=@G\n@J0\nI; cxx(yy)\nHH=1\n2JH@G\n@\u0001; (11)\nczz\nHH=@G\n@JH\u0000\u0001\nJH@G\n@\u0001;czz\nIH=1\n4@G\n@JI: (12)\nThe knowledge of rigorous results for the local magnetization\nmHand pair correlation functions cxx(yy)\nHHandczz\nHHcorresponding\nto the Heisenberg spin pairs gives the opportunity to rigorously\ncalculate an interesting physical quantity called concurrence ac-\ncording to the formula [28–31]:\nC=max8>>><>>>:0;4jcxx(yy)\nHHj\u00002s\u00121\n4+czz\nHH\u00132\n\u0000m2\nH9>>>=>>>;: (13)\nThe quantity (13) represents a feasible measure of bipartite\nentanglement of the Heisenberg spins forming the horizontal\ndimers at zero as well as nonzero temperatures.\n33. Results and discussion\nIn this section, we will discuss a diversity of zero-tempera-\nture spin arrangements, magnetization process and bipartite en-\ntanglement of the particular version of the quantum spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain with the antiferromag-\nnetic exchange interactions JH>0,JI>0 and J0\nI>0. Without\nloss of generality, we will restrict our analysis to the special\ncase of the orthogonal-dimer chain with the isotropic Heisen-\nberg intra-dimer interaction (the case \u0001 =1), which exhibits all\ngeneric features of the more general quantum spin-1 =2 Ising-\nHeisenberg orthogonal-dimer chain with the anisotropic XXZ\nHeisenberg intra-dimer interaction with \u0001,1. To reduce a\nnumber of free parameters, we will also assume the same mag-\nnetic fields acting on the Ising and Heisenberg spins hI=hH=\nh, which corresponds to setting the same g-factors for these\nspins from the physical point of view.\n3.1. Ground-state phase diagram\nLet us start by analyzing possible zero-temperature spin ar-\nrangements of the considered quantum spin chain. By compar-\ning the eigenvalues (4a)–(4d) for all available configurations of\nthe Ising spins from ith and ( i+1)st vertical dimers one can\nidentify in total six di \u000berent ground states specified below:\n(i) The ferromagnetic (FM) phase – the unique classical phase\nwith the perfect ferromagnetic arrangement of all the Ising\nand Heisenberg spins:\njFMi=NY\ni=1\f\f\f\"\n\"E\ni\nj\"\"i i: (14)\nThe energy is: EFM=N\n4\u0000JH+J0\nI+4JI\u00008h\u0001;\n(ii) The frustrated ferrimagnetic (FI) phase – the macroscop-\nically degenerate (2N) ferrimagnetic phase with the anti-\nferromagnetic spin arrangement on the vertical Ising di-\nmers and the ferromagnetic spin arrangement on the hor-\nizontal Heisenberg dimers:\njFIi=NY\ni=1\f\f\f\"\n#E\ni\u0010\nor\f\f\f#\n\"E\ni\u0011\n\nj\"\"i i: (15)\nThe energy is: EFI=N\n4\u0000JH\u0000J0\nI\u00004h\u0001;\n(iii) The singlet ferrimagnetic (SFI) phase – the unique quan-\ntum phase with the ferromagnetic spin arrangement of the\nvertical Ising dimers and the fully entangled singlet state\nof the horizontal Heisenberg dimers:\njSFIi=NY\ni=1\f\f\f\"\n\"E\ni\n1p\n2\u0010\nj\"#i i\u0000j#\"i i\u0011\n: (16)\nThe energy is: ESFI=\u0000N\n4\u00003JH\u0000J0\nI+4h\u0001;(iv) The singlet antiferromagnetic (SAF) phase – the macro-\nscopically degenerate (2N) antiferromagnetic phase with\nthe perfect antiferromagnetic spin arrangement of the ver-\ntical Ising dimers and the fully entangled singlet state of\nthe horizontal Heisenberg dimers:\njSAFi=NY\ni=1\f\f\f\"\n#E\ni\u0010\nor\f\f\f#\n\"E\ni\u0011\n\n1p\n2\u0010\nj\"#i i\u0000j#\"i i\u0011\n:(17)\nThe energy is: ESAF=\u0000N\n4\u00003JH+J0\nI\u0001;\n(v) The modulated ferrimagnetic (MFI) phase – the macro-\nscopically degenerate (2N=2) phase characterized by a reg-\nular alternation of the ferromagnetically and antiferro-\nmagnetically ordered vertical Ising dimers and the sing-\nlet-like state of the horizontal Heisenberg dimers:\njMFIi=N=2Y\ni=1\f\f\f\"\n\"E\n2i\u00001\n\u0010\nsin'1j\"#i 2i\u00001\u0000cos'1j#\"i 2i\u00001\u0011\n\n\f\f\f\"\n#E\n2i\u0010\nor\f\f\f#\n\"E\n2i\u0011\n\n\u0010\ncos'1j\"#i 2i\u0000sin'1j#\"i 2i\u0011\n:(18)\nThe energy is: EMFI=\u0000N\n4\u0012\nJH+2q\nJ2\nI+J2\nH+2h\u0013\n;\n(vi) The modulated antiferromagnetic (MAF) phase – the two-\nfold degenerate phase characterized by a regular alterna-\ntion of two kinds of fully polarized vertical Ising dimers\nand the other singlet-like state of the horizontal Heisen-\nberg dimers:\njMAFi=N=2Y\ni=1\f\f\f\"\n\"E\n2i\u00001\n\u0010\nsin'2j\"#i 2i\u00001\u0000cos'2j#\"i 2i\u00001\u0011\n\n\f\f\f#\n#E\n2i\n\u0010\ncos'2j\"#i 2i\u0000sin'2j#\"i 2i\u0011\n:(19)\nThe energy is: EMAF=\u0000N\n4\u0012\nJH+2q\n4J2\nI+J2\nH\u0000J0\nI\u0013\n:\nNote that the two-site ket vectors with vertically written ar-\nrows in the eigenvectors (14)–(19) determine spin arrangements\nwithin the vertical Ising dimers, while the ones with horizon-\ntally written arrows determine spin arrangements within the\nhorizontal Heisenberg dimers. Up (down) arrow refers to the\nspin state 1 =2 (\u00001=2) in both kinds of ket vectors. The mix-\ning angles'1and'2in the last two eigenvectors (18) and (19),\nwhich determine a degree of quantum entanglement within the\nhorizontal Heisenberg spin dimers in the MAF and MFI phases,\nrespectively, are given by the relation tan (2'n)=JH=(nJI)\n(n=1;2).\nThe overall ground-state behavior of the investigated spin-\n1=2 Ising-Heisenberg orthogonal-dimer chain is depicted in\nFig. 2 in the parameter planes J0\nI=JI\u0000JH=JIforh=JI=0\n(panel a) and J0\nI=JI\u0000h=JIfor three representative values of the\ninteraction ratio JH=JI=1=4,p\n2=2, 1 (panels b-d). Black solid\nlines in the displayed phase diagrams indicate first-order (dis-\ncontinuous) phase transitions between the coexisting phases.\n4/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s74\n/s72/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s77/s65/s70\n/s97/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s77\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s98/s70/s73\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s73\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s99/s70/s77\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s77\n/s70/s73\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s100/s48/s46/s48/s48/s48/s48/s46/s49/s48/s53/s48/s46/s50/s49/s48/s48/s46/s51/s49/s53/s48/s46/s52/s50/s48/s48/s46/s53/s50/s53/s48/s46/s54/s51/s48/s48/s46/s55/s51/s53/s48/s46/s56/s52/s48/s48/s46/s57/s52/s53/s49/s46/s48/s48/s48/s67Figure 2: The ground-state phase diagrams of the spin-1 =2 Ising-Heisenberg orthogonal-dimer chain in the J0\nI=JI\u0000JH=JIparameter plane by assuming zero magnetic\nfield h=JI=0 (panel a) and in the J0\nI=JI\u0000h=JIparameter plane for three representative values of the interaction ratio JH=JI=0:25 (panel b), JH=JI=p\n2=2\n(panel c) and JH=JI=1 (panel d). The figures are supplemented with a density plot of the concurrence Cmeasuring a bipartite entanglement within the horizontal\nHeisenberg dimers.\nThey were analytically calculated by comparing the ground-\nstate energies corresponding to the eigenvectors listed in\nEqs. (14)–(19). As one can see from Fig. 2a, only two quan-\ntum phases SAF and MAF emerge as possible ground states at\nzero magnetic field h=JI=0. The SAF phase is stable in the\nparameter region J0\nI=JI>p\n4+(JH=JI)2\u0000JH=JI, where the\npredominant intra-dimer coupling J0\nImaintains the antiparallel\nspin alignment of the vertical Ising dimers and the intra-dimer\ninteraction JHis strong enough to create the fully entangled\nsinglet-dimer state on all horizontal Heisenberg dimers. In the\nrest of the parameter space, the peculiar MAF phase with a two-\nfold broken translational symmetry due to a regular alternation\nof two kinds of fully polarized vertical Ising dimers and two\nalternating kinds of singlet-like states of the horizontal Heisen-\nberg dimers appears. It is noteworthy that the total magnetiza-\ntion is zero within both zero-field ground states SAF and MAF.\nThe situation becomes more complex after turning on the\nexternal magnetic field. Besides the SAF and MAF phases,\nthree ferrimagnetic phases SFI, MFI and FI with nonzero mag-\nnetization can be observed in addition to the fully polarized\nFM phase due to a mutual interplay between the applied mag-netic field hand the coupling constants JH,J0\nI,JI. It is obvious\nfrom Figs. 2b-d that the parameter regions corresponding to the\nquantum phases SAF and SFI (MAF) are gradually extended\n(reduced) upon increasing value of the interaction ratio JH=JI\npromoting existence of the singlet-dimer state on the horizon-\ntal bonds, while the classical FI and FM phases are contrarily\nshifted towards higher magnetic fields. As a result, both spin\narrangements inherent to SAF and SFI phases simultaneously\nappear in a zero-temperature magnetization process for mod-\nerate values of the interaction ratio J0\nI=JI2\u0000p\n4+(JH=JI)2\u0000\nJH=JI;2JH=JI\u0001after a relative strength of the Heisenberg intra-\ndimer coupling exceeds the value JH=JI=p\n2=2 (see Fig. 2d).\nLast but not least, the evolution of the parallelogram-shaped pa-\nrameter space corresponding to the MFI phase, which emerges\nat moderate values of the interaction ratio J0\nI=JIand the mag-\nnetic field h=JIfaithfully follows the trend of adjacent phases\nMAF, SFI, SAF, and FI: the phase boundaries MFI–SFI and\nSAF–MFI are gradually prolonged, while the ones MAF–MFI,\nMFI–FI are gradually shortened upon increasing of the interac-\ntion ratio JH=JI. For the particular value JH=JI=p\n2=2, the\nparameter region corresponding to the MFI phase has the shape\n5of a rhombus with the shorter diagonal parallel to the field-axis\n(see Fig. 2c).\n3.2. Magnetization process at zero and nonzero temperatures\nThe observed diversity of the ground states suggests vari-\nous magnetization scenarios at zero temperature. In fact, the\nzero-temperature magnetization curves of the studied spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain may exhibit the zero\nplateau as well as intermediate magnetization plateaus at one-\nquarter, one-half and three-quarters of the saturation magnetiza-\ntion according to the Oshikawa-Yamanaka-A \u000feck rule [32, 33]\nas long as the period doubling of a magnetic unit cell is consid-\nered. In accordance with this rule, the plateau at zero magneti-\nzation is pertinent either to SAF or MAF ground state, the in-\ntermediate 1 /4-plateau corresponds to the MFI ground state, the\nintermediate 1 /2-plateau relates either to the SFI or FI ground\nstate, while the last possible intermediate 3 /4-plateau does not\nemerge in general. A comprehensive view of the situation is\nprovided by three-dimensional (3D) plots of the total magne-\ntization mnormalized with respect to its saturation value msat,\nwhich are depicted in Fig. 3 against the magnetic field h=JIand\nthe interaction ratio J0\nI=JIby assuming either zero temperature\nkBT=JI=0 (left panels) or su \u000eciently small but finite tempera-\nturekBT=JI=0:1 (right panels). For the sake of a comparison,\nthe interaction ratio JH=JIis fixed to the same values as used\nfor a construction of the ground-state phase diagrams depicted\nin Figs. 2b-d. Obviously, the zero-temperature magnetization\ncurves plotted in left panels of Fig. 3 reflect up to six di \u000berent\nsequences of the field-driven phase transitions depending on a\nmutual interplay between the intra- and inter-dimer coupling\nconstants JH,JI,J0\nI:\n(i) MAF!SFI!FM ,\n(ii) MAF!MFI!SFI!FM ,\n(iii) MAF!MFI!FI!FM ,\n(iv) SAF!MFI!FI!FM ,\n(v) SAF!FI!FM ,\n(vi) SAF!MFI!SFI!FM .\nIn agreement with the aforementioned ground-state analysis,\nthe sequences of the field-induced phase transitions MAF !\nSFI!FM, MAF!MFI!SFI!FM, SAF!MFI!FI!\nFM and SAF!FI!FM emerge in the zero-temperature mag-\nnetization curves for any value of the interaction ratio JH=JI,\nwhile the ones MAF !MFI!FI!FM and SAF!MFI\n!SFI!FM can be identified in the magnetization process\nonly for JH=JI<p\n2=2 and JH=JI>p\n2=2, respectively (see\nFigs. 3a1 and 3c1). Of course, the actual magnetization plateaus\nand discontinuous magnetization jumps at the critical fields,\nwhich correspond to individual field-induced phase transitions,\ncan be detected merely at zero temperature, because any finite\ntemperature completely wipes a discontinuity in the magnetiza-\ntion and also diminishes the perfect plateaus from the respective\nisothermal magnetization curves (see right panels of Fig. 3).In general, the staircase character of the magnetization curves\nis gradually smoothing upon increasing of temperature until it\ncompletely vanishes due to su \u000eciently strong thermal fluctua-\ntions.\n3.3. Bipartite quantum entanglement\nTo gain a better insight into a degree of bipartite quantum\nentanglement emergent within the pure states of the horizontal\nHeisenberg spin dimers, the zero-temperature phase diagrams\ndisplayed in Fig. 2 are supplemented with the corresponding\ndensity plots of the concurrence calculated according to Eq. (13).\nAs expected, the concurrence Cbecomes non-zero in all quan-\ntum ground states SAF, MAF, SFI, and MFI, while it equals\nzero within two classical ground states FM and FI phases with a\nfull alignment of the horizontal Heisenberg dimers towards the\nmagnetic field [see the respective eigenvectors (14) and (15)].\nThe quantum phases SAF, SFI, MAF and MFI generally show\na di\u000berent strength of the bipartite quantum entanglement as\nevidenced by zero-temperature asymptotic values of the con-\ncurrence:\nCSAF=CSFI=1; CMAF=JH=JIp\n4+(JH=JI)2;\nCMFI=JH=JIp\n1+(JH=JI)2: (20)\nIt can be understood from Eq. (20) as well as the density plots\nshown in Fig. 2 that the Heisenberg dimers residing on the hor-\nizontal bonds are fully entangled only within the SAF and SFI\nground states. On the other hand, the strength of the bipar-\ntite quantum entanglement within the MAF and MFI ground\nstates basically depends on a relative strength of the Heisenberg\nand Ising intra-dimer interactions. More specifically, the higher\nvalue the interaction ratio JH=JItakes, the more strongly en-\ntangled the horizontal Heisenberg dimers are. It could be gen-\nerally concluded that the Heisenberg dimers generally display\na stronger quantum entanglement in the MFI phase than in the\nMAF phase when assuming the same value of the interaction\nratio JH=JI. In contrast to the ground states SAF and SFI with\na perfect quantum entanglement of the horizontal Heisenberg\ndimers, the perfect bipartite quantum entanglement cannot be\nreached neither in MAF nor in MFI phase for any finite value\nof the interaction ratio JH=JI, because the Heisenberg spin pairs\nreside in the outstanding singlet-like states instead of a perfect\nsinglet-dimer state [cf. the eigenvectors (18) and (19) with the\nones (16) and (17)].\n3.4. Bipartite thermal entanglement\nLast but not least, let us turn our attention to a detailed ex-\namination of the bipartite thermal entanglement, which refers\nto a bipartite entanglement emergent within the mixed states of\nthe horizontal Heisenberg dimers at nonzero temperatures. The\noverall picture of this issue can easily be created from density\nplots of the concurrence Calong with its magnetic-field and\ntemperature dependencies, which are depicted in Figs. 4–6 for\nthe particular value of the interaction ratio JH=JI=p\n2=2 and\nfour di \u000berent values of the interaction ratio J0\nI=JI=1=4;1;2\n6Figure 3: 3D plots of the total magnetization mreduced with respect to its saturation value msatas a function of the magnetic field h=JIand the interaction ratio J0\nI=JI\nfor three fixed values of the interaction ratio JH=JI=1=4 (panels a1, a2), JH=JI=p\n2=2 (panels b1, b2) and JH=JI=1 (panels c1, c2) at two di \u000berent temperatures\nkBT=JI=0 (left panels) and kBT=JI=0:1 (right panels). The curves of distinct colors refer to di \u000berent magnetization scenarios, which can be observed in the\nindividual 3D plots.\n7/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s83/s70/s73/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s77/s65/s70\n/s97/s70/s77\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s83/s65/s70/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s70/s77 /s77/s65/s70 /s77/s70/s73 /s83/s70/s73\n/s98\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s83/s65/s70 /s77/s70/s73\n/s99/s70/s77 /s70/s73\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s48/s46/s48/s48/s48/s48/s46/s49/s48/s53/s48/s46/s50/s49/s48/s48/s46/s51/s49/s53/s48/s46/s52/s50/s48/s48/s46/s53/s50/s53/s48/s46/s54/s51/s48/s48/s46/s55/s51/s53/s48/s46/s56/s52/s48/s48/s46/s57/s52/s53/s49/s46/s48/s48/s48\n/s83/s65/s70/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s83/s65/s70\n/s100/s67\n/s70/s73Figure 4: The density plots of the concurrence Cin the h=JI\u0000kBT=JIparameter plane for the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=1=4\n(panel a), J0\nI=JI=1 (panel b), J0\nI=JI=2 (panel c), J0\nI=JI=3 (panel d).\nand 3. As one could expect, the displayed data for the con-\ncurrence at low enough temperatures kBT=JI.0:05 faithfully\nresemble zero-temperature asymptotic values, which were dis-\ncussed in above by the ground-state analysis. For the inter-\naction ratios J0\nI=JI=1=4 and 1 the concurrence Cfirst in-\ncreases upon increasing of the magnetic field near the critical\nfields hc1=JI\u00190:7071 (for J0\nI=JI=1=4) and hc1=JI\u00190:3966,\nhc2=JI\u00191:0176 (for J0\nI=JI=1) due to a strengthening of the bi-\npartite entanglement within the horizontal dimers near the field-\ninduced phase transitions MAF !SFI and MAF!MFI,\nMFI!SFI, respectively. The second (third) field-induced\nphase transition SFI !FM, which can be found for both partic-\nular values of the interaction ratio J0\nI=JIat the same saturation\nfield hc2(3)=JI\u00191:7071 is responsible for a sudden drop of the\nconcurrence Cto zero, which confirms a breakdown of the bi-\npartite entanglement (see Figs. 4a,b and 5a,b). On the other\nhand, the bipartite entanglement of the horizontal Heisenberg\ndimers generally weakens for J0\nI=JI=2 along the whole mag-\nnetization process. In fact, the first rapid decrease of the con-\ncurrence observable around the critical field hc1=JI\u00190:4824 is\nattributable to the field-induced transition SAF !MFI, whilethe second abrupt decline of the concurrence associated with a\ncomplete breakdown of the concurrence emerges near the crit-\nical field hc2=JI\u00190:9319 of the field-driven phase transition\nMFI!FI (see Figs. 4c and 5c). Finally, the bipartite ther-\nmal entanglement disappears upon strengthening of the mag-\nnetic field according to the most standard scheme for the high-\nest value of the interaction ratio J0\nI=JI=3 illustrated in Figs. 4d\nand 5d. A breakdown of the concurrence already appears in a\nvicinity of the first critical field hc1=JI\u00190:7071, which corre-\nsponds to the field-induced phase transition from the quantum\nground state SAF to the classical one FI (see Fig. 4d and also\nFig. 5d).\nOf course, an increase in temperature causes a gradual\nsmoothing of abrupt changes of the concurrence observable at\nlow enough temperatures in a proximity of the critical fields, be-\ncause the bipartite entanglement between the Heisenberg spin\npairs is in general suppressed by thermal fluctuations above\nall quantum ground states (see Fig. 5). However, it should\nbe also mentioned that a small temperature rise may eventu-\nally invoke a gentle strengthening of the thermal entanglement.\nAs a matter of fact, the concurrence Cmay exhibit an out-\n8/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48/s67\n/s104/s32 /s47/s32/s74\n/s73/s97/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s83/s70/s73\n/s98\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s99/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s100Figure 5: The magnetic-field dependencies of the concurrence Cfor the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=1=4 (panel a), J0\nI=JI=1\n(panel b), J0\nI=JI=2 (panel c), J0\nI=JI=3 (panel d) by assuming four di \u000berent values of temperature kBT=JI.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s32 /s104/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s49\n/s32/s32/s48/s46/s54\n/s32/s32/s48/s46/s55/s48/s55/s49\n/s32/s32/s49/s46/s48\n/s32/s32/s49/s46/s55/s48/s55/s49\n/s32/s32/s49/s46/s56\n/s32/s32/s50/s46/s53/s67\n/s107\n/s66/s84/s32 /s47/s32/s74\n/s73/s97/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s32 /s104/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s49\n/s32/s32/s48/s46/s54\n/s32/s32/s48/s46/s55/s48/s55/s49\n/s32/s32/s48/s46/s56\n/s32/s32/s49/s46/s53/s67\n/s107\n/s66/s84/s32 /s47/s32/s74\n/s73/s98\nFigure 6: The temperature dependencies of the concurrence Cfor the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=0:25 (panel a), J0\nI=JI=3\n(panel b) by assuming a few di \u000berent values of the external magnetic field h=JI.\nstanding temperature-induced rise on account of thermal ex-\ncitations from a less entangled quantum ground state towardsa more entangled excited state (see for instance the low-field\nparts of solid red and dashed green curves corresponding to\n9the temperatures kBT=JI=0:05 and 0:2 in Figs. 5a,b and the\ndashed green curve for the magnetic field h=JI=0:6 in Fig. 6a.\nFurthermore, the density plots of the concurrence along with\nthe magnetic-field and temperature dependencies depicted in\nFigs. 4–6 clearly evidence that the thermal entanglement of the\nhorizontal Heisenberg dimers, although relatively weak, is also\ndetectable at nonzero temperatures above the classical FI and\nFM ground states. This peculiar finding can be repeatedly ex-\nplained in terms of a thermal activation of the entangled low-\nlying excited states related to some of the quantum phases SFI,\nMFI, MAF or SAF.\n4. Conclusion\nIn the present work we have introduced and exactly solved\na spin-1 /2 Ising-Heisenberg orthogonal-dimer chain, which is\ncomposed of regularly alternating Ising and Heisenberg dimers\nplaced in an external magnetic field. After tracing out spin\ndegrees of freedom of the Heisenberg dimers, the considered\nquantum spin chain has been rigorously treated by making use\nof the classical transfer-matrix approach. It is shown that the\nground-state phase diagram involves in total six di \u000berent ground\nstates. In addition to the classical ferromagnetic phase emergent\nabove the saturation field one also encounters two ground states\nwith zero total magnetization referred to as the singlet and mod-\nulated antiferromagnetic phases, two ground states with the to-\ntal magnetization equal to a half of the saturation value referred\nto as the frustrated ferrimagnetic phase and the singlet ferrimag-\nnetic phase and, finally, one peculiar ground state with the total\nmagnetization equal to a quarter of the saturation value referred\nto as the modulated ferrimagnetic phase. It is also evidenced\nthat the diversity of the ground states gives rise to six di \u000ber-\nent magnetization scenarios depending on a mutual interplay of\nthree considered coupling constants.\nA particular attention has been paid to quantification of the\nbipartite entanglement within the pure and mixed states of the\nhorizontal Heisenberg dimers at zero and nonzero temperatures\nwith the help of concurrence. Surprisingly, it turns out that\nthe bipartite entanglement may be reinforced by increasing of\ntemperature or even upon strengthening of the magnetic field,\nwhich is in contrast with general expectations. In addition, the\nbipartite thermal entanglement has been identified also above\ntwo classical phases: the ferromagnetic phase and the frustrated\nferrimagnetic phase. This unexpected finding can be ascribed to\nthermal excitations driving the investigated quantum spin chain\nfrom a pure classical ground state to a mixed state incorporating\nlow-lying excited state(s) closely connected to other quantum\nground states.\nAlthough the magnetic structure of the investigated spin-1 /2\nIsing-Heisenberg orthogonal-dimer chain was inspired by a het-\nerobimetallic backbone of the coordination polymer [Dy 2Cu2]n,\nthe substantial di \u000berence between the Land ´e g-factors of Dy3+\nand Cu2+magnetic ions ( gDy\u001920 vs. gCu\u00192:2) precludes a\nstraightforward comparison of the obtained theoretical results\nwith the available experimental data [21, 22]. The investiga-\ntion of the di \u000berence of the respective Land ´e g-factors alongwith anisotropy of the couplings constants is accordingly left as\nfuture task for our forthcoming study.\nAcknowledgment\nThis work was financially supported by the grant of The\nMinistry of Education, Science, Research and Sport of the Slo-\nvak Republic under the contract No. VEGA 1 /0105/20 and by\nthe grant of the Slovak Research and Development Agency un-\nder the contract No. APVV-16-0186.\nReferences\n[1] H. Kageyama, K. Yoshimura, R. Stern, N. V . Mushnikov, K. Onizuka, M.\nKato, K. Kosuge, C. P. Slichter, T. Goto, Y . Ueda, Phys. Rev. Lett. 82\n(1999) 3168.\n[2] K. Onizuka, H. Kageyama, Y . Narumi, K. Kindo, Y . Ueda, T. Goto, J.\nPhys. Soc. Jpn. 69(2000) 1016.\n[3] H. Kageyama, Y . Narumi, K. Kindo, K. Onizuka, Y . Ueda, T. Goto, J.\nAlloys Compd. 317–318(2001) 177.\n[4] S.E. Sebastian, N. Harrison, P. Sengupta, C.D. Batista, S. Francoual, E.\nPalm, T. Murphy, N. Marcano, H.A. Dabkowska, B.D. Gaulin, Proc. Natl.\nAcad. Sci. USA 105(2008) 20157.\n[5] Y . H. Matsuda, N. Abe, S. Takeyama, H. Kageyama, P. Corboz, A. Ho-\nnecker, S. R. Manmana, G. R. Foltin, K. P. Schmidt, F. Mila, Phys. Rev.\nLett. 111(2013) 137204.\n[6] B.S. Shastry, B. Sutherland, Physica B +C108(1981) 1069.\n[7] S. Miyahara, K. Ueda, Phys. Rev. Lett. 82(1999) 3701.\n[8] J. Dorier, K.P. Schmidt, F. Mila, Phys. Rev. Lett. 101(2008) 250402.\n[9] M. Nemec, G.R. Foltin, K.P. Schmidt, Phys. Rev. B 86(2012) 174425.\n[10] P. Corboz, F. Mila, Phys. Rev. Lett. 112(2014) 147203.\n[11] T. Verkholyak, J. Stre ˇcka, F. Mila, K.P. Schmidt, Phys. Rev. B 90(2014)\n134413.\n[12] G.R. Foltin, S.R. Manmana, K.P. Schmidt, Phys. Rev. B 90(2014)\n104404.\n[13] D.A. Schneider, K. Coester, F. Mila, K.P. Schmidt, Phys. Rev. B 93(2016)\n241107(R).\n[14] S. Wessel, I. Niesen, J. Stapmanns, B. Normand, F. Mila, P. Corboz, A.\nHonecker, Phys. Rev. B 98(2018) 174432.\n[15] A. Wietek, P. Corboz, S. Wessel, B. Normand, F. Mila, A. Honecker,\nPhys. Rev. Research 1(2019) 033038.\n[16] J. Richter, N.B. Ivanov, Czech. J. Phys. Suppl. 46(1996) 1919.\n[17] N.B. Ivanov, J. Richter, Phys. Lett. A 232(1997) 308.\n[18] J. Richter, N. B. Ivanov, J. Schulenburg, J. Phys.: Condens. Matter 10\n(1998) 3635.\n[19] J. Schulenburg, J. Richter, Phys. Rev. B 65(2002) 054420.\n[20] J. Schulenburg, J. Richter, Phys. Rev. B 66(2002) 134419.\n[21] S. Ueki, A. Okazawa, T. Ishida, T. Nogami, H. Nojiri, Polyhedron 26\n(2007) 1970.\n[22] A. Okazawa, T. Nogami, H. Nojiri, T. Ishida, Chem. Mater. 20(2008)\n3110.\n[23] H.G. Paulinelli, S. M. de Souza, O. Rojas, J. Phys.: Condens. Matter 25\n(2013) 306003.\n[24] T. Verkholyak, J. Stre ˇcka, Phys. Rev. B 88(2013) 134419.\n[25] T. Verkholyak, J. Stre ˇcka, Acta Phys. Polonica A 126(2014) 22.\n[26] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic\nPress, New York, 1982.\n[27] J. Stre ˇcka, M. Ja ˇsˇcur, Acta Physics Slovaca 65(2015) 235.\n[28] W. K. Wooters, Phys. Rev. Lett. 80(1998) 2245.\n[29] L. Amico, A. Osterloh, F. Plastina, R. Fazio, Phys. Rev. A 69(2004)\n022304.\n[30] L. Amico, R. Fazio, A. Osterloh, V . Vedral, Rev. Mod. Phys. 80(2008)\n517.\n[31] A. Osterloh, Int. J. Mod. Phys. B 27(2013) 1345018.\n[32] M. Oshikawa, M. Yamanaka, I. A \u000feck, Phys. Rev. Lett. 78(1997) 1984.\n[33] I. A \u000feck, Phys. Rev. B 37(1998) 5186.\n10"    },    {        "title": "0811.2455v1.Half_metallic_ferrimagnet_formed_by_substituting_Fe_for_Mn_in_semiconductor_MnTe.pdf",        "content": "arXiv:0811.2455v1  [cond-mat.mtrl-sci]  15 Nov 2008physica statussolidi, 4 November2018\nHalf-metallic ferrimagnet formed\nby substituting Fe for Mn in\nsemiconductorMnTe\nLi-Fang Zhu1,2*, Bang-Gui Liu1,2.\n1Institute of Physics,Chinese Academy of Science, Beijing1 00190, China\n2BeijingNational Laboratory forCondensed Matter Physics, Beijing100190, China\nReceivedXXXX, revisedXXXX,accepted XXXX\nPublishedonline XXXX\nPACS75.90.+w, 75.50.Pp, 75.47.-m, 75.30.-m\n∗Corresponding author: e-mail lfzhu@aphy.iphy.ac.cn , Phone +86-10-82649438, Fax+86-10-62553698\ne-mailbgliu@aphy.iphy.ac.cn , Phone +86-10-82649437, Fax+86-10-62553698\nA ternary ferrimagnetic half-metal, constructed through s ubstituting 25% Fe for Mn in zincblende semiconductor\nMnTe, is predicted in terms of accurate first-principles cal culations. It has a large half-metallic (HM) gap of 0.54eV\nand its ferrimagnetic order is very stable against other mag netic fluctuations. The HM ferrimagnetism is formed\nbecause the complete moment compensation in the antiferrom agnetic MnTe is replaced by an uncomplete one in\ntheFe-substitutedMnTe.Thisshouldmakeanovelapproacht onewHMmaterials.Thehalf-metalcouldbefabricated\nbecauseFe hasgoodaffinitywithMn,andusefulforspintroni cs.\nCopyrightlinewillbe provided by the publisher\n1 Introduction Half-metallic(HM)ferromagnetshave\nattracted much attention because they have band gaps at\nthe Fermi energy for one electronic spin channel and are\nmetallicfortheotherchannel[1,2].AlotofHMferromag-\nnetic(FM)materialshavebeenfound[3,4,5,6,7,8,9].Ac-\ncurate first-principles calculations have revealed HM fer-\nromagnetism in binary transition metal chalcogenidesand\npnictides in the zincblende and wurtzite structures[10,11 ,\n12,13,14,15]. It is excitingthat a Singaporegroup,stimu-\nlated by the theoretical prediction of zincblende CrTe (z-\nCrTe)[14,15], has fabricated z-CrTe samples of 100 nm\nthickness[16]. It has also been reported that half-metalli c\nferrimagnets can be formed by introducing Cr antisites in\nCrAs or CrSb[17]. It is still highly desirable to search for\nnovelsemiconductor-compatiblehalf-metalswithhighCur ie\ntemperatureforpotentialspintronicapplications[18].\nMagneticmaterialswithandbasedonzincblendestruc-\ntureareveryinterestingtospintronicapplications.Zinc ble-\nndeMnTe(z-MnTe)isoneofafewantiferromagnetic(AF-\nM)semiconductors.AlthoughMnTecrystallizesintoaNiAs\nphase, the metastable z-MnTe has been grown by molecu-\nlarbeamepitaxy(MBE)growthtechnique[19]andsemibulk\n(about 1 micrometer thick) film samples of z-MnTe have\nbeen fabricated[20] because z-MnTe is only 0.02eV performulaunithigherintotalenergythantheNiAs-typeMnTe.\nTernaryCr-dopedNiAs-typemanganesetellurides,Mn 1−x-\nCrxTe, withxbeing up to 14%, have been fabricated, in\nwhichthesubstitutionofCrforMnleadstoachangefrom\nan AFM semiconductor of MnTe to a FM (or ferrimag-\nnetic) semiconductor of Mn 1−xCrxTe[21]. Therefore, z-\nMnTe should be an interesting novel approach to explore\npromisingmagneticsemiconductorsandHM compounds.\nInthispaper,weperformfirst-principlesstudyonstruc-\ntural, electronic, and magnetic properties of the 25%-Fe-\ndoped z-MnTe. The substitution of Fe for Mn results in a\ntransition from the AFM semiconductor of z-MnTe to the\nferrimagnetichalf-metalof Mn 3FeTe4. We understandthe\nmechanism of the magnetism and the magnetic transition\nthrough investigating the atomic and electronic structure s\nofMn3FeTe4incomparisonwiththoseofz-MnTe.\nThe remaining part of this paper is organized as fol-\nlows. In next section we present our computational detail.\nIn the third section we shall present our optimized results\nof crystal structuresand investigatethe stability of the f er-\nrimagnetism against magnetic fluctuations. In the fourth\nsection we shall present the electronic structures and dis-\ncuss the mechanism for the half-metallic ferrimagnetism.\nCopyrightlinewillbe provided by the publisher2 Li-Fang Zhu and Bang-Gui Liu: Half-metallicferrimagnet f ormed bysubstituting Fefor Mninsemiconductor MnTe\nFinally we shall make some discussions and give our con-\nclusion.\n2 Computationaldetail Toperformthecalculations,\nwe use the package WIEN2K[22], which is based on full-\npotential linearized augmented plane wave method within\nthedensity-functionaltheory(DFT)[23].ThePerdew-Bur-\nke-Ernzerhof1996version[24] of the generalizedgradient\napproximation(GGA)isusedfortheexchange-correlation\npotential. Full relativistic effects are calculated for co re\nstates, and the scalar relativistic approximationis used f or\nvalence states. We investigate the effect of the spin-orbit\ncoupling,butstillpresenttheresultswithoutspin-orbit cou-\npling in the following because it does not affect our main\nconclusions. For different magnetic structures we use dif-\nferent but appropriate k points in the first Brillouin zones\nandmaketheexpansionupto l=10inmuffintins. Rmt×Kmax\nis set to 8.5 for z-MnTe and to 7.0 for Mn 3FeTe4with-\nout affecting our conclusions. The self-consistent calcul a-\ntions are considered to be converged when the integrated\ncharge differenceper formula unit between input and out-\nputchargedensityislessthan0.0001.\n3 Optimizedcrystalstructures Recentinelasticne-\nutron-scatteringexperimenthasrevealedthatthestablem a-\ngnetic structure of z-MnTe is collinear type-III AFM or-\nder of Mn spins in a double conventionalunit cell[25,26],\nrather than early type-I AFM order in single conventional\nunitcell[27]ornoncollineartype-IIIAFMordersuggested\nin terms of previous neutron-diffractionresult[20]. Ther e-\nfore, we consider only collinear spin configurationsin the\nfollowing.\n(d1)\n (d0) (e0)\nFigure 1 (color online). AFM-I ( d0) and AFM-III ( e0)\nstructures of zincblende MnTe and the most stable struc-\nture (d1) of Mn 3FeTe4. The black (red) ball denotes Fe\n(with arrow), the grey (blue) one Mn (with arrow) or Te.\nThearrowsrepresentthespinsonthesites.\nFive spin configurations a0,b0,c0,d0ande0can be\nconstructedforz-MnTe. a0isaFMstructurewithfourMn\nmomentsbeinginparallel. b0andc0,obtainedbyreversing\none Mn moment respectivelyat the face-centerand on theTable1Thespacegroups(SG),themagneticorders(MO),\nthe lattice constants ( aora/c), the relative energy Er(de-\nfined with respect to the lowest structure for the same for-\nmula), the absolute value of total magnetic moment ( M),\nand the Kohn-Sham gaps ( Eg) or the HM gaps ( Eh).Er\nandMare normalizedin termsof those of d1for compar-\nison.\nz-MnTe\n(a0) (b0) (c0) (d0) ( e0)\nSG 216 215 215 111 122\nMO FM FM FM AFM AFM\na(/c˚A) 6.393 6.314 6.314 6.290 6.290/12.580\nEr(eV) 0.712 0.196 0.196 0.020 0\nM(µB) 20.000 10.00 10.00 0.000 0.000\nEg(eV) −0.90 0.90 1.30 1.35\nMn3FeTe4\n(a1) (b1) (c1) (d1) ( e1)\nSG 215 215 111 111 35\nMO FM FM FM FM AFM\na(/c˚A) 6.305 6.244 6.251 6.223 8.822/12.476\nEr(eV) 0.764 0.114 0.222 0 0.125\nM(µB) 18.226 11.000 9.000 1.000 0.000\nEh(eV) −0.21 0.16 0.54 −\nvertex of a0, are ferrimagnetic structures and equivalent\nwith each other. d0ande0are the type-I AFM (AFM-I)\nstructurewithsingleconventionalcell[27]andthetype-I II\nAFM (AFM-III) structure with double unit cells[25], re-\nspectively, as shown in Fig. 1. By substituting Fe for Mn\non the vertex in the single MnTe cell of a0∼d0, we get\ncorrespondinglyfour FM (or ferrimagnetic) structures a1,\nb1,c1, andd1of Mn3FeTe4.d1is the most stable among\nthem andis shown in Fig. 1. Generallyspeaking,to get an\nAFM structure we construct a supercell of two unit cells\nandmakethemomentsinoneunitcelloppositetothosein\nthe other. a1∼d1are all possible FM (or ferrimagnetic)\nstructures one can construct without enlarging the mag-\nnetic unit cell. We construct all possible AFM structures\nbased on them, and the results for the most stable AFM\nstructure e1(astherepresentative)areshownin Table1.\nAll the above structures, both FM and AFM, are opti-\nmizedfully.Themomentandelectronicstructuresare cal-\nculated with the lattice constants of the optimized struc-\ntures. Our calculated results are summarized in Table 1. It\nis clear that the most stable structuretendsto havea small\nequilibrium lattice constant. As is shown in Table 1, the\ntwo FM structures ( a0anda1) and the four ferrimagnetic\nstructures ( b0,c0,b1andc1), having large magnetic mo-\nments, areunfavorablein total energy.For z-MnTe,AFM-\nIIIe0and AFM-I d0, with the total moments being 0, are\nfavorable in total energy,and e0is 5meV per formula unit\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nTable 2The partial magnetic moments ( µB) projected in\nthemuffin-tinspheresofMn1,Mn2,Fe,andTeatomsand\nintheinterstitialregion(Inter)andthetotalmoment(Tot al)\ninthemoststablestructuresz-MnTe e0andMn 3FeTe4d1.\nz-MnTee0Mn3FeTe4d1\nMn1 4.180 4.130\nMn2 -4.180 -4.114\nFe − 3.157\nTe 0.000 0.036\nInter 0.000 -0.095\nTotal 0.000 -1.000\nlower than d0, being in agreement with experimental fact\nthate0istheground-statephaseofz-MnTewithasemicon-\nducting gap of about 3.2eV[19]. The most stable structure\nforMn 3FeTe4, however,is notanyAFM structure,but the\nferrimagneticstructure d1withanabsolutetotalmomentof\n1.000µB. It is lower by 0.125eV per formula unit in total\nenergythanthelowestAFM structure e1.\nWesummarizethepartialmagneticmoments( µB)pro-\njected in the muffin-tin spheres of Mn1, Mn2, Fe, and Te\natomsandintheinterstitialregioninthemoststablestruc -\ntures z-MnTe e0and Mn 3FeTe4d1in Table 2. The cor-\nrespondingtotal magnetic moments also are presented for\ncomparison.ItisworthnotingthattherearetwoMn1atoms\nand two Mn2 ones in z-MnTe e0, but we have one Mn1\natom, one Fe atom, and two Mn2 atoms in Mn 3FeTe4d1,\nasshowninFig.1.Itisobviousthatthepartialsubstitutio n\nof Fe for Mn leads to the transferring of a little magnetic\nmomentsfromtheMnatomstotheTeatomsandtheinter-\nstitial region. Mn 3FeTe4d1has a total moment of -1.000\nµBbecause Fe has one more delectron, or one µBless\nmagneticmoment,thanMn.\n4 Electronicstructuresandmagneticmechanism\nThe spin-dependent density of states (DOS) of the AFM-\nIII MnTe are presented in Fig. 2( a). The primitive cell of\nAFM-III MnTe consists of 2 Mn1 (with spin up), 2 Mn2\n(with spin down), and 4 Te atoms. The valence bands are\nformedby 10 dand 12pstates. The 10 lowest conduction\nbands originate from Mn dstates. The Mn moments are\ncoupledwithasuperexchangeinteractionthroughthenear-\nest Te atoms, which yields the antiferromagnetism. The\nspin exchange splitting is about 4.7eV, as shown in Fig.\n2(a).\nThespin-dependentdensityofstates(DOS)andenergy\nbandsof the Mn 3FeTe4are presentedin Fig. 2( b) and Fig.\n3,respectively.TheFermilevel EFissettozero.Thefilled\nbands between -5eV and -0.6eV, consisting of 10 dstates\nand12pstatesforeachspinchannel,aresimilartothoseof\ntheMnTe.Themaindifferenceisthattherearepartly-filled\nmajority-spin bands across the Fermi level in the case of\ntheMn 3FeTe4.Theminority-spinbandsstill haveagapof/s49/s50/s54/s48/s54/s49/s50/s49/s50/s54/s48/s54/s49/s50\n/s45/s52 /s45/s50 /s48 /s50/s49/s50/s54/s48/s54/s49/s50/s32/s116/s111/s116/s97/s108\n/s32/s70/s101\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114/s32\n/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s32/s116/s111/s116/s97/s108\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115\n/s40 /s99 /s41/s40/s98 /s41 /s40/s97 /s41 \n/s32/s32/s116/s111/s116/s97/s108\n/s32/s67/s114\n/s32/s77 /s110/s49\n/s32/s77 /s110/s50\n/s32/s84/s101\n/s32/s105/s110/s116/s101/s114\nFigure 2 (coloronline).Spin-dependenttotal(thicksolid)\nand partial DOS (state/eV per formula unit) for the AFM-\nIIIMnTe( a),themoststable d1structureofMn 3FeTe4(b),\nand the most stable structure of Mn 3CrTe4(c). The upper\nhalf of each panel is DOS for majority spin and the lower\none for minority spin. The partial DOS are those in Fe/Cr\n(dot),Mn1(dash),Mn2(longdash),andTe(dotdash)muf-\nfin tinsandininterstitial region(thinsolid).\n1.12eV,alittlesmallerthantheKohn-Shamgap,1.35eV,of\ntheMnTe.TheMn 3FeTe4has0.54eVasitsHMgapwhich\nisdefinedasthesmallerof Ec\nmin-EFandEF-Ev\nmax,where\nEc\nminis the bottom of the minority-spinconductionbands\nandEv\nmaxthetopoftheminority-spinvalenceones[14,15].\nThe 25% Fe substitution for Mn results in a cell con-\nsistingof2Fe,6Mn,and8Teatoms.Thespinorientations\ncannotremainthesameasthoseofAFM-IIIMnTebecause\nFe has one more delectron than Mn. Instead, the mag-\nneticorderisreorganizedsothatthe16-atomcellisdivide d\nintotwoequivalentsmaller8-atomoneswhichwouldhave\nAFM-I structure if we neglect the difference between Fe\nand Mn. As a result, we obtain a ferrimagnetic order be-\ncause the Fe moment cannot completely compensate the\nopposite Mn moment. The substitution does not substan-\ntiallychangethevalencebands,butmovessomemajority-\nspindstates downwards with respect to those of the z-\nMnTebecausetheFe dstatesarealittlelowerthanthoseof\nCopyrightlinewillbe provided by the publisher4 Li-Fang Zhu and Bang-Gui Liu: Half-metallicferrimagnet f ormed bysubstituting Fefor Mninsemiconductor MnTe\nΓX M ΓZ R A Z E F Energy (eV)  0.0  1.0  2.0  3.0\n -1.0\n -2.0\n -3.0\n -4.0\n -5.0\nΓX M ΓZ R A Z   0.0  1.0  2.0  3.0\n -1.0\n -2.0\n -3.0\n -4.0\n -5.0\nFigure 3 Spin-dependent energy bands (plotted with cir-\ncles) of the d1structure of Mn 3FeTe4. A larger circle im-\nplies more Fe dcharacter. The left panel is for majority-\nspinandthe rightoneforminority-spin.\nMndonesinenergy.Themajority-spinbandsattheFermi\nlevel, belonging to a doublet, are half-filled because there\nis only one electron for them. The HM ferrimagnetism is\nachievedbecausewestillhaveagapacrosstheFermilevel\nforminority-spinchannel.\nWehavestudiedsimilar25%-Cr-dopedMnTe,Mn 3Cr-\nTe4. Its stable structure also exhibits HM ferrimagnetism.\nTheresultsforMn 3CrTe4areconsistentwithNakamura et\nal’s through doping 75% Mn into z-CrTe [28]. The spin-\ndependentdensityof states forthe moststable structureof\nMn3CrTe4are also given in Fig. 2( c). Cr has four delec-\ntrons, one less than Mn. The Cr dstates are a little higher\nthan those of Mn in energy, which results in the partially\noccupiedCrimpuritybandsinthemajority-spinbandsand\ntheopengapin theminority-spinbands.\nBycomparingtheDOSsoftheMn 3FeTe4andMn 3Cr-\nTe4in Fig. 2, we can explain the origin of their ferrimag-\nnetismuniformlyaccordingtothenumberof delectronsin\nthetransitionmetalatomsandtheenergylevelsof dstates.\nThe substitution of Fe for Mn or Cr for Mn changes the\ndistribution of dstates at the fermi level and results in the\nferrimagnetism.\n5 Discussion and conclusion All of our presented\nresults are calculated with GGA, although local density\napproximation (LDA) yields almost the same results. It\nis worth noting that a developed single-ion implantation\ntechnique recently was used to implant dopant ions one-\nby-oneintoa semiconductor[29].Thatis, boththenumber\nand the position of the dopantatoms in the semiconductor\nare precisely controlled. As a result, the promising half-\nmetals predicted in this paper could be realized by using\nsuchtechniques.\nIn summary, we have predicted a ternary half-metal\nMn3FeTe4, constructedbysubstitutingFe forMnin semi-\nconductorz-MnTe,intermsofouraccuratefirst-principlescalculations. The substitution results in a transition fro m\nthe AFM semiconductor MnTe to the HM ferrimagnet of\nthe Mn 3FeTe4. The HM ferrimagnetism is stable against\nantiferromagneticfluctuations.ThelargeHMgapimpliesa\npossiblehighCurietemperature[30].TheMn 3FeTe4could\nbefabricatedexperimentallysoonbecauseofthegoodaffin-\nity ofFeto Mn,andit couldbeusedin spintronics.\nAcknowledgements ThisworkissupportedbyNatureSci-\nenceFoundationofChina(GrantNos.10874232,10774180,90 406010,\nand60621091),bytheChineseAcademyofSciences(GrantNo. KJC-\nX2.YW.W09-5),andbyChineseDepartmentofScienceandTech -\nnology (GrantNo. 2005CB623602).\nReferences\n[1] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M.\nDaughton, S. von Molnar, M.L. 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B 67, 220403 (2003).\nCopyrightlinewillbe provided by the publisher"    },    {        "title": "1101.3645v3.Mott_transition_and_ferrimagnetism_in_the_Hubbard_model_on_the_anisotropic_kagomé_lattice.pdf",        "content": "arXiv:1101.3645v3  [cond-mat.str-el]  16 Apr 2013Mott transition and ferrimagnetism in the Hubbard model on t he anisotropic kagom´ e\nlattice\nA. Yamada1, K. Seki1, R. Eder2, and Y. Ohta1\n1Department of Physics, Chiba University, Chiba 263-8522, J apan\n2Karlsruhe Institut of Technology, Institut f¨ ur Festk¨ orp erphysik, 76021 Karlsruhe, Germany\n(Dated: November 12, 2018)\nMott transition and ferrimagnetism are studied in the Hubba rd model on the anisotropic kagom´ e\nlattice using the variational cluster approximation and th e phase diagram at zero temperature and\nhalf-fillingis analyzed. The ferrimagnetic phaserapidly g rows as thegeometric frustration is relaxed,\nand the Mott insulator phase disappears in moderately frust rated region, showing that the ferri-\nmagnetic fluctuations stemming from the relaxation of the ge ometric frustration is enhanced by the\nelectron correlations. In metallic phase, heavy fermion be havior is observed and mass enhancement\nfactor is computed. Enhancement of effective spatial anisot ropy by the electron correlations is also\nconfirmed in moderately frustrated region, and its effect on h eavy fermion behavior is examined.\nPACS numbers: 71.30.+h, 71.10.Fd, 71.27.+a\nEffect of geometric frustration is one of the most\nimportant subjects actively studied in the field of\nstrongly correlated electron systems. For instance,\nthe heavy fermion behavior in LiV 2O41,2with py-\nrochlore lattice structure, and the spin liquid states\nin the triangular-lattice organic materials κ-(BEDT-\nTTF)2X3–5and herbertsmithite ZnCu3(OH) 6Cl2with\nkagom´ e lattice structure6,7have attracted a lot of at-\ntentions.\nWhen spatial anisotropy is introduced in systems with\ngeometricfrustration,theinterplaybetweenthespinfluc-\ntuations and Mott transition appears as a new feature\nand provides unique phenomena which take place nei-\nther in the unfrustrated nor fully frustrated systems. A\nreentrant behavior of the Mott transition observed in the\nκ-(DEBT-TTY) 2Cu[N(CM) 2]Cl under pressure3,5is an\ninterestingexample realizedonan anisotropictriangular-\nlattice, where that behavior stems from the enhancement\nof the antiferromagnetic fluctuations due to the electron\ncorrelations.8,9\nAs for the kagom´ e lattice, which is a prototype of frus-\nFIG. 1: (Color online) (a) Anisotropic kagom´ e lattice.\nIn our lattice geometry, the three sites 1 ,2, and 3 form\nan equilateral triangle of the unit length and the dashed\nlines are along the xdirection. Inside the hexagon (dotted\nline) and square (dash-dotted line) are the 12- and 6-site\nclusters, respectively, which will be used in our analysis.\n(b) The first Brillouin zone of the anisotropic kagom´ e latti ce.trated systems, a fully frustrated case has been theoreti-\ncally studied in detail,10–13however the issues related to\nthe anisotropy have been considered only recently. The\nMott transition and magnetic properties near the transi-\ntion havebeen studied using the cellular dynamicalmean\nfield theory14, where the Mott transition point was an-\nalyzed and enhancement of spatial anisotropy and spin\ncorrelations were observed. Such enhancement may give\nrisetothe extensionoftheordered(ferrimagnetic)phase.\nTherefore, if the Mott transition itself persists without\nbeing veiled by the ferrimagnetic phase remains to be\nexamined. Also, the effect of the enhanced anisotropy on\nthe heavy fermion behavior is worth being studied.\nIn this paper, we investigate the ferrimagnetism and\nMott transition on the anisotropic kagom´ e lattice using\nthe variational cluster approximation (VCA)15–17, which\nis formulated based on a rigorous variational principle\nand exactly takes into account the short-range correla-\ntions. We study the phase diagram at zero temperature\nand half-filling. We show that, in moderately frustrated\nregion,theferrimagneticphaserapidlygrowsdowntothe\nmetal-insulator phase boundary, indicating that the spin\ncorrelations stemming from the relaxation of the frus-\ntration is enhanced by the electron correlations and the\nMott insulator (MI) phase disappears. In the metallic\nphase, heavy fermion behavior is observed and the mass\nenhancement of the quasiparticle is computed. Effective\nspatial anisotropy becomes also larger due to the elec-\ntroncorrelations,inagreementwiththepreviousstudy.14\nThis effect givesrise to an enhancement ofthe anisotropy\nof the effective masses of the quasiparticles.\nThe Hamiltonian of the Hubbard model on the\nanisotropic kagom´ e lattice (see Fig. 1) reads\nH=−/summationdisplay\ni,j,σtijc†\niσcjσ+U/summationdisplay\nini↑ni↓−µ/summationdisplay\ni,σniσ,(1)\nwheretij=tbetween the sites 1 and 2, 3 and tij=t′\nbetweenthesites2and3, Uistheon-siteCoulombrepul-\nsion, and µis the chemical potential. The annihilation\n(creation) operator for an electron at site iwith spin σis2\ndenoted as cjσ(c†\niσ) andniσ=c†\niσciσ. The system corre-\nsponds to the fully frustrated kagom´ e lattice at t′/t= 1,\nandfrustrationbecomesweakerwithdecreasing t′/t. The\nend memberat t′/t= 0is adecoratedsquarelattice. The\nenergy unit is set as t= 1 hereafter.\nWe use VCA15–17to examine the phase diagram and\nbehavior of the quasiparticles in the metallic phase at\nzero temperature. VCA is an extension of the cluster\nperturbationtheory15basedontheself-energy-functional\napproach.17This approach uses the rigorous variational\nprinciple δΩt[Σ]/δΣ = 0 for the thermodynamic grand-\npotential Ω twritten as a functional of the self-energy Σ\nΩt[Σ] =F[Σ]+Trln( −(G−1\n0−Σ)−1).(2)\nIn the above expression, F[Σ] is the Legendre transform\nof the Luttinger-Ward functional18and the index tde-\nnotes the explicit dependence of Ω ton all the one-body\noperators in the Hamiltonian. The stationary condition\nforΩt[Σ]leadstotheDyson’sequation. AllHamiltonians\nwith the same interaction part share the same functional\nform ofF[Σ], and using that property F[Σ] can be eval-\nuated from the exact solution of a simpler Hamiltonian\nH′, though the space of the self-energies where F[Σ] is\nevaluated is now restricted to that of H′. In VCA, one\nuses forH′a Hamiltonian formed ofclusters that are dis-\nconnected by removing hopping terms between identical\nclusters that tile the infinite lattice. A possible symme-\ntry breaking is investigated by including in H′the cor-\nresponding Weiss field that will be determined by mini-\nmizing the grand-potential Ω t. Rewriting F[Σ] in terms\nof the grand-potential Ω′≡Ω′\nt[Σ] and Green function\nG′−1≡G′\n0−1−Σ of the cluster Hamiltonian H′, the\ngrand-potential is expressed as\nΩt(t′) = Ω′−/integraldisplay\nCdω\n2πeδω/summationdisplay\nKlndet/parenleftbig\n1+(G−1\n0−G′\n0−1)G′/parenrightbig\n(3)\nand is now a function of t′. The functional trace has be-\ncome an integral over the diagonal variables (frequency\nandsuperlatticewavevectors)ofthelogarithmofadeter-\nminant over intra-cluster indices. The frequency integral\nis carried along the imaginary axis and δ→+0. The\nstationary solution of Ω t(t′) and the exact self-energy of\nH′at the stationary point, denoted as Σ∗, are the ap-\nproximate grand-potential and self-energy of Hin VCA,\nand physical quantities, such as expectation values of the\none-body operators, are calculated using the Green func-\ntionG0−1−Σ∗. In VCA, the restriction of the space of\nthe self-energies Σ into that of H′is the only approxi-\nmation involved and short-range correlations within the\ncluster are exactly taken into account by exactly solving\nH′.\nIn our analysis, the 6- and 12-site clusters in Fig. 1(a)\nare used to set up the cluster Hamiltonian H′. These\nclustershaveevennumberofsitessothatasingletground\nstate is possible. To study the ferrimagnetism, the Weissfield\nHF=hF/summationdisplay\nisign(i)(ni↑−ni↓) (4)\nwith sign( i) =−1 for the site 1 and sign( i) = 1 for\nthe sites 2 and 3, is also included. In the stationary\npoint search of Ω( µ′,hF), which we denote as the grand-\npotential per site, the Weiss field hFand the cluster\nchemical potential µ′are treated as the variational pa-\nrameters, where the latter should be included for the\nthermodynamic consistency.19During the search, the\nchemical potential of the system µis also adjusted so\nthat the electron density nis equal to 1 within 0.1%. In\ngeneral, a stationary solution with hF/ne}ationslash= 0 corresponding\nto the ferrimagnetic state and that with hF= 0 corre-\nspondingtothe paramagneticstateareobtained, and the\nground-state energies per site E= Ω+µnare compared\nto determine which solution (ferrimagnetic or paramag-\nnetic) is stable. The density of state\nD(ω) = lim\nη→0/integraldisplayd2k\n(2π)23/summationdisplay\nσ,a=1{−1\nπImGaσ(k,ω+iη)}(5)\nis also calculated to examine the gap. To be precise,\nD(ω) is calculated for η= 0.2, 0.1, and 0.05, andη→0\nlimit is evaluated by the standard extrapolation method.\nThe numerical error after this extrapolation is estimated\nto be of order 10−3, so the gap is identified as the re-\ngion ofωaroundω≃0 where the extrapolated D(ω) is\nless than 10−2. We also compute the ferrimagnetic order\nparameter per site\nM=3/summationdisplay\na=1(/an}bracketle{tna↑/an}bracketri}ht−/an}bracketle{tna↓/an}bracketri}ht)\nand the double occupancy per site\nDocc.=1\n33/summationdisplay\na=1/an}bracketle{tna↑na↓/an}bracketri}ht=dE\ndU\nwhere/an}bracketle{tnaσ/an}bracketri}htand/an}bracketle{tna↑na↓/an}bracketri}htare the expectation values of\nnaσandna↑na↓, respectively, with a=1, 2, and 3 being\nthe sites in Fig. 1(a).\nIn Fig. 2, we show the phase diagram at zero temper-\nature and half-filling obtained from this analysis using\nthe 12-site cluster. The results obtained using the 6-site\ncluster are also shown to quantitatively see the cluster\nsize dependence. The critical interaction strength UF\nseparating the ferrimagnetic and MI phases rapidly de-\ncreasesinthemoderatelyfrustratedregion t′= 0.5∼0.7,\nshowingthat the ferrimagneticfluctuations due to the re-\nlaxation of the geometric frustration is enhanced by the\nelectron correlations. (At t′= 0.75UF>20 for the 6-\nsite cluster.) In this region of t′, the ferrimagnetic phase\nis an insulator since there is a gap, and the transition\nbetween the ferrimagnetic and paramagnetic (including\nMI) phases is a level crossing (first order) because the3\n 0 5 10 15\n 0.5  0.6  0.7  0.8  0.9  1U\nt'Ferrimagnetic insulator Mott insulator\nParamagnetic metal\nFIG. 2: (Color online) Phase diagram of the Hubbard model\non the anisotropic kagom´ e lattice at zero temperature and\nhalf-filling as a function of t′andUobtained by VCA, where\nthe 12-site cluster is used (the crosses and circles). Lines are\nguides to the eye. The triangles and squares are the results\nobtained using the 6-site cluster. The crosses and triangle s\ncorrespond to the ferrimagnetic and paramagnetic transiti on\npoints and circles and squares are for the Mott transition\npoints.\nferrimagnetic solutions exist also U < U Feven though it\nis energetically disfavored there. The critical interaction\nstrength UMIseparating the MI and metallic phases is\nslightly smaller than the noninteracting band width W,\nwhereW= 6 att′= 1 and W= 4√\n2≃5.66 att′= 0.\nUMIdecreasesasthe geometricfrustrationisrelaxed, and\nthe slope becomes steeper in moderately frustrated re-\ngion. For the 12-site results, at t′= 0.5,UF= 4.0 while\nUMI= 4.1 so the MI phase has disappeared. Taking into\naccount the drastic growth of ferrimagnetic phase and\nthe fact that Wremains almost the same, the decrease\nofUMIaccording to the relaxation stems from the fer-\nrimagnetic fluctuations. As for the Mott transition, we\ncould not find out the Mott insulator and paramagnetic\nmetal coexisting region of Uat half-filling within our two\ncontrolling parameters µandµ′. Also as will be shown\nlater the Mott gap changes continuously as a function of\nU. Therefore, we could not find out an indication of the\ndiscontinuity at the Mott transition in this analysis. To\nsupplement this analysis, we show in Fig. 3 the double\noccupancy Docc.as a function of Ufor the 12-site clus-\nter, which also looks continuous at the transition point.\nIn Ref. 14 this transition is reported to be first order.\nFirst order Mott transitions are obtained in other models\nin the variational cluster approach with bath degrees of\nfreedom and treating the hybridization between the bath\nsites and cluster sites as a variational parameter.20,21In\nthese analyses, the coexisting metal and insulator solu-\ntions, leading to the first order transition, differ by the\nvalue of these hybridization parameters, and these situa-\ntions will be similar to the case of Ref. 14. Our analysis\ndoes not have bath degrees of freedom and technically\nthis will be the origin of the difference. It remains to be\nclarified which is the correct picture.\nNext we consider the cluster size dependence of our 0.08 0.1 0.12 0.14 0.16 0.18 0.2\n 3.5  4  4.5  5  5.5  6  6.5Docc.\nUt' = 1.0\nt' = 0.8\nt' = 0.6\nFIG. 3: (Color online) The double occupancy Docc.as a\nfunction of Ufort′= 1.0, 0.8, and 0 .6. The 12-site cluster\nisused. Thethreearrows indicatetheMott transitionpoint s.\n 0 2 4∆(a) t' = 1.0\n12 sites\n6 sites(b) t' = 0.8\n12 sites\n6 sites\n01\n0 2 4 6 8∆\nU(c)  t' = 0.6\n12 sites∆\nM\n0 2 4 6 8 10 0 1\nM\nU(d)  t' = 0.6\n6 sites∆\nM\nFIG. 4: (Color online) Mott gap ∆ as a function of Uat (a)\nt′= 1.0, (b)t′= 0.8, and (c), (d) t′= 0.6. These parameter\nregions correspond to the three vertical lines in Fig. 2. For\nt′= 0.6, order parameter Mis also included and the vertical\nlines separate the ferrimagnetic and MI phase.\nresults. In general, UMIis larger for larger clusters,\nsince the kinetic energyofthe cluster Hamiltoniancan be\nlarger for larger clusters. As for UF, when spin correla-\ntions are highly suppressed due to the frustration, cluster\nwavefunctions with small ferrimagneticfluctuations play\nan important role to examine near the true minimum of\nthe effective potential, so UFis smaller for larger clus-\nters. When the geometric frustration is moderate and\nspin correlations are not largely suppressed, the differ-\nence of cluster kinetic energies due to the cluster size be-\ncomes more important to determine the phase boundary,\nsoUFbecomes larger for larger clusters. Our result is\nconsistent with this general argument on the cluster size\ndependence. Quantitatively, UFis almost the same for\nthe 12- and 6-site clusters at t′= 0.6, andUFis smaller\nfor the 12-site clusters for t′>0.6. Relatively large dif-\nference of UFbetween the 12- and 6-site cluster results in\nstrongly frustrated region indicates strong suppression of\nthe spin correlations. The difference of UMIbetween the4\nFIG. 5: Spectral density at t′= 0.75 for (a), (b) U= 8\n(ferrimagnetic state), (c) U= 6 (MI state), and (d) U= 3\n(metallic state) along the dotted line in Fig. 1(b). The\nLorentzian broadening with η= 0.15tis used in all the\ncases. In (a) and (b), the solid lines are the mean-field\nSDW dispersion for the same values of U,t′andµ. In\n(d), the solid lines are the noninteracting band structure.\nIn(a), (b), and (c)thepeaks are scaled by5compared to(d).\n12-site and 6-site analysis is less than 20% of W. The\nbehavior of our UMIaccording to the relaxation of the\nfrustration is qualitatively consistent with the previous\nresults14, though our values for UMIare relatively small\ncompared to those in Ref. 14. At present the origin of\nthese discrepancies are not clear to us.\nIn Fig. 4 we show the Mott gap ∆ and ferrimagnetic\norderparameter Masfunctionsof Ufort′= 1.0,0.8, and\n0.6 corresponding to the three vertical lines in Fig. 2. ∆\nmonotonically decreases as Udecreases in all cases and\nMis always smaller for the 12-site cluster at t′= 0.6.\nThe gap ∆ Fin the ferrimagnetic phase is slightly larger\nthan ∆ if Uis the same. For example, for U= 10,\n∆ = 4.35 att′= 0.8 while ∆ F= 4.91 att′= 0.7 and\n∆F= 5.54 att′= 0.5 in the 12-site analysis.\nIn Fig. 5 we show the spectral weight function ρ(ω,k)\ncalculated using the 12-site cluster for solutions corre-\nsponding to (a), (b) the ferrimagnetic phase (up and\ndown spin parts are plotted separately), (c) the MI\nphase, and (d) the metallic phase at t′= 0.75. In (a) 1 1.1 1.2 1.3  r(a)\nt' = 0.6\nt' = 0.8\nt' = 1.0t' = 0.4\nt' = 0.5\n12\n m*/m(b)  t' = 1.0 x\ny\n12\n0 1 2 3m*/m\nU(c)   t' = 0.8x\ny\n0 1 2 312\n m*/m\nU(d)  t'= 0.6x\ny\nFIG. 6: (Color online) (a) Ratio ras a function of Ufor\nt′= 1.0, 0.8, and 0.6. The two arrows indicate the value of r\nfor noninteracting band at t′= 0.5 (r= 1.24) and at t′= 0.4\n(r= 1.30). (b)∼(d) Mass enhancement factor m∗/min the\nxandydirections as functions of Ufor (b)t′= 1.0, (c)\nt′= 0.8, and (d) t′= 0.6. The lines are obtained using the\n12-site cluster and symbols (squares, triangles, and circl es)\nare the results with the 6-site cluster. In (a) the squares,\ntriangles, and circles correspond to t′= 1.0, 0.8, and 0 .6,\nrespectively. In (b) ∼(d) the triangles correspond to the x\ndirection while the squares correspond to the ydirection.\nand (b), the mean-field spin-density-wave (SDW) disper-\nsion is also included (solid lines) to see its general fea-\ntures, where M= 0.92 in the mean-field solution while\nM= 0.72 in VCA. In (d), the noninteracting band struc-\nture is also plotted (solid lines) for comparison. In (a)\nand (b), the dispersion is largely affected due to the elec-\ntron correlations compared to the mean-field solution.\nIn (c), the SDW dispersion disappears and the spectral\nfunction displaysa Mott gap acrossall wavevectors. The\ngap is smaller compared to (a) and (b) since Uis smaller.\nComparingwith (d), in (c) it looksthat the lowestenergy\nband in (d) is shifted downwards and the second band\nsplits into lower and upper Hubbard bands while the top\nflat band remains almostthe same. In (d), we notice that\nthe spectral function is consistent with the Fermi liquid\nstateandthe interactingbandsslightlyshrinktowardthe\nFermi surface, leading to heavy fermion behavior.\nTo study it in detail, we consider well below the MI\ntransition line where Fermi liquid natures are confirmed\nfrom the behavior of the spectral function and com-\npute the mass enhancement factor m∗/malong the x\nandydirections in kspace, where the xdirection cor-\nresponds to the direction of t′hopping in real space.\nNear the Fermi surface the position of the peak ( ω,k) =\n(ωF+δω,kF+δk) ofρ(ω,k) changes according to the re-\nlation|δω|= (kF/m∗)δkand the band mass m∗is calcu-\nlatedusingthisrelation. Wealsocomputetheratioofthe\nFermi momenta in the xandydirections, r=kyF/kxF.\nAst′decreases, the noninteracting Fermi surface slightly\nshrinks in the xdirection and slightly evolves in the y\ndirection10, soris a measure of the anisotropy includ-\ning the effect of the electron correlations. Even though5\nprecise values of these quantities may depend on the lat-\ntice geometry, we show in Fig. 6 randm∗/min thex\nandydirections as functions of Ufort′= 1.0, 0.8, and\n0.6 in our lattice geometry, to see general features about\nthe effect of the electron correlations on these quantities.\nThe lines are the results obtained using the 12-site clus-\nter and symbols are the results with the 6-site cluster.\nAtt′= 0.6,rrapidly grows around U≃3 for the 12-\nsite cluster. This tendency is also observed for the 6-site\ncluster, where rturns to grow around U≃1.5. The\nrapid growth of rindicates that the effective anisotropy\nis enhanced due to the electron correlations in moder-\nately frustrated region. In fact, for the 12-site cluster,\nthe value of ratt′= 0.6 andU= 3.5 is equal to that\nof noninteracting band at t′= 0.4. The analysis of the\neffective anisotropy was also done in Ref. 14 by consid-\nering the renormalization of the hoping parameters and\nour results are qualitatively consistent with their analy-\nsis. This enhancement of effective anisotropy indicates\nthat the spin fluctuations due to the relaxation of the\ngeometric frustration are also enhanced by the electron\ncorrelations. Within our analysis, this indication is con-\nsistent with the rapid growth of the ferrimagnetic phase\naboveUMI, and it is demonstrated by the analysis of\nthe spin correlations.14As is shown in Fig. 6(b) ∼(d), the\nheavy fermion behavior is observed in all cases for the\n12-site cluster. For the 6-site cluster sizable mass en-\nhancements are not observed and the 6-site cluster may\nnot be large enough for subtle analysis related to thespectral function. The growth of raffects also m∗/m\nsincekFenters into the calculation of m∗. In general,\nm∗is enhanced in the ydirection and suppressed in the\nxdirection since the Fermi surface shrinks in the xdirec-\ntion and evolves in the ydirection. This appears largely\nin moderately frustrated region due to the rapid growth\nofr, as is observed around U∼3.5 for the 12-site clus-\nter results in Fig. 6(d). Therefore the anisotropy of the\neffective masses is enhanced in moderately frustrated re-\ngion.\nIn summary we have investigated the ferrimagnetism\nand Mott transition on the anisotropic the kagom´ e lat-\ntice using VCA. The phase diagram at zero temperature\nand half-filling is determined. The ferrimagnetic phase\nrapidly grows in moderately frustrated region and the\nMI phase disappears there. In the metallic phase, heavy\nfermion behavior is studied and the mass enhancement\nis computed. Enhancement of spatial anisotropy due to\nthe electron correlations is also observed for moderately\nfrustrated region and its effect on the heavy fermion be-\nhavior is discussed. Thus, the interplay between the spin\ncorrelationsand Mott transitionis quantitatively studied\nabove and below the metal-insulator transition.\nOne of us (A.Y.) would like to thank N. Fukui,\nK. Kurasawa, H. Mikami, and H. Nakada for useful dis-\ncussions on numerical analysis. This work was supported\nin part by Kakenhi Grant No. 22540363of Japan. A part\nof computations was done at Research Center for Com-\nputational Science, Okazaki, Japan.\n1S. Kondo, D. C. Johnston, C. A. Swenson, F. Borsa1, A.\nV. Mahajan, L. L. Miller, T. Gu, A. I. Goldman, M. B.\nMaple, D. A. Gajewski, E. J. Freeman, N. R. Dilley, R. P.\nDickey, J. Merrin, K. Kojima, G. M. Luke, Y. J. Uemura,\nO. Chmaissem, and J. D. Jorgensen, Phys. Rev. Lett. 78,\n3729 (1997).\n2P. E. J¨ onsson, K. Takenaka, S. Niitaka, T. 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Bulut, W. Koshibae, and S. Maekawa, Phys. Rev. Lett.\n95, 037001 (2005).\n12T. Ohashi, N. Kawakami, and H. Tsunetsugu, Phys. Rev.\nLett.97, 066401 (2006).\n13Y. Udagawa andY. Motome, Phys. Rev. Lett. 104, 106409\n(2010); J. Phys.: Conf. Ser. 200, 012214 (2010).\n14Y. Furukawa, T. Ohashi, Y. Koyama, and N. Kawakami,\nPhys. Rev. B 82, 161101 (2010).\n15D. S´ en´ echal, D. Perez, and M. Pioro-Ladri´ ere, Phys. Rev.\nLett.84, 522(2000); D.S´ en´ echal, D.Perez, andD.Plouffe,\nPhys. Rev. B 66, 075129 (2002).\n16M. Potthoff, M. Aichhorn, and C. Dahnken, Phys. Rev.\nLett.91206402 (2003); C. Dahnken, M. Aichhorn, W.\nHanke, E. Arrigoni, and M. Potthoff, Phys. Rev. B 70,\n2451100 (2004).\n17M. Potthoff, Eur. Phys. J. B 32, 429 (2003).\n18L. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417\n(1960).\n19M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke,\nPhys. Rev. B 74, 024508 (2006).\n20M. Potthoff, Eur. Phys. J. B 36, 335 (2003).\n21M. Balzer, M. Kyung, D. S´ en´ echal, A. M. Tremblay, M.\nPotthoff, Eur. Phys. Lett. 85, 17002 (2009)."    },    {        "title": "2010.13643v4.Effect_of_the_Uniform_Random_External_Magnetic_Field_with_Spatio_temporal_Variation_on_Compensation_in_Ising_Spin_1_2_Trilayered_Square_Ferrimagnet.pdf",        "content": "arXiv:2010.13643v4  [cond-mat.stat-mech]  12 Oct 2021Effect of the Uniform Random External Magnetic Field\nwith Spatio-temporal Variation on Compensation in Ising\nSpin-1/2 Trilayered Square Ferrimagnet\nSoham Chandra∗1\n1Department of Physics, Presidency University, 86/1 Colleg e Street, Kolkata -700 073, India\nAbstract\nTrilayered spin-1/2 Ising ferrimagnets are interesting thinsystems for compensation phenomenon . In this work,\na Metropolis Monte Carlo study is performed on the magnetic a nd thermodynamic response of such a system on\nsquare Bravais lattice, driven by uniform random external m agnetic field with spatio-temporal variations. In two\ndistinct configurations, the surface layers are made up of A a nd the mid-layer is made up of B atoms in a ABA\ntype stacking while in AABtype stacking, the top-layer and the mid-layer is made up of A -atoms while the bottom\nlayer is made up of B-atoms. The magnetic coupling between th e like atoms (A-A and B-B) is ferromagnetic\nwhile between the unlike atoms (A-B), it is antiferromagnet ic. For the time-dependent external uniform random\nfield, the mean is always set to zero and the standard deviatio n is varied until spin-field energy is comparable to\nthe dominant cooperative energy of the system. The findings s how that the observed compensation and critical\npoints shift and steady-state magnetic behaviours shift be tween N-, L-, P- and Q- etc. type of ferrimagnetic\nbehaviours, depending upon the strength of external unifor m random field. The compensation phenomenon even\nvanishes after crossing a finite threshold of standard devia tion of the magnetic field for particular choices of the\nother controlling parameters. Thus islands of ferrimagnet ic phase without compensation appear within the phase\narea with compensation of field-free case, in the 2D Hamilton ian parameter space. For both the configurations,\nthe areas of such islands even grow with increasing standard deviation of the external field, σ, obeying a scaling\nrelation of the form: f(σ,A(σ)) =σ−bA(σ) withbABA= 1.958±0.122 and bAAB= 1.783±0.118 .\nKeywords: Spin-1/2 Ising square trilayer; Uniform random external magnetic field; Spatio-temporal variation in\nfield; Metropolis Monte Carlo simulation; Compensation temperature ; No-compensation islands\n∗E-mail addresses: soham.rs@presiuniv.ac.in ; sohamc07@g mail.com\n11 Introduction\nThe Random Field Ising Model (RFIM) was introduced\nby Larkin [1] in 1970. In spite of its simplicity, such\nsystems exhibit many interesting static and dynamic be-\nhaviour [2]. From intuitive domain wall arguments, Imry\nand Ma [3] & Grinstein and Ma [4], suggested that lower\ncritical dimension for RFIM, dl= 2 which was consoli-\ndated by rigorous mathematics [5], and by Monte Carlo\n(MC) simulations [6]. In [7], the dependence of the criti-\ncal properties of the RFIM on the form of the distribution\nfunction of the random field has been emphasized. We\nhave witnessed non-trivial results for different field dis-\ntributions, e.g. the existence of a tricritical point in the\nstrong disorder regime of the system, present only in the\nbimodal case [7–9]. Random field type phenomenology\ncan be found in a variety of experimentally accessible dis-\nordered systems, such as: (a) structural phase transitions\nin random alloys [10]; (b) frustration introduced by the\ndisorder in interacting many body systems and several as-\npects of electronic transport in disordered insulators [11];\n(c) commensurate charge-density-wave systems with im-\npurity pinning [12]; (d) systems near the metal-insulator\ntransition [13,14]; (e) melting of intercalates in layered\ncompounds such as TiS2[15] and (f) binary fluid mixtures\nin random porous media [16]. These examples obviously\nlend credibility to such a simple model and attract the\nattention of the experimentalists. An interesting article\nby Sethna et al. [17] and references therein throw light\non RFIM in the context of hysteresis. The simulational\nresults and subsequent analyses explain how this model\nmay describe some real phenomena e.g. different kinds of\nnoises in magnets.\nMermin-Wagner theorem [18] had conceived the ab-\nsence of intrinsic long range ferromagnetic (FM) and an-\ntiferromagnetic (AFM) order in 2D magnetic materials.\nContradicting, recent experiments have established the\npresence of intrinsic 2D magnetism: (a) In atomically thin\nCr2Ge2Te6[19] and CrI3[20], we find the existence of\nfinite low temperature long-range FM order; (b) In mono-\nlayerVSe2, strong FM order is observed at room tem-\nperatures [21]; (c) Strong magnetic anisotropy is respon-\nsible for the observed long-range AFM order in atom-\nically thin FePS 3[22, 23]. Apart from intrinsic mag-\nnetism, the focus also is on tunablemagnetism. Success-\nful techniques in inducing FM order in intrinsically non-\nmagnetic low-dimensional materials are charge or carrier\ndoping [24–27]. Application of electric field [28] in ex-\nperiments and strain engineering [29] by numerical calcu-\nlations are also established to be successful in AFM to\nFM transition in monolayers. Another successful method\nin switchable magnetism is Li-intercalation. In a recent\nwork by DFT based calculations [30], induced ferrimag-\nnetism by intercalation with Li,MgandLi−Mgmix-\nture in pristine, naturally AFM, FeO2monolayer (pre-\ndicted by computational exfoliation from its bulk [31,32])\nis studied. Interested readers may find a few more in-\nteresting theoretical and computational works on 2D FM\nand AFM materialsin References[33–36]. That iswhy, re-\ncentscientificandtechnologicalinterestsinthebehaviorof\nthinsystems, where one dimension is significantly reduced\nthan the other two, are growing. Many interesting results\nof magnetism in thin systems (e.g., ribbons and films)under RFIM have come up because of numerous exper-\niments [37–43] in this direction. Sophisticated experimen-\ntal techniques, e.g. atomic layer deposition (ALD) [44],\npulsed laser deposition (PLD) [45], molecular beam epi-\ntaxy (MBE) [46] and metalorganic chemical vapor depo-\nsition (MOCVD) [47] have made growth of bilayered [48],\ntrilayered [49] and multilayered [50,51] systems a reality.\nEquilibrium (field-free and in presence of static fields)\nstudies by numerical methods on the layered Ising ferri-\nmagnetic systems on different lattice geometries has been\nperformed in the recent past [52–56]. In some of such sys-\ntems, for certain combinations of the coupling strengths,\nwe find two temperatures with zero bulk magnetization.\nOne is the Critical temperature with zero sublattice mag-\nnetizations, consequently zero bulk magnetization. The\nother temperature, the Compensation temperature , lower\nthan the criticaltemperature, has vanishingbulk magneti-\nzation but the sublattice magnetizations has non-zero val-\nues. In [55](a), by MFA and EFA and in [55](b), by MC\nsimulations with Wolff single cluster Algorithm, the au-\nthors have shown that under certain range of interaction\nstrengths, different temperature dependencies of sublat-\ntice magnetisations cause the compensation point to ap-\npear in both ABA and AAB configurations. In [56](a),\nit was hinted that there may exist underlying mathemati-\ncal relations between Inverse absolute of reduced residual\nmagnetisation (IARRM for brevity, which may be con-\nsidered as an interesting physical quantity for such kind\nof systems) and parameters of the trilayered, s= 1/2,\nIsing system and in [56](b), certain functional forms de-\nscribingthe systematicsof compensation, are proposedfor\nboth the AAB and ABA configurations, which agree fairly\nwell with accepted numerical results of [55](b). In [56](c),\nfor a triangular trilayered spin-1/2 ferrimagnet, the mag-\nnetic description by traditional Monte Carlo Simulation is\nshown to be in very good agreement with the description\nprovided by IARRM and Temperature interval between\nCritical and Compensation temperatures (TICCT). The\nresults in [56](c) actually strengthens the conjecture pro-\nposed in [56](b).\nDisorder is crucial in the description of spin models\nas departures from ideal systems is only natural. In the\nsystem of our study, disorder may appear in different as-\npects, such as: (a) in the number of spins interacting with\na particular one may be different (dilution or creation of\nbond(s); interactions with other than nearest neighbours\netc.) (b) the interaction strength between pairs of spins\nmay be different (distances between them may become\ndifferent at some of the sites) (c) change in the nature of\nspins in the ordered structure (A atoms getting doped by\nB or vice versa or by atoms with spin values other than\n1/2). Such varioustypes ofdisordersmayhaveeffects that\nvary with time and be uncorrelated among the lattice sites .\nA uniform random external magnetic field, with spatio-\ntemporal variations, may effectively model such kinds of\ndisorderedtrilayeredferrimagneticsystemswithadynamic\nHamiltonian. In the literature, such studies are yet to be\nnumerically performed on the system of this study. The\nscope of the present work is to shed light on the effects,\nthat a site-dependent and time-varying uniform random\nexternalfield mayhave,onthe compensationphenomenon\nof the ABA and AAB type trilayered spin-1 /2 Ising ferri-\nmagnets. It will be interesting to observe how the phase\n2separation curve reacts to the different strengths of the\nexternal field. This eventually leads to interesting kind\nof phase diagrams in the Hamiltonian parameter space,\nwhich for the present case, has the standard deviation (in-\nterchangably, strength) of the external uniform random\nfield as another controlling variable.\nThe rest of the paper is organized as follows. The\nmodel and the technical details of the simulation scheme\nisdescribedinSection2. The analysisofsimulationaldata\nand results are presented in Section 3. Section 4 contains\nthe summary of the work.\n2 Model and Simulation Protocol\nThe ferrimagnetic Ising superlattice in this study (with\neach site having spin value, s= 1/2), contains three mag-\nnetic sub-layers on square lattice with the following de-\ntails:\n(a) Each alternate layer is exhaustively composed of by\neitherAorBtype ofatomswith no coupling between\nspins on top and bottom layers [Fig.-1].\n(b) The system has mixed interactions between atoms\nthroughout the bulk:\nA-A→Ferromagnetic\nB-B→Ferromagnetic\nA-B→Anti-ferromagnetic\n(c) Ateachofthesites, i,oneverylayer,the z-component\nof spins, Sz\nicouples with a uniform random external\nmagnetic field, hi(t). This external field varies in\ntime at a particular lattice site and at a frozen time\ninstant, the values of this field are different from one\nsite to another.\nAs we have considered the spins to interact Ising-like,\nin-plane as well as inter-plane, the time dependent Hamil-\ntonian for the trilayered ferrimagnetic system is:\nH(t) =−J11/summationdisplay\n<t,t′>Sz\ntSz\nt′−J22/summationdisplay\n<m,m′>Sz\nmSz\nm′\n−J33/summationdisplay\n<b,b′>Sz\nbSz\nb′−J12/summationdisplay\n<t,m>Sz\ntSz\nm−J23/summationdisplay\n<m,b>Sz\nmSz\nb\n−/summationdisplay\nihi(t)Sz\ni (1)\nwhere/an}bracketle{tt,t′/an}bracketri}ht,/an}bracketle{tm,m′/an}bracketri}ht,/an}bracketle{tb,b′/an}bracketri}htdenote nearest-neighbor pairs\nin the top, mid and bottom layers respectively and /an}bracketle{tt,m/an}bracketri}ht,\n/an}bracketle{tm,b/an}bracketri}htare, respectively, pairs of nearest-neighbor sites in\nadjacent layers, top & mid and mid & bottom layers. At\nthe right of Equation [1], the first three terms are for the\nintra-planar ferromagnetic contributions. The fourth and\nfifth terms arise out of the nearest neighbour inter-planar\ninteractions, between top and mid layersand mid and bot-\ntom layers, respectively. The sixth term is the spin-field\ninteraction term of all the spins to the external random\nmagnetic field, at time instant t. The summation index,\ni, runs over all the spins in the system. To satisfy the\ntype of interactions, we need: JAA>0 ,JBB>0, and\nJAB<0. For an ABA type system, J11=J33=JAA;\nJ22=JBBandJ12=J23=JAB. For an AAB type sys-\ntem,J11=J22=J12=JAA;J33=JBBandJ23=JAB.We’ve considered periodic boundary conditions in-plane\nand open boundary conditions along the vertical.\nThe Metropolis single spin-flip algorithm [57,58] was\nemployed to simulate the model. Each of the three planes\nhasL2sites where the linear size, Lis 100. Each site is\nlabelled by an integer index, say i, and the z-components\nof spin projections, Sz\ni(Sz\ni=±1) contribute to the inter-\nactions. At each site i, alocal, time-varying random field\nhicouples with the spin. In [55](b), for L/greaterorequalslant60, the au-\nthors found the value of the compensation temperature to\nbe practicallyconstantfor the system ofthis study. So the\nsize of the lattice, considered here, is sufficient in obtain-\ning statistically reliable results. The system was initiated\natahigh temperatureparamagneticphase, with randomly\nselected half of the total spin projections, Sz\ni= +1 and\nthe rest with Sz\ni=−1 (Using 1 instead of 1 /2 fixes up the\nenergy scale). At a fixed temperature T, the Metropolis\nrate [59,60] , of Equation [2], governs the spin flipping\nfromSz\nito−Sz\ni:\nP(Sz\ni→ −Sz\ni) = min{1,exp(−∆E/kBT)}(2)\nwhere the associated change in internal energy in flipping\nthei-th spin projection from Sz\nito−Sz\ni, is ∆E. Simi-\nlar 3L2individual, random single-spin updates constitute\nOne Monte Carlo sweep (MCS) of the entire system and\nthisone MCS is the unit of time in this study.\nAt every temperature step, the system goes through\n105MCS. The last configuration of the system at the just\nprevious temperature acts as the starting configuration.\nFor the first 5 ×104MCS, the system is allowed to reach\nequilibrium (which is sufficient for equilibration [Refer to\nFigure 5 and discussions therein]) in a field-free environ-\nment. After that the external field is switched on and\nkept switched on for the next 5 ×104MCS. So for the sys-\ntem, the exposure time interval in the field, δis 5×104.\nThe temperatures of the systems are measured in units\nofJBB/kB. The tactics for observation, at first, include\nfixing the standard deviation (sd) or randomness of the\nfield. For each of the fixed values of the sd of the field,\nthe system was observed for seven equidistant values of\nJAA/JBB, from 0.04 to 1.0 with an interval of 0 .16. For\neach fixed value of JAA/JBB,JAB/JBBwas varied from\n−0.04 to−1.0 with a decrement of −0.16 at each step.\nFor each combination of JAA/JBBandJAB/JBB, the\ntime averages of the following quantities are calculated at\neach of the temperature steps ( T) and fields, in the fol-\nlowing manner:\n(1) Sublattice magnetisations for top, mid and bot-\ntom layers calculated, identically, at time instant say, t,\nafter equilibration, denoted by Mq(T,t), by:\nMq(T,t) =1\nL2L/summationdisplay\nx,y=1/parenleftbig\nSz\nq(T,t)/parenrightbig\nxy(3)\nThen we get the time averaged sublattice magnetizations\nat temperature, T, as:\n/an}bracketle{tMq(T)/an}bracketri}ht=1\nδ/integraldisplayt0+δ\nt0Mq(T,t)dt (4)\nwhereqis to be replaced by t,morbfor top, mid and bot-\ntom layers. The order parameter ,OT, for the trilayer\nat temperature, Tis defined as:\nOT=1\n3(/an}bracketle{tMt(T)/an}bracketri}ht+/an}bracketle{tMm(T)/an}bracketri}ht+/an}bracketle{tMb(T)/an}bracketri}ht) (5)\n3(a)\n (b)\nFigure 1: (Colour Online) Miniaturised versions (3 ×4×4) of (a) ABA and (b) AAB square trilayered ferrimagnet\nwith two types of theoretical atoms, AandB. Each of the sublattices of the ferrimagnetic systems are formed on\nsquare lattice. The actual simulation is carried out on a system with Nsites= 3×100×100 .\n(2)After attaining equilibrium, we calculate fluctuation\nof the order parameter, ∆O(T) at temperature, Tas\nfollows [61]:\n∆O(T) =/radicalBigg\n1\nδ/integraldisplayt0+δ\nt0[M(T,t)−OT]2dt(6)\nwhereM(T,t) is the total magnetisation of the whole sys-\ntem, at temperature, T, calculated at the ( t-th time in-\nstant). On the similar lines, /an}bracketle{tE/an}bracketri}htT, the time averaged\nvalue of cooperative energy, per site at temperature, T,\nis determined for the two configurations by:\n/an}bracketle{tE/an}bracketri}htABA\nT=−1\n3L2δ/integraldisplayt0+δ\nt0dt[JAA(/summationdisplay\n<t,t′>Sz\ntSz\nt′\n+/summationdisplay\n<b,b′>Sz\nbSz\nb′)+JBB/summationdisplay\n<m,m′>Sz\nmSz\nm′\n+JAB(/summationdisplay\n<t,m>Sz\ntSz\nm+/summationdisplay\n<m,b>Sz\nmSz\nb)] (7)\nand\n/an}bracketle{tE/an}bracketri}htAAB\nT=−1\n3L2δ/integraldisplayt0+δ\nt0dt[JAA(/summationdisplay\n<t,t′>Sz\ntSz\nt′\n+/summationdisplay\n<m,m′>Sz\nmSz\nm′/summationdisplay\n<t,m>Sz\ntSz\nm)\n+JBB/summationdisplay\n<b,b′>Sz\nbSz\nb′+JAB/summationdisplay\n<m,b>Sz\nmSz\nb] (8)\nThese relations enable us to calculate the fluctuation of\nthe cooperative energy per site at temperature, T, by the\nfollowing formula:\n∆E(T) =/radicalBigg\n1\nδ/integraldisplayt0+δ\nt0[E(T,t)−/an}bracketle{tE/an}bracketri}htT]2dt(9)\nwithE(T,t) being the instantaneous cooperative energy,\nper site, for the system at time instant tand at tem-\nperature, T, residing within the exposure interval of δ.The sharp peaks in the fluctuations, allow us to detect\nthe pseudo-critical temperatures. Around this tempera-\nture close range simulations were performed with temper-\nature interval of 0 .02 to land up on the reported critical\ntemperatures with an accuracy of, ∆ Tcrit= 0.04 . The\ncompensation temperature ( < Tcrit), where the average\nmagnetisation again becomes zero, is determined by lin-\near interpolation from the two neighbouring points across\nthe zero of magnetization in the plots of order parameter\nvs. temperature [e.g. Figure 2(a)]. The errors associated\nwith the magnetizations and fluctuations are estimated by\nthe Jackknife method [60].\n3 Results and discussions\n3.1 Characteristics of the External field\nThe local, uniform random external magnetic field values\nhi(t) at any site, iat time instant t, are drawn from the\nfollowing symmetric probability distribution:\nPuniform(hi(t)) =/braceleftBigg\n1√\n12σ|hi(t)| ≤√\n3σ\n0|hi(t)|>√\n3σ(10)\nHere,σis thestandard deviation (sd) of the uniform\nrandom distribtion.\nThe external field is also considered to have the follow-\ning characteristics:\n(a) The values of the external field are uncorrelated for\ndifferent sites at a particular time instant. Also at\na particular lattice site, the values of the external\nfield are uncorrelated for different time instants. So\nthese conditions can be conveniently written as :\nhp(t)hq(t′) =a(t)δpqδ(t−t′), where p,qare two\ndifferent lattice sites and t,t′are two different time\ninstants.\n(b) The following conditions are also trivially met:\n(i) After t=t0(when the field is switched ON),\nthe spatial mean (equivalently, bulk average)\n4of the above symmetric distribution of the ran-\ndom field, at any fixed time instant t, is zero\nas: /summationdisplay\nphp(t) = 0\n.\nConsequently,\n/summationdisplay\np,qhp(t)hq(t)δpq= 3L2σ2\n.\n(ii) At the p-th site, the temporal mean of the lo-\ncal field hp(t) over the exposure interval, δ, is\nalsozero:/an}bracketle{thp(t)/an}bracketri}ht=1\nδ/integraltextt0+δ\nt0hp(t)dt= 0 . It\nhas been checked the duration of the exposure\ninterval satisfies the above condition.\nThe reliability of implementation of such a probabil-\nity distribution at a few randomly chosen time instants\nwithin the exposure interval, is then checked by the Cu-\nmulativeDistributionFunction(CDF),theKernelDensity\nEstimate (KDE) and the Histogram. Interested readers\nare referred to [62] for derivations of KDE and to [63] for\ndiscussions on CDF. The Histogram is the most intuitive\nand traditional one in this regard. While the properly\nnormalised CDF’s show half of the total events (drawing\nof the values of fields) happen just before reaching 0, for\nall the sampled time instants. Kernel density estimates\nindicate the number density in field intervals at all the\nselected time instants.\n3.2 Thermodynamic Response\nThe zero-field magnetic response [55,56] for the ABA and\nAAB type systems showed us, if JAA/JBBis kept fixed\nand themagnitude ofJAB/JBBis increased or vice-versa,\nboth, the critical and the compensation temperatures for\nsuch systems, increases. This feature of shift of compensa-\ntion and critical temepratures is retained when we apply\nan external uniform random field with desired character-\nistics of Section 3.1 [Refer to Figure 2]. For any combi-\nnation of coupling strengths, with increase in the value of\nstandard deviation (or randomness) of the external field\ndistribution, the compensation and critical, both the tem-\nperatures decrease [Refer to Figure 3]. With increment\nin the randomness of the external field, the magnitude\nof decrement for the compensation temperatures is much\nmore pronounced than the corresponding magnitude of\ndecrement of critical temperature, with or without com-\npensation. The nature ferrimagnetic magnetisation ver-\nsus temperature dependences change with a change in the\nstandard deviation of the field and even a field driven ab-\nsence of compensation phenomenon can also be observed\nin Figures 3(a)&(b) for ABA and AAB configrations re-\nspectively. This salient feature is very robust, as it is al-\nways present irrespective of which coupling ratio is kept\nfixed among JAA/JBBandJAB/JBB.\nNowafewcommentsregardingthe field-driven changes\nin the nature of the magnetization curves of Figure 3, are\nin order after references [64–66]. In Figure 3(a) for the\nABA configuration, the magnetic response changes from\ntype-N(σ= 0) to type- P(σ= 0.20,0.40,0.60,0.76) to\ntype-Q(σ= 1.00). While going from from type- Ntotype-P, the curve passes through type- Lwherethe mag-\nnetization at the lowest temperature is zero . Exactly sim-\nilar conclusions are drawn from the Figure 3(e) for the\nAAB configuration. For the ABA configurations, in Fig-\nure 3(b), the magnetic response is confined to type- Nand\nin Figures 3(c)&(d), the magneticresponsesare all type- Q\nfor all the fields of observation. Now for the AAB configu-\nration, in Figure 3(f), the transition happens from type- N\n(σ= 0.00,0.20,0.40,0.60) to type- P(σ= 0.76,1.00) via\ntype-L. In Figure 3(g), all the magnetic responses are of\ntype-Pand in Figure 3(h), all the magnetic responses are\nof type-Q, for the AAB configuration.\nNow it is worthy of looking at the fluctuation of the\norder parameter and fluctuation of the cooperative energy\nper siteat this stage, for how they react according to the\nrandomness of the external field as a function of temper-\nature. In Figure 4, the zero-field cases are also shown to\nemphasise on the departures in presence of the field. In\nthe zero field cases, both the fluctuations of order param-\neter and energy, had a plateau with a smeared peak at the\nposition of compensation. Here the field strength is taken\nfor 7 different values along with the zero-field one, which\nbears necessary informations. We can see the compensa-\ntion and critical temperatures move towards lower tem-\nperature values, with increase in the field strength. As we\nmake the strength of the external field larger in compari-\nson to the cooperative part of the Hamiltonian, the shifts\nin both the temepratures become readily detectable. Even\none can clearly see the compensation gradually vanishing\nas the smeared peaks at the low temperature segments\nflatten out. This is a signature of field-driven vanishing\nof compensation. At the lowest temperature, the increase\nof both the fluctuations readily suggests considerable loss\nof magnetic ordering with the increase of the strength or\nrandomness of the external field.\nThe question arises now: What exactly does lead to\nthefield driven absence of compensation ? According to\nthe definition, the total magnetisation can change signa-\nture even in the close vicinity of 0K so the compensation\ntemperature can even be equal to the absolute zero. Let\nus observe the Magnetisation of individual layers and the\ntotal magnetisation of the bulk, as a function of time at\na very low temperature T= 0.01. Here, three particular\ncombinations of coupling ratios: JAA/JBBandJAB/JBB\nfor three values of sd of the external uniform randommag-\nnetic field are investigated. The results for both the ABA\nand AAB configurations are in Figure 5. One can see\nin all these cases that about and till t= 5×104MCS,\nthe sublattice as well as total magnetisation of the system\nhas attained equilibrium (as in this time interval, there\nis no time dependent part in the Hamiltonian). Start-\ning from a completely randomised high temperature state,\nthis equilibrium is achieved by slowcooling (in steps of\n∆T= 0.05). As the external field at t=t0= 5×104\nMCS, is switched ON all the sublattice magnetisations\nand correspondingly the total magnetisation, starts to re-\nact in response to the external field. The steady state, is\nachieved rather quickly, and the system behaves similarly\nin the rest of the exposure interval. So the choice of ex-\nposure interval is potent in revealing the characteristics of\nthe system. We observe whenever the per site in-plane in-\nteraction energies are comparable to the magnitude of per\nsite spin-field interaction energies [Refer to the top panel\n5-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00ABA (a)\nJAA/JBB=0.20\nσ=0.52\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00ABA (b)\nJAB/JBB=-0.20\nσ=0.52\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTJAB/JBB=-0.04\nJAB/JBB=-0.20\nJAB/JBB=-0.36\nJAB/JBB=-0.52\nJAB/JBB=-0.68\nJAB/JBB=-0.84\nJAB/JBB=-1.00AAB (c)\nJAA/JBB=0.20\nσ=0.60-0.4-0.2 0 0.2 0.4\n 0 1 2 3 4 5OT\nTJAA/JBB=0.04\nJAA/JBB=0.20\nJAA/JBB=0.36\nJAA/JBB=0.52\nJAA/JBB=0.68\nJAA/JBB=0.84\nJAA/JBB=1.00AAB (d)\nJAB/JBB=-0.20\nσ=0.60\nFigure 2: (Colour Online) Order parameter (i.e. time averaged total magnetisation) versus reduced temperature for:\n(a) ABA: JAA/JBB= 0.20 and variable JAB/JBBforσ= 0.52; (b) ABA: JAB/JBB=−0.20 and variable JAA/JBB\nforσ= 0.52; (c) AAB: JAA/JBB= 0.20and variable JAB/JBBforσ= 0.60; (d) AAB: JAB/JBB=−0.20andvariable\nJAA/JBBforσ= 0.60 . In all the cases Compensation and Critical temperatures shift towards higher temperature\nend with increase in any of the coupling ratios. The field-driven vanish ing of compensation is also present for the\nweakest combination of coupling strengths.\nandσ= 1.00in the middle panelofFigure5] oftheA- lay-\ners, the A-layers significantly lose magnetic ordering. In\ntop panelofFigure5forABA, whenthe spin-fieldenergies\nper site are greater than the cooperative energy per site,\nit causes the surface layers to become nearly completely\nrandomatevensuchalowtemperature. Consequentlythe\ntotal magnetisation doesn’t change sign below the Critical\ntemperature which leads to the absence of compensation\nphenomenon.\nFor the ABA system, the surface layers behave almost\nidentically as the mid B-layer affects their behaviour in\nidentical way through nearest neighbour Ising interaction.\nBut in AAB system, the bottom B-layer provides mag-\nnetic stability to the mid A-layer. Thus the magnetic be-\nhaviour of the mid A-layer is a little less affected than the\ntop A-layer. In top panel of Figure 5 for AAB, when the\nspin-field energiesper sitearegreaterthanthe cooperative\nenergy per site of the A-layers, like in the previous case,\nthe A layers lose much of the magnetic ordering at such\na low temperature and the top A-layer is more affected.\nConsequently the combined magnetisation of A-layers is\nunable to cancel the magnetisation of the bottom B-layer.\nIt leads to the absence of compensation phenomenon for\nthis configuration.\nSuch type of an event can be termed as field-drivenvanishing of compensation in the Ising spin-1/2 trilayers.\nIt is a dynamic phenomenon as the Hamiltonian’s time\ndependent part is responsible for it.\n3.3 Lattice Morphology\nAs it is said before, the increase in the fluctuations in the\nlow temperature segments in Figure 4 signifies the loss of\nmagnetic ordering in the system as a whole with increase\nin the randomness of the external field. But, how do the\nmagnetisations of the individual layers react to the exter-\nnal field? The magnetization versus time plots [Figure 5]\nfor the sublayers as well as the bulk has shown the rele-\nvant energy scales where the compensation becomes ab-\nsent. The spin-density plots or the lattice morphology at\nteperature T= 0.01 are now used to establish the reason.\nFor ABA type composition: in Figure 6(a): with\nJAA/JBB= 0.04 andJAB/JBB=−0.04, and with the\nintroduction of the external field with a very low value\n(σ= 0.20), the compensation vanishes. In the steady\nstate, as the spin-field terms take effect the local spin\nconfigurations of the top and bottom layers lose most of\nthe magnetic ordering [Please refer to the instantaneous\nvalues below the spin-density plots]. It is similar for the\nhigher values of standard deviations. In Figure 6(b): with\n6-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (a)\nJAA/JBB=0.04\nJAB/JBB=-0.04 N,(L),P,Q-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (b)\nJAA/JBB=0.04\nJAB/JBB=-1.00 N\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (c)\nJAA/JBB=1.00\nJAB/JBB=-0.04Q -0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00ABA (d)\nJAA/JBB=1.00\nJAB/JBB=-1.00\nQ\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (e)\nJAA/JBB=0.04\nJAB/JBB=-0.04 N,(L),P,Q-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (f)\nJAA/JBB=0.04\nJAB/JBB=-1.00 N,(L),P\n-0.4-0.2 0 0.2 0.4\n 0 1 2 3 4 5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (g)\nJAA/JBB=1.00\nJAB/JBB=-0.04\nP-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4  5OT\nTσ=0.00\nσ=0.20\nσ=0.40\nσ=0.60\nσ=0.76\nσ=1.00AAB (h)\nJAA/JBB=1.00\nJAB/JBB=-1.00\nQ\nFigure 3: (Colour Online) Magnetic response of the trilayered syste m for a few selected cases with: (a)-(d) ABA\nand (e)-(h) AAB. The shift of both, the compensation (where it is p resent) and critical temperatures towards the low\ntemperature ends and shift of the magnetic behaviours between N ,L,P,Q etc. type of ferrimagnetism, with increase\nin standard deviation of the uniform random external magnetic field , are clearly visible in all these plots. The type\nL within brackets is explicitly not seen in the plots but encountered in- transition. In (a) and (e): we witness the\nfield-driven vanishing of compensation from σ= 0.20 and upwards. Where, the errorbars are not visible, they are\nsmaller than the area of the point-markers. All these plots are obt ained for a system of 3 ×100×100 sites.\n7 0 0.01 0.02 0.03\n 0  1  2  3  4∆O\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00ABA (a)\nJAA/JBB=0.04\nJAB/JBB=-0.04\n 0 0.004 0.008 0.012 0.016\n 0  1  2  3  4∆E\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00ABA (b)\nJAA/JBB=0.04\nJAB/JBB=-0.04\n 0 0.003 0.006 0.009 0.012\n 0  1  2  3  4∆O\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00ABA (c)\nJAA/JBB=0.04\nJAB/JBB=-1.00\n 0 0.004 0.008 0.012 0.016\n 0  1  2  3  4∆E\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00ABA (d)\nJAA/JBB=0.04\nJAB/JBB=-1.00\n 0 0.01 0.02 0.03\n 0  1  2  3  4∆O\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00AAB (e)\nJAA/JBB=0.04\nJAB/JBB=-0.04\n 0 0.004 0.008 0.012 0.016\n 0  1  2  3  4∆E\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00AAB (f)\nJAA/JBB=0.04\nJAB/JBB=-0.04\n 0 0.004 0.008 0.012 0.016\n 0  1  2  3  4∆O\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00AAB (g)\nJAA/JBB=0.04\nJAB/JBB=-1.00\n 0 0.004 0.008 0.012 0.016\n 0  1  2  3  4∆E\nTσ=0.00\nσ=0.40\nσ=0.70\nσ=1.00AAB (h)\nJAA/JBB=0.04\nJAB/JBB=-1.00\nFigure 4: (Colour Online) Temperature dependence of Fluctuation o f order parameter, ∆ Oand Fluctuation of\ncooperativeenergyper site, ∆ E, for: ABA in (a)-(d) and AAB in (e)-(h) with JAA/JBB= 0.04andJAB/JBB=−0.04\nand with JAA/JBB= 0.04 andJAB/JBB=−1.00. Where, the errorbars are not visible, they are smaller than the\narea of the point-markers. All these plots are obtained for a syst em of 3×100×100 sites. The nature of the curves\nprominently shows the shift of critical temperatures and even rea son for absence of compensation can be understood\nfrom the low temperature segment of the curves.\nJAA/JBB= 0.20andJAB/JBB=−0.36,within theexpo-\nsure interval to the field, as the spin-field terms take effect\nthe local spin configurations of the top and bottom layers\nlose most of the magnetic ordering for σ= 1.0. Here, thepersitecooperativeenergyofthesurfaceA-layersarecom-\nparable with the spin-field energy per site. The changes\nin magnetization of the surface layers are not-detectable\nwhenσis 0.2 or 0.5, i.e. when per site spin-field ener-\n8-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.04\nJAB/JBB=-0.04\nT=0.01Field: OFF Field: ON\n(A) Top panel: JAA/JBB= 0.04andJAB/JBB=−0.04\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=0.20\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n(B) Middle panel: JAA/JBB= 0.20andJAB/JBB=−0.36\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=0.50\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)ABA:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.20\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=0.50\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01\n-1.2-0.6 0 0.6 1.2 1.8\n 0 25000  50000  75000  100000Magnetisations\nt (MCS)Mt(t)\nMm(t)\nMb(t)\nM(t)AAB:σ=1.00\nJAA/JBB=1.00\nJAB/JBB=-0.36\nT=0.01Field: OFF Field: ON\n(C) Bottom panel: JAA/JBB= 1.00andJAB/JBB=−0.36\nFigure 5: (Colour Online) Plots of Magnetisations for individual layers and average magnetisation of the bulk\nversus time in MCS where Mt(t): Magnetization of the top layer; Mm(t): Magnetization of the mid layer; Mb(t):\nMagnetizationofthe bottom layerareallfunctions oftime, t, in units ofMCS. Forthe ABAstacking,the magnetisation\ncurves for the surface layers (orange and green) overlap for th e most of the times.\n9gies are much lower compared to the cooperative energy.\nThe same argument can be extended for the mid-layer\nwhere no detectable changes are seen from equilibrium to-\nwardssteadystatevaluesofmagnetisationforallthe three\nstandard deviations as the energies of spin-field terms are\nmuch lesser than the exchange energies ( JBBis taken as\nthe dominant coupling strength, equal to 1, for all cases).\nNow we are in a position to understand why, in some\ncases, there is a change in the magnitude of total magneti-\nsation at the lowest temeperature. For example, in Figure\n3(b), evenwhen compensationis presentwith JAA/JBB=\n0.04 andJAB/JBB=−1.00, forσ= 0.20 onwards, we see\nchangesinthenatureofevolution(with respecttotemper-\nature) as well the absolute value. The nature of evolution\nis governed by the competition of energy scales between\nspin-external field and in-plane cooperative energiesof the\nsurface layers. But from σ= 0.76 onwards, the strength\nof the external field is such that the spin-field energy per\nsite is comparableto the combined cooperativeenergy (in-\nplane ferromagnetic and inter-plane anti-ferromagnetic)\nper site in the surface layers. This leads to a pronounced\nrandomisation, similar in nature to the Figure 6(c). The\nonly difference is, the surface layers retain much of the\nmagnetic ordering(for being partiallyrandomised) to can-\ncel out the magnetization ofthe midlayerat a shifted com-\npensation temperature ( < Tcomp(σ= 0.00)). In these two\ncases, the two competing energy scales are again compa-\nrable.\nFigure 6(d) supports the above argument in favour of\ndifferent nature ofevolutionoftotal magnetisation[Figure\n3(b)] which comes out to be the competition between the\nspin-fieldandcooperativeenergies. Here,with JAA/JBB=\n0.04 ;JAB/JBB=−1.00 andσ= 1.00, the per site co-\noperative energy in surface A-layers is comparable to the\nper site coupling energy with the external field.\nAt this point, in light of the explanations provided in\nthe Section 3.2 and above for the ABA configuration, we\ncanunderstandthebehaviourofspin-densityplotsofAAB\nconfiguration in Figure 7, for a few interesting selected\ncases.\n3.4 Phase Diagram and Scaling\nFor a fixed standard deviation or strength of the exter-\nnal uniform random field with spatio-temporal variation,\nin both the configurations: ABA and AAB spin-1/2 Ising\ntrilayered ferrimagnet on square lattice, we have the fol-\nlowing common observations:\n(a) In presence of the field and in those cases with compen-\nsation, compensationtemperaturemergeswith the critical\ntemperatureforhighervaluesof JAA/JBBwhen|JAB/JBB|\nis fixed or vice-versa, just like in the zero field case. This\nimplies the phase diagrams, in presence of the field, can\nbe drwan by following similar procedures as in the refer-\nences [55,56]. In Figure 8, the areas coloured in pale-blue\nare where compensation is present(marked by P) and the\narea(s) in white are the regions where compensation is ab-\nsent (marked by A). The qualitative features of the phase\nseparation curves are same like that in the field-free cases.\nThese curves divide the entire area of the parameter space\ninto: onecontainsferrimagneticphaseswithcompensation\nand the others contain ferrimagnetic phases without com-\npensation.(b) But in cases where the external field is present [From\nσ=0.2, onwards], we see the field makes compensation\ndisappear for a certain range of values of the coupling\nstrengths [Refer to Figure 8]. This observation is a novel\none. In the phase diagrams, the range of values of such\ncoupling strengths for which compensation is absent are\nconfined in areas within the region where compensation\nis present. The appearance of such closed areas resemble\nlike islands orenclaves. The magnitude of areas of such an\nisland also grows as the randomness of the external field\nincreases .\nThe islands of no compensation starts to appear from\nthelowcouplingstrengthpartofthephasediagrams(when\nσ= 0.20). Then the area of the No-Compensation island\n(NCI) gradually increases (reaches higher values of cou-\npling ratios) with increase in the standard deviation of\nthe field. In Figure 9, we see the plots of absolute area\nand rate of increase of absolute area versus the sd of the\nappliedfield. With referencetoFigure2, let’sdiscuss how\nthe magnitude of the area of NCIs are determined:\n(a) Variation of the magnetization at the lowest simula-\ntional temeperature with coupling strengths is noted\nwhere external field destroys compensation at any of\nits strengths.\n(b) The variation in (a) is linearly approximated to find\nout the value of the coupling strength where the\nmagnetizationatthelowesttemperatureiszero. Only\nabove this approximated magnitude of relative cou-\npling strength, we should expect compensation.\n(c) Intermediatepointsandtheleadingandtrailingparts\nof the separation curves are again linearly approxi-\nmated. This provides us with a closed curve.\n(d) Monte Carlointegration is now employed[67] to find\nout the fractional area under the curve obtained in\n(c).\n(e) Centraldifferenceformulaisusedtofindouttherate\nof increase of the area of NCIs.\nA few comments with respect to the nature of the\ncurves of absolute area versus the sd are in order. First,\nthe curve (in RED), comes out to be mostly a superlinear\none for the ABA configuration and mixture of a superlin-\near and sublinear ones for the AAB configuration. The\nplots of the slopes (in BLUE) also confirms it. Second,\nwhere the spin-field energies per site are greater than the\nin-plane cooperative energies per site, the compensation\ndisappers. For higher strengths, the spin-field interaction\nenergies can greatly influence the higher ends of the coop-\nerative energies. For the islands, the area unavailable for\ncompensation for both the configurations are mostly af-\nfected at the lower in-plane coupling strength region. The\nregions with lowest in-plane coupling ratios are affected\nby all the strengths of the external random field, start-\ning from σ= 0.20. as the cooperative energies in this\nregion are smaller than spin-field terms for most of the\nfield-strengths.\nNow an interesting exercise can be carried out on how\nthe magnitude of the area of NCIs scale with the sd of the\nfield. This would be an 1 Dscaling and let’s assume the\nfollowing scaling function:\nf(A(σ),σ) =σ−bA(σ) (11)\n10Top layer Mid layer Bottom layer\n(a)ABA:JAA/JBB=0.04;JAB/JBB=−0.04andσ=0.50\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−0.039 Mm(tmorph) = +1.00 Mb(tmorph) =−0.056\n(b)ABA:JAA/JBB=0.20;JAB/JBB=−0.36andσ=0.50\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−1.00 Mm(tmorph) = +1.00 Mb(tmorph) =−1.00\n(c)ABA:JAA/JBB=0.20;JAB/JBB=−0.36andσ=1.00\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−0.402 Mm(tmorph) = +1.00 Mb(tmorph) =−0.375\n(d)ABA:JAA/JBB=0.04;JAB/JBB=−1.00andσ=1.00\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−0.649 Mm(tmorph) = +1.00 Mb(tmorph) =−0.638\nFigure 6: For ABA configuration : Lattice morphologies of top layer (at Left) ;mid layer (at Middle) and\nbottom layer (at Right) att=tmorph= 105MCSfor a few selected coupling strengths and external field. The\ndestruction of compensation in the cases (a) and (c) is due to the s ignificant reduction of magnetic ordering in the top\nand bottom layers i.e. surface layers . In the case (d), the increase in the value of total magnetisation f rom−0.333 (in\nfield-free cases) to −0.096 (with σ= 1.00) at the lowest temperature is due to the partial loss of magnetic order and\nit is responsible for the change in the nature of evolution of total ma gnetisation of the system.\nThe scaling exponent comes out to be, for ABA: bABA=\n1.958±0.122 and for AAB: bAAB= 1.783±0.118. The\nestimate oferrorquoted in scalingexponent, b, is obtained\nby the standard deviation among all the sets of data. The\nagreement is a little poor at the two extreme ends ( σis\neither 0.20 or 1.00 ) for both the configurations. At thelow ends, the effect of the field is not much pronounced\nwhile at the highest end, the absolute values of the area\nof NCIs tend to saturate.\n11Top layer Mid layer Bottom layer\n(a)AAB:JAA/JBB=0.04;JAB/JBB=−0.04andσ=0.50\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) = +0.003 Mm(tmorph) =−0.053 Mb(tmorph) = +1.00\n(b)AAB:JAA/JBB=0.20;JAB/JBB=−0.36andσ=0.50\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−1.00 Mm(tmorph) =−1.00 Mb(tmorph) = +1.00\n(c)AAB:JAA/JBB=0.20;JAB/JBB=−0.36andσ=1.00\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−0.101 Mm(tmorph) =−0.407 Mb(tmorph) = +1.00\n(d)AAB:JAA/JBB=0.20;JAB/JBB=−1.00andσ=1.00\n 20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection  20 40 60 80 100\n 20 40 60 80 100y\nx\n-1 1\nspin-projection\nMt(tmorph) =−0.208 Mm(tmorph) =−0.954 Mb(tmorph) = +1.00\nFigure 7: For AAB configuration : Lattice morphologies of top layer (at Left) ;mid layer (at Middle) and\nbottom layer (at Right) att=tmorph= 105MCSfor a few selected coupling strengths and external field. The\ndestruction of compensation in the cases (a) and (c) is due to the s ignificant reduction of magnetic ordering in the top\nand bottom layers i.e. surface layers . In the case (d), the increase in the value of total magnetisation f rom−0.333 (in\nfield-free cases) to −0.096 (with σ= 1.00) at the lowest temperature is due to the partial loss of magnetic order and\nit is responsible for the change in the nature of evolution of total ma gnetisation of the system.\n4 Summary\nFrom the zero-field cases [56](b), it is already established\nfrom lattice morphologies for both the configurations of\nFigure 1 that magnetic ordering develops considerably at\nthe compensation temperatures in the A-layers (for cases\nwith compensation and weaker JAA) and sublayers are al-most completely ordered at the lowest temperature. With\nthe increase in the field-strength in temeperatures much\nlower than Tcrit(nearly athermal), for both the ABA and\nAAB configurations, magnetic ordering gradually dimin-\nishes in the A-layers in its steady state (Refer to Figures\n3,4,6 and 7; when the magnitudes of per site spin-field en-\n12(a)\n (b)\n(c)\n (d)\nFigure 8: (Colour Online) Phase diagram for the: ABA trilayered ferr imagnetic system when: (a) σ= 0.52; (b)\nσ= 1.00 and AAB trilayered ferrimagnetic system when: (c) σ= 0.52; (d)σ= 1.00, in presence of the uniform\nrandom external magnetic field. A: Compensation is absent; P: Com pensation is present. With increase in the\nstandard deviation of the external field, the magnitude of the are a of the no-compensation island have grown. The\nblue segment of the phase separation curves are obtained via linear extrapolation. All these plots are obtained for a\nsystem of 3 ×100×100 sites. Where the errorbars are not visible, they are smaller tha n the point markers.\n(a)\n0.000.100.200.300.40\n0.000.250.500.751.00|Aσ| , |dAσ|/dσ \nσ (sd)|Aσ|\n|dAσ|/dσ\nABA(b)\n0.000.150.300.450.60\n0.000.250.500.751.00|Aσ| , |dAσ|/dσ \nσ (sd)|Aσ|\n|dAσ|/dσ\nAAB\nFigure 9: (Colour Online) Plots of: Magnitude of the area of the no-c ompensation islands versus standard deviation\nof the field (in RED) and the rate of increase in the magnitude of the a rea of the no-compensation islands versus\nstandard deviation of the field (in BLUE) for (a) ABA and (b) AAB con figurations. All these plots are obtained for\na system of 3 ×100×100 sites.\n13ergies is comparable to in-plane cooperative energies per\nsite). This leadsto havingamuch lowermagnitude ofsub-\nlattice magnetisations in the A-layersthan in the field-free\ncase. In moderate-to-high field strengths, such low sub-\nlattice magnetisations in the A-layers are unable to cancel\nout the magnetization in the mid B-layer at even in the\nnearly athermal case. This is the reason why compen-\nsation disappears when the field strength is comparably\nhigher than in-plane coupling strengths of the surface A-\nlayers. This can also be judged from the Figure 4, where\nthe plateau around compensation point flattens and loses\nits smeared peak (which happens at the location of com-\npensation) with increasing strength of the random field.\nEven at the lowest point of tempearture in the simulation,\nboth the fluctuations increase from zero with increase in\nthe standarddeviation ofthe external field. This indicates\ngradual increase in the randomness in the orientations of\nspin projections in the surface A-layers. The origin of\ndynamical field-driven vanishing of compensation can be\nattributed to the diminishing of magnetic ordering in the\nA-layers when the spin-field interaction energy is larger\nthan in-plane coupling, for the trlayeredspin-1/2Isingfer-\nrimagnetic systems of Figure 1. Figure 8 shows how the\nenclave of No-Compensation comes up within the phase\ndiagram and evolves with the increase of the strength of\nthe external uniform random field. A proposed scaling re-\nlation is found to hold good [Equation 11] with exponent\nbABA= 1.958±0.122 and bABA= 1.783±0.118. So the\ntemporally varying and site dependent external random\nfield,hi(t)’s in Equation 1 can reveal novel features in the\nphase diagrams [Figure 8] and represent systems where\nthe Hamiltonian is itself a dynamic one. The coupling\nconstants in the Ising model is traditionally understood\nto be translationally invariant. Under such a constraint,\nif competing ferromagnetic and anti-ferromagnetic inter-\nactions are taken into account, the Ising model exhibits\na remarkable complexity. For an example, the equilib-\nrium studies of the system studied in the current article\ncan be mentioned [55,56]. In reality, compositional disor-\nder, impurities, vacancies, lattice dislocations etc. lead to\nmodifications in the Hamiltonian, which, with Ising me-\nchanics, may be characterized by changes which are no\nlonger translationally invariant, but random quantities,\ncharacterized by their probability distributions. It would\nbe interesting to study how the system behaves when it\nis exposed to other types of random magnetic fields with\nprobabilty distribution functions. 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Statist. 33(3), 1065\n(1962).\n63. Deisenroth M. P., Aldo Faisal A., and Ong C. S.,\nMathematics for Machine Learning (CambridgeUni-\nversity Press, New York, 2020).\n64. N´ eel L., Ann. Phys. (Paris) 3, 137 (1948).\n65. Chikazumi S., Physics of Ferromagnetism (Oxford\nUniversity Press, Oxford, 1997).\n66. Streˇ cka J., Physica A 360, 379 (2006).\n67. see e.g. Krauth W., Statistical Mechanics: Algo-\nrithms and Computations (Oxford University Press,\nNew York, 2006).\n15"    },    {        "title": "0903.2331v1.Half_metallic_ferrimagnetism_in_the__Sc___1_x__V__x__C_and__Sc___1_x___V__x__Si_alloys_adopting_the_zinc_blende_and_wurtzite_structures_from_first_principles.pdf",        "content": "arXiv:0903.2331v1  [cond-mat.mtrl-sci]  13 Mar 2009Half-metallicferrimagnetisminthe[Sc 1−xVx]Cand[Sc 1−xVx]Sialloys\nadoptingthezinc-blendeandwurtzitestructuresfromfirst -principles\nK.¨Ozdo˜ ganaE. S ¸a¸ sıo˜ glub,cI. Galanakisd,∗\naDepartment of Physics, Gebze Institute of Technology, Gebz e, 41400, Kocaeli, Turkey\nbInstitut f¨ ur Festk¨ orperforschung, Forschungszentrum J ¨ ulich, D-52425 J¨ ulich, Germany\ncFatih University, Physics Department, 34500, B¨ uy¨ uk¸ cek mece,˙Istanbul, Turkey\ndDepartment of Materials Science, School of Natural Science s, University of Patras, GR-26504 Patra, Greece\nAbstract\nEmploying first-principles calculations we study the struc tural, electronic and magnetic properties of the [Sc 1−xVx]C\nand [Sc 1−xVx]Si alloys. In their equilibrium rocksalt structure all all oys are non-magnetic. The zincblende and\nwurtzite structures are degenerated with respect to the tot al energy. For all concentrations the alloys in these lattic e\nstructures are half-metallic with the gap located in the spi n-down band. The total spin moment follows the Slater-\nPauling behavior varying linearly between the -1 µBof the perfect ScC and ScSi alloys and the +1 µBof the perfect\nVC and VSi alloys. For the intermediate concentrations V and Sc atoms have antiparallel spin magnetic moments\nand the compounds are half-metallic ferrimagnets. At the cr itical concentration, both [Sc 0.5V0.5]C and [Sc 0.5V0.5]Si\nalloys present zero total spin-magnetic moment but the C-ba sed alloy shows a semiconducting behavior contrary to\nthe Si-based alloys which is a half-metallic antiferromagn et.\nKey words: Electronic structure, Half-metals, Ferrimagnets\nPACS:75.47.Np, 75.50.Cc, 75.30.Et\n1. Introduction\nHalf-metallicferromagnetshaveattractedconsid-\nerable attention during the last decade due to their\npotential applications in magnetoelectronic devices\n[1]. The term “half-metal” was initially introduced\nby de Groot and collaborators in 1983 to denote\nthe peculiar behavior exhibited by a Heusler com-\npounds: NiMnSb [2]. The have found using first-\nprinciples calculations that the majority-spin band\nwasmetallicwhiletheminority-spinbandwassemi-\n∗Corresponding author. Phone +30-2610-969925, Fax +30-\n2610-969368\nEmail addresses: kozdogan@gyte.edu.tr (K.¨Ozdo˜ gan),\ne.sasioglu@fz-juelich.de (E. S ¸a¸ sıo˜ glu),\ngalanakis@upatras.gr (I. Galanakis).conducting leading to 100%spin-polarizationof the\nelectrons at the Fermi level. This behaviorwaslater\non confirmed both by infrared absorption [3] and\nby spin-polarized positron annihilation [4] experi-\nments.\nAlthough Heusler alloys have attracted a lot of\ninterest as potential half-metallic systems, the dis-\ncovery of Akinaga and his collaborators in 2000 has\nshown the possibility to grow new half-metallic sys-\ntems in metastable structures usually adopted by\nthinfilms[5].TheyhaveshownthattheCrAs/GaAs\nmultilayers are ferromagnets and surprisingly the\nX-ray diffraction measurements suggest that CrAs\nadoptsthelatticestructureofGaAsandgrowsinthe\nmetastable zincblende structure. Moreover SQUID\nmeasurementshaveshownthatCrAsexhibitsanin-\nteger total spin magnetic moment of 3 µBper unit\nPreprint submitted to Comp. Mater. Sci. 14 September 2021cell [5]. These findings have intensified the interest\non transition-metalpnictides and chalcogenideslike\nCrAs and CrSe which crystallize either in the zinc-\nblende orwurtzite structures ofbinary semiconduc-\ntors and an extended review can be found in refer-\nence [6]. Galanakis and Mavropoulos have studied\nusing first-principles calculations several such com-\npounds and have determined the lattice constants\nfor which half-metallicity is present [7]. Moreover\nthey haveexplainedthe gapin terms ofthe p−dre-\npulsion; the porbitalsofthe spatomhybridize with\nthet2gorbitals of the transition metal atoms creat-\ningthreebondingandthreeantibondingstates.The\ngap is created between these states. The egorbitals\nofthetransitionmetalatomareverylocalizedinen-\nergy since they do not hybridize with other orbitals\nandtheyareplacedabovetheFermilevelinthespin-\ndown band. Their relative position with respect to\nthe antibonding p-t2gorbitals depends on each sys-\ntem. It has been also shown in the same reference\nthatexceptthethreebonding p-t2gbandsbelowthe\nFermi level exists also a deep sband. Thus, since\nthereareexactlyfourspin-downoccupiedbands,the\ntotal spin moment, Mt, follows a Slater-Pauling be-\nhaviorand it equalsin µBunits:Mt=Zt−8 where\nZtthe total number of valence electrons in the unit\ncell.\nAs we mentioned above most of the studies con-\ncern cases where the spatom belongs to the Vth\n(pncitides) or the VIth (chalcogenides) column of\nthe periodical table. The case where the spatom\ncomes from the IVth column of the periodical ta-\nble has attracted much less attention in literature\nand to the best of our knowledge only the cases\nof MnC [8,9,10], MnSn [9] and MnSi [11,12,13]\nin the zincblende lattice have been studied. In\nthis manuscript we study using the full–potential\nnonorthogonal local–orbital minimum–basis band\nstructure scheme (FPLO) [14] within the local\ndensity approximation (LDA) [15] the case of\n[Sc1−xVx]C and [Sc 1−xVx]Si alloys for xtaking the\nvalues 0, 0.1, 0.2, ...,0.9, 1. The disorder is simu-\nlated using the coherent potential approximation\n[16]. ScC and ScSi compounds have 7 electrons per\nunit cell and, if they are half metals, they should\nshow a total spin moment of -1 µB. On the other\nhand VC and VSi have 9 electrons and, if they are\nhalf-metals, should exhibit a total spin moment of\n+1µB. Thus within these families of alloys we can\nstudy the transition at the x= 0.5 concentration\nwhere the total spin moment changes sign. In sec-\ntion2westudytheperfectcompoundstodetermine153045 6002468Energy (eV)\nRS\nZB\nWZ\n12 24 36 4803691225 50 75 10002468\n20 40 6080\nVolume  (Å3)02468ScC                                          ScSi\nVC                                              VSi\n4.2 4.95.5 4.76.84.8\n6.35.6\n5.13.64.43.9\nFig. 1. (Color online) Calculated total energy as a function\nof the volume of the unit cell for ScC, ScSi, VC and VSi\nin the rocksalt (RS), zincblende (ZB) and wurtzite (WZ)\nstructures. The zero of the total energy is defined as the\nenergy of the global equilibrium volume and with arrows we\nrepresent the corresponding equilibrium lattice constant ; for\nthe WZ structure which is not cubic we give the in-plane\nlattice parameter aand the c/aratio is for all calculations\nthe ideal (8\n3)1\n2for which the nearest environment in the WZ\nstructure is the same with its cubic ZB analogue. In the RS\nand ZB there is one transition-metal atom and one sp atom\nper unit cell. In the WZ structure there are two atoms of\neach chemical kind but we have divided the energy by two\nto compare it directly to the other two cases.\nthe equilibrium lattice constants and we show that\nall four ScC, ScSi, VC and VSi crystallize in the\nnon-magnetic rocksalt structure. In section 3 we\ncontinue our study with the case of the magnetic\nzincblende and wurtzite structures which are de-\ngenerated with respect to their total energy and we\nstudy both the electronic and magnetic properties.\nWe show that for the intermediate concentrations\nthe total spin moment scales linearly with the con-\ncentrationandtheScandVatomshaveantiparallel\nspin magnetic moments. We also try to explain why\nforx= 0.5 the C-based alloy is a semiconductor\nwhile the Si-based alloysis a half-metallic antiferro-\nmagnet [17]. The latter property is highly desirable\nfor applications since such materials create van-\nishing stray fields and thus minimize energy losses\nin devices. Finally in section 4 we summarize and\npresent our conclusions.\n2-202DOS (states/eV)Total\nSc (V)\nC (Si)\n-3 -2 -1 0 1-202\n-6-4 -2 0 2\nE − EF (eV)ScC                                                                            VC\nScSi                                                                           VSiRS\nFig. 2. (Color online) Total and atom-resolved density of\nstates (DOS) for all four perfect compounds in the rock-salt\nstructure at their equilibrium lattice constants. We prese nt\nupwards the DOS for the spin-up states and downwards for\nthe spin-down. In this structure we have converged to a\nnon-magnetic state for all four alloys. The Fermi level has\nbeen chosen as the zero of the energy axis.\n2. Total energy calculations\nWe will start the presentation of our results\ndiscussing the equilibrium lattices and lattice con-\nstants. We have taken into account three different\nlattices: (i) the rocksalt (RS), (ii) the zincblende,\nand (iii) the wurtzite (WZ) structures, and we\npresent in figure 1 the calculated total energy as\na function of the volume of the unit cell. Before\nproceeding with the discussion and presentation of\nour results we should focus on the characteristics of\nthe three structures under study. We have chosen\nthe RS structure since it is adopted by the majority\nof the binary compounds between transition-metal\nandspatoms when the stoichiometry is 1:1. The\nlattice is actually a fcc with two atoms as basis set,\none atom at (0 0 0) and the second atom at (1\n21\n21\n2) in Wyckoff coordinates. Thus the RS is a close\npacked structure. Contrary to RS both ZB and WZ\nstructures are open structures. In the ZB structure\nthe lattice is again a fcc with four sites as basis set\nalong the diagonal, but now two out of the four\nsites are empty. The WZ is the hexagonal analogue\nof the ZB structure. In our WZ calculations we\nhave varied only the in-plane lattice parameter a\nand we have considered that the c/aratio is for all\ncalculations the ideal (8\n3)1\n2for which the nearestenvironment in the WZ structure is the same with\nits cubic ZB analogue. Finally we should mention\nthat in the RS and ZB there is one transition-metal\natom and one sp atom per unit cell while in the\nWZ structure there are two atoms of each chemical\nkind. We havedivided in the WZ caseby twoallthe\nproperties, which are calculated per unit cell (total\ndensity of states, total spin magnetic moment and\ntotal energy), in order to compare them directly to\nthe other two cases.\nFor all four perfect compounds presented in fig-\nure 1 the RS structure is the equilibrium lattice and\nthe energy difference between the equilibrium RS\nlattice constant and the ZB-WZ equilibrium lattice\nconstantsisbetween4and8eVwhichareverylarge\nenergy differences. Thus in the form of single crys-\ntals all ScC, ScSi, VC and VSi prefer to crystallize\nin the RS lattice. As expected the equilibrium vol-\nume is much smaller in the RS case compared to\nthe other two since in the former one no voids ex-\nist. Surprisinglyourresultssuggestthat the ZBand\nWZ structure are degenerated in all cases and thus,\nwhen grown as thin films on top of semiconductors,\nthe unit cell can easily deform itself. In the same\nfigure we have also denoted the equilibrium lattice\nconstants for all three structures (for the WZ one\nwe give the in-plane lattice parameter). We remark\nthat the trends depend on the chemical elements.\nWhen wesubstitute Si forC, the equilibrium lattice\nparameters increases. C and Si have both four va-\nlence electrons but C has six electrons in total (the\natomic configuration is 1 s22s22p2) while Si has 14\nelectrons (the atomic configuration is 1 s22s22p6\n3s23p2)andthusoccupiesmorespace.ScandVbe-\nlong in the same series in the periodic table. Sc has\nthree valence electrons (the valence electrons in the\natomicconfigurationarethe4 s2and3d1)andVhas\ntwo more delectrons and thus in total five valence\nelectrons. Although V has more valence electrons\nthan Sc, it is well known that for the early transi-\ntion metal atomsasthe valenceincreasesthe lattice\nparameter decreases [18] and this is also the cases\nhere.VCandVSi correspondtosmallerequilibrium\nlattice parameters than ScC and ScSi, respectively.\nFinallyweshoulddiscusstheelectronicproperties\nin the case of the RS structure before proceeding in\nthe next section with the ZB and WZ structures. In\nthe RS lattice all four perfect compounds are non-\nmagnetic as can be seen from the density of states\n(DOS) presented in figure 2. In the case of the ScC\nand ScSi alloys the Fermi level crosses the valence\nband, which is created by the bonding states due to\n3the hybridization between the Sc d- and the C(Si)\np-orbitals,andthe atom-resolvedDOS hasasimilar\nshape for both the Sc and C(Si) atoms. When we\nsubstitute V for Sc we populate also partially the\nconduction band created by the antibonding p−d\nstates. As can be seen in the figure, the conduction\nband has its main weight mainly at the vanadium\natom since the latter one offers a lot of empty d-\nstates with respect to the empty pstates of the sp\natoms.Finallyweshouldnotethatforallfouralloys\nthere is one sband lying very deep in energy and\nwhich is separated by more than 3 eV from the bot-\ntom of the valence band and thus we do not present\nit in figure 2.\n3. Ferrimagnetism in the zincblende and\nwurtzite structures\nAs we showed in the previous section all com-\npoundsunderstudyarenon-magneticintheequilib-\nriumRSlatticeandthusarenotusefulforspintronic\napplications. Contrary to the RS case, we found all\nfour ScC, ScSi, VC and VSi to be magnetic in both\nthe ZB and WZ structures. Although these struc-\nturesarenotstableones,itispossibletooccurwhen\nthesealloysaregrownasthinfilmsormultilayerson\ntopofbinarysemiconductorsadoptingthesamelat-\ntice structure. This is possible with the new up-to-\ndate experimental techniques like Molecular Beam\nEpitaxy or Pulsed Layer Deposition and in fact the\nformermethodwasemployedtogrowtheCrAsfilms\nwith the ZB structure on top of the GaAs semicon-\nductor in reference [5].\nWewillstartourdisucssionfromthepropertiesof\nthe perfect compounds and in table 1 we have gath-\nered the atom-resolvedand total spin magnetic mo-\nmentsandinfigure3wepresenttheassociatedDOS.\nAll presented calculations have been performed at\nthe equilibrium lattice constants presented in figure\n1. Our first remark concerns the difference between\nthe ZB andWZ structures.As we cansee in table 1,\nthe spin magnetic moments only scarcely vary be-\ntweenthetwolatticestructuresandthesameoccurs\nalso for the total and atom-resolvedDOS presented\nin figure 3 (note that we have divided the total spin\nmoment and DOS in the WZ structure by two to\nmake it comparable to the ZB case). Thus we will\nconcentrate our discussion to the ZB case and re-\nfer to the WZ structure only if the difference with\nrespect to the ZB lattice is significant. ScC has in\ntotal 7 valence electrons per unit cell and thus inTable 1\nTotal and atom-resolved spinmagnetic moments in µBforall\ncompounds under study at the equilibrium lattice constants\nfor the zincblende (ZB) and wurtzite (WZ) structures. For\nthe intermediate concentrations we have considered that th e\nlattice constant scales linearly with the concentration. W e\nhave scaled the atom-resolved spin moments to one atom.\nThe total spin moment is the sum (1 −x)∗mSc+x∗mV+\nmC(Si)+mint, where mintrefers to the interstitial region\n(empty sites). We do not present mintseparately since it is\nnegligible with respect to the atomic spin moments. In the\nZB there is one transition-metal atom and one sp atom per\nunit cell, while in the WZ structure there are two equivalent\natoms of each chemical kind and thus we present half the\ntotal spin moment in the unit cell to compare it directly to\nthe ZB case.\n[Sc1−xVx]C\nzincblende wurtzite\nxmScmVmCmTotalmScmVmCmTotal\n0-0.179 - -0.821 -1.000 -0.186 - -0.814 -1.000\n0.1-0.177 0.994 -0.740 -0.799 -0.201 1.122 -0.731 -0.800\n0.2-0.170 0.783 -0.620 -0.599 -0.202 0.968 -0.632 -0.600\n0.3-0.144 0.533 -0.458 -0.399 -0.183 0.764 -0.501 -0.400\n0.4∼-0∼0∼-0∼-0-0.126 0.466 0.310 -0.200\n0.5∼-0∼0∼-0 0 ∼-0∼0∼-0 0\n0.6-0.064 0.646 -0.162 0.200 -0.090 0.755 -0.217 0.200\n0.7-0.086 0.899 -0.204 0.400 -0.108 0.986 -0.258 0.400\n0.8-0.098 1.047 -0.218 0.600 -0.117 1.122 -0.274 0.600\n0.9-0.107 1.144 -0.219 0.800 -0.122 1.215 -0.282 0.800\n1- 1.192 -0.192 1.000 - 1.194 -0.194 1.000\n[Sc1−xVx]Si\nzincblende wurtzite\nxmScmVmSimTotalmScmVmSimTotal\n0-0.349 - -0.651 -1.000 -0.367 - -0.633 -1.000\n0.1-0.383 2.030 -0.658 -0.799 -0.403 1.868 -0.624 -0.800\n0.2-0.423 1.914 -0.644 -0.599 -0.436 1.778 -0.607 -0.599\n0.3-0.451 1.764 -0.613 -0.399 -0.457 1.656 -0.577 -0.399\n0.4-0.448 1.558 -0.554 -0.199 -0.456 1.504 -0.527 -0.199\n0.5-0.405 1.325 -0.460 -0.001 -0.429 1.352 -0.461 0.001\n0.6-0.413 1.324 -0.429 0.200 -0.417 1.308 -0.418 0.200\n0.7-0.426 1.333 -0.405 0.400 -0.414 1.302 -0.387 0.400\n0.8-0.439 1.337 -0.381 0.600 -0.413 1.305 -0.361 0.600\n0.9-0.451 1.335 -0.356 0.800 -0.414 1.310 -0.338 0.800\n1- 1.330 -0.330 1.000 - 1.288 -0.288 1.000\n4-303 Total\n-303\n-303DOS (states/eV)\n-303Sc\nV\nC\n-2 0 2\nE − EF (eV)-303\n-2 0 2x=0.2ZB - [Sc1-xVx]C \nx=0.5\nx=0.8\nx=1x=0 -303 Total\n-303\n-303DOS (states/eV)\n-303Sc\nV\nC\n-2 0 2\nE − EF (eV)-303\n-2 0 2x=0WZ - [Sc1-xVx]C\nx=0.2\nx=1x=0.5\nx=0.8\n-606Total\n-606\n-606DOS (states/eV)\n-606Sc\nV\nSi\n-2 0 2\nE − EF (eV)-606\n-2 0 2x=0ZB - [Sc1-xVx]Si \nx=0.2\nx=0.5\nx=0.8\nx=1-606Total\n-606\n-606DOS (states/eV)\n-606Sc\nV\nSi\n-2 0 2\nE − EF (eV)-606\n-2 0 2x=0WZ - [Sc1-xVx]Si  \nx=0.2\nx=0.5\nx=0.8\nx=1\nFig. 3. (Color online) Total and atom-resolved DOS for [Sc 1−xVx]C (upper panel) and [Sc 1−xVx]Si (lower panel) in both\nzincblende (left) and wurtzite (right) lattice structures for several values of the concentration x. The lattice parameter has\nbeen assumed to vary linearly between the equilibrium latti ce constants of the ScC(ScSi) and VC(VSi) alloys. We have sca led\nthe atom-resolved DOS to one atom. In the case of the wurtzite structure the unit cell contains double the atoms of the\nzincblende unit-cell and thus we have scaled the total DOS in the wurtzite structure by 0.5 to make them comparable. In all\ncases the deep lying sstates are not shown.\n5order to be half-metal it should exhibit a spin mag-\nnetic momentof-1 µBaccordingtothe Mt=Zt−8\nSlater-Pauling rule. This is actually the case since,\nas can be seen in table 1, Sc carries as spin moment\nof around -0.18 µBand C atoms carry a spin mo-\nment ofaround-0.82 µB. Although it seemsstrange\nthat the spin moment is mainly concencentrated to\nthespatoms, this phenomenon can be easily un-\nderstood if we look at the DOS in figure 3. In the\nspin-down band all four states are occupied. These\nstates,aswehavealreadydiscussedin section1and\nas it is shown in reference [7], consist of a deep ly-\nings-state and three bonding states due to the hy-\nbridizationbetween the pelectronsofthe C and the\nt2gd-electronsofthe Sc atom.Thesebonding states\nare located mainly at the C atom since the pstates\nofClielowerinenergywithrespecttothe t2gstates\nofSc.Sinceintotalwehavesevenelectronsandfour\nare already accomodated in the spin-down bands,\nonly three have to be accomodated in the spin-up\nband.Thusinthespin-upbandagainonlythebond-\ning hybrids with their main weight at the C atom\nare partially occupied and thus the spin moment is\nmainly concentratedat the sp-atom.As can be seen\nin the same graph the unoccupied states are mainly\nof Sc character since they are consisted of the anti-\nbonding p−t2gstates and the localized egd−states\nof Sc. When we subsitute V for Sc, the two extra\nelectronsfillexclusivelyspin-upstatessincethehalf-\nmetallicity is preserved and the Fermi level is again\nwithin the spin-down gap. Thus now all the boind-\ning spin-up states are occupied and one electron oc-\ncupies partially the localized spin-up egstates of V\n(in total these states can occupy two electrons per\nspin). As a result the Fermi level falls within a large\npeak in the spin-up band and almost divides it in\ntwoequalparts.The spin-up peak ofthe egstatesis\nclearly separated by the antibonding p−t2gstates\nas can be seen in the DOS. As a result of the above\ndiscussion V has now a spin magnetic moment of ∼\n1.19µBand C a spin magnetic moment of ∼-0.19\nµBresulting to a total spin moment of +1 µBin\nagreement to the Slater-Pauling behavior. The spin\nmoment ofV can be decomposed to 1 µBdue to the\negstate,and0.19 µBwhichcontrobalancesthe-0.19\nµBof C and come from the small inbalance in the\ndistribution of the bonding p−t2gstates between\nthe two chemical species in the spin-up and spin-\ndown bands. In the case ofthe Si compound the sit-\nuation is similar with only noticeable difference the\nfactthatSicarriesinthecaseofScSiasmallerabso-\nlutevalueofthespinmagneticmomentwithrespectto C in ScC while the situation is inversed when we\ncomparethe VSi totheVC alloy.ScSiandVSi com-\npounds have significantly larger equilibrium lattice\nconstants with respect to the ScC and VC alloys as\nshown in figure 1. Thus the hybridization between\nthe Sip-states and the t2g-states of Sc(V) is less in-\ntense than in the case of the C-based alloys result-\ningin bandswhicharemorenarrowin energywidth\nbut more intense (the scale in the vertical DOS axis\nin figure 3 is double for the Si-based alloys). Due to\nthis small change in hybridization a variation in the\nspin magnetic moments occurs with respect to the\nC-based compounds.\nWe have also performed calculations for the in-\ntermediate concentrations xand in table 1 and fig-\nure 3 we have gathered the atom-resolved and to-\ntal spin magnetic moments and DOS. The atom-\nresolved properties have been scaled to one atom.\nWehaveassumedthatthelatticeconstantsscalelin-\nearly with the concentration xbetween the perfect\ncompounds as occurs experimentally for the qua-\nternary Heusler alloys which have similar structure\n[19]. In orderforthese alloysto be half-metallic, the\nSlater-Pauling rule should be again valid where as\nvalence of the transition-metal site we consider the\n(1−x)∗ZSc+x∗ZV, whereZSc= 3 and ZV= 5\nthe valence of the Sc and V atoms respectvely. As\nwe dope ScC and ScSi with V, the bonding p−t2g\nstatesinthe spin-upbandstarttomovelowerin en-\nergy with respect to the Fermi level and the peak of\ntheegstates starts appearing at the Fermi level as\ncan be clearly seen for the case of x= 0.8. Thus al-\nmost all intermediate compounds follow the Slater-\nPauling behavior as can be seen from the total spin\nmoments in table 1 and the electronic and magnetic\nproperties of the compounds vary in a continuous\nwaywiththeconcentration.Thecompoundsarefer-\nrimagnets since the V atoms have a spin moment\nantiparallel to the one of the Sc and C(Si) atoms.\nThe only exception of compounds which are not\nmagnetic is the case of [Sc 1−xVx]C forx=0.4 and\n0.5 in the ZB lattice and x=0.5 in the WZ struc-\nture. We will start our discussion from the case of\nx=0.5. [Sc 0.5V0.5]C is not magnetic both in the ZB\nand WZ structures and as shown in the DOS pre-\nsentedinfigure3itisactuallyasemiconductorsince\nthere are exactly eight valence electrons which oc-\ncupy the bonding p−t2gstates in both the spin-up\nand spin-down bands. The energy gap is about 0.4\neV. We canclearlydistinguish in theDOS the occu-\npied bonding hybrids, followed by the egbands just\nabove the Fermi level with most of their weight at\n60246DOS (states/eV)Total\nSc\nV\nSi\n-2 -1 0 1 2\nE − EF (eV)0246ZB - [Sc0.5V0.5]Si\nWZ - [Sc0.5V0.5]Si\nFig. 4. (Color online) Total and atom-resolved DOS for\n[Sc0.5V0.5]Si in both zincblende and wurtzite structures per-\nforming non-spinpolarized calculations. The atom-resolv ed\nDOS have been scaled to one atom.\nthe V atoms and which are clearly separated in en-\nergy from the antibonding p−t2gstates which are\neven higher in energy (the deep-lying sstates are\nnot shown). In the case of x=0.4 in the ZB struc-\nture the Fermilevelfalls in aregionofsmallspin-up\nDOS (the DOS is not presented here) and due to\nthe Stoner theorem the alloy is not magnetic con-\ntrarytotheWZstructurewheretheStonertheorem\nis satisfied even marginally resulting in a magnetic\ncompound.\nContrary to [Sc 0.5V0.5]C the [Sc 0.5V0.5]Si alloy is\nmagneticandinreallityitisaso-calledhalf-metallic\nantiferromagnet[17,20]sincethe totalspin-moment\nis exactly zero as in conventional antiferromagnets\nbutthedifferentconstituentsaremagneticasshown\nin table 1. The question which arises is why the Si-\nalloy shows such a different behavior from the C-\nalloy. To elucidate the origin of the difference we\nhave performed non-magnetic calculations for the\n[Sc0.5V0.5]SialloyinboththeZBandWZstructures\nand we present the DOS in figure 4. The Fermi level\nfalls within a deep of the DOS in the ZB structure\nand within a small gap in the WZ structure, thus\ndue to Stoner theorem in both cases magnetism is\nnot favorable.But Stoner theorem has a limited ap-\nplication in alloys where the situation is more com-\nplicatedwithrespecttocrystalsmadeoutofasingle\nchemical element. Below the Fermi level the bands\nare concentrated in a small energy range and just\nabove the Fermi level there is a very intense peakdue to the V egstates making the non-magnetic so-\nlution unstable and the system prefers to be mag-\nnetic in order to lower its total energy.\n4. Summary and conclusions\nWe have employed first-principles calculations\nandhavestudiedthestructural,electronicandmag-\nnetic properties of the [Sc 1−xVx]C and [Sc 1−xVx]Si\nalloys. In their equilibrium rocksalt structure all al-\nloysarenon-magnetic.The zincblende andwurtzite\nstructures are degenerated with respect to the total\nenergy and show magnetism. For all concentra-\ntions we found that the alloys in these two lattice\nstructures are half-metallic with the gap located\nin the spin-down band. Moreover they are ferri-\nmagnets since V and Sc atoms have antiparallel\nspin-magnetic moments. The total spin moment\nfollows the Slater-Pauling behavior varying linearly\nbetweenthe-1 µBoftheperfectScCandScSialloys\nand the +1 µBof the perfect VC and VSi alloys.\nAt the critical concentration, both [Sc 0.5V0.5]C and\n[Sc0.5V0.5]Si alloys present zero total spin-magnetic\nmoment but the C-based alloy shows a semicon-\nducting behaviorsince the Stoner criterionfor mag-\nnetism is not satisfied. Contrary, the [Sc 0.5V0.5]Si\ncompounds is a half-metallic antiferromagnet and\nthus very interesting for potential application in\nspintronics.\nWe have shown that also in the case of artificial\nbinary compounds between transition-metal and\nspatoms we can tune their magnetic properties in\na continuous way by mixing atoms of neighboring\nchemical species. Following this procedure we were\nable to demonstrate the existence of half-metallic\nferrimagnetism in the studied alloys (and in an ex-\ntreme case even half-metallic antiferromagnetism)\nwhich is highly desirable for spintronic applications\nsince ferrimagnets create smaller stray fields than\nferromagnets and consequatively lead to smaller\nenergy losses in devices.\nAcknowledgementsAuthors acknowledge the\ncomputer support of the Leibniz Institute for Solid\nState and Materials Research Dresden, and the as-\nsistance of Ulrike Nitzsche in using the computer\nfacilities.\n7References\n[1] I.ˇZuti´ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76\n(2004) 323.\n[2] R.A. de Groot. F.M. Mueller, P.G. van Engen, K.H.J.\nBuschow, Phys. Rev. Lett. 50 (1983) 2024.\n[3] M.N. Kirillova, A.A. Makhnev, E.I. Shreder, V.P.\nDyakina, N.B. Gorina, Phys. St. Sol. (b) 187 (1995) 231.\n[4] K.E.H.M. Hanssen, P.E. Mijnarends, L.P.L.M. Rabou,\nK.H.J. Buschow, Phys. Rev. B 42 (1990) 1522.\n[5] H. Akinaga, T. Manago, M. Shirai, Jpn. J. Appl. Phys.\n39 (2000) L1118.\n[6] Ph. Mavropoulos and I. Galanakis, J. Phys.: Condens.\nMatter 19 (2007) 315221.\n[7] I. Galanakis and Ph. Mavropoulos, Phys. Rev. B 67\n(2003) 104417.\n[8] M.C. Qian, C.Y. Fong, L.H. Yang, Phys. Rev. B 70\n(2004) 052404.\n[9] J. Li, Y. Li, X. Dai, H. Liu, X Yu, Physica B 403 (2008)\n2473.\n[10] J.E. Pask, L.H. Yang, C.Y. Fong, W.E. Peakett, S. Dag,\nPhys. Rev. B 67 (2003) 224420.\n[11] H. Wu, M. Hortamani, P. Kratzer, M. Scheffler, Phys.\nRev. Lett. 92 (2004) 237202.\n[12] M. Hortamani, H. Wu, P. Kratzer, M. Scheffler, Phys.\nRev. B 74 (2006) 205305.\n[13] M. Hortamani, L. Sandratskii, P. Kratzer, I. Mertig, M.\nScheffler, Phys. Rev. B 78 (2008) 104402.\n[14] K. Koepernik, B. Velicky, R. Hayn, H. Eschrig, Phys.\nRev. B 58 (1998) 6944.\n[15] J.P.Perdew andY.Wang, Phys.Rev.B45(1992) 13244.\n[16] K. Koepernik and H. Eschrig, Phys. Rev. B 59 (1999)\n1743.\n[17] H. van Leuken and R.A. de Groot, Phys. Rev. Lett. 74\n(1995) 1171.\n[18] V.L. Moruzzi and C.B. Sommers, in Calculated\nElectronic Properties of Ordered Alloys: A Handbook;\nThe Elements and Their 3d/3d and 4d/4d Alloys (World\nScientific Publishing Company, Singapore), 1995, pp\n421.\n[19] P.J. Webster and K.R.A. Ziebeck, in Alloys and\nCompounds of d-Elements with Main Group Elements.\nPart 2., edited by H.R.J. Wijn, Landolt-B¨ ornstein, New\nSeries, Group III, Vol. 19,Pt.c (Springer-Verlag, Berlin) ,\n1988, pp 75-184.\n[20] I. Galanakis, K. ¨Ozdo˜ gan, E. S ¸a¸ sıo˜ glu, B. Akta¸ s, Phys.\nRev. B 75 (2007) 172405.\n8"    },    {        "title": "2301.08300v1.Ab_initio_comparison_of_spin_transport_properties_in_MgO_spaced_ferrimagnetic_tunnel_junctions_based_on_Mn__3_Ga_and_Mn__3_Al.pdf",        "content": "Ab initio comparison of spin-transport properties in MgO-spaced\nferrimagnetic tunnel junctions based on Mn 3Ga and Mn 3Al\nM. Stamenova,1,\u0003P. Stamenov,1and N. Baadji2\n1School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland.\n2Laboratoire de Physique des Mat\u0013 eriaux et ses applications & D\u0013 epartement de Physique,\nFacult\u0013 e des Sciences, Universit\u0013 e Mohamed Boudiaf, M'sila, 28000, Alg\u0013 erie\nWe report on \frst-principles spin-polarised quantum transport calculations (from NEGF+DFT)\nin MgO-spaced magnetic tunnel junctions (MTJs) based on two di\u000berent Mn-based Heusler fer-\nrimagnetic metals, namely Mn 3Al and Mn 3Ga in their tetragonal DO 22phase. The former is a\nfully compensated half-metallic ferrimagnet, while the latter is a low-moment high-spin-polarisation\nferrimagnet, both with a small lattice mismatch from MgO. In identical symmetric and asymmetric\ninterface reconstructions across a 3-monolayer thick MgO barrier for both ferrimagets, the linear\nresponse (low-voltage) spin-transfer torque (STT) and tunneling magneto-resistance (TMR) e\u000bects\nare evaluated. A larger staggered in-plane STT is found in the Mn 3Ga case, while the STT in\nMn3Al vanishes quickly away from the interface (similarly to STT in ferromagnetic MTJs). The\nroles are reversed for the TMR, which is practically 100% in the half-metallic Mn 3Al-based MTJs\n(using the conservative de\fnition) as opposed to 60% in the Mn 3Ga case. The weak dependence on\nthe exact interface reconstruction would suggest Mn 3Ga-Mn 3Al solid solutions as a possible route\ntowards optimal trade-o\u000b of STT and TMR in the low-bias, low-temperature transport regime.\nI. INTRODUCTION\nAmong the Mn-based Heusler ferrimagnets are\na number of topical binary materials, combining\nlow moments with high Curie temperature and\nspin-polarisation [1{4], as well as high anisotropy\nwith low Gilbert damping [5, 6] { hence holding\npromise for the emerging \feld of Antiferromagnetic\nSpintronics[7, 8]. Ferrimagnetic electrodes, and es-\npecially, the compensated ones, in magnetic-tunnel\njunction (MTJ) devices lead to the reduction of\nthe demagnetizing \feld and, more importantly, the\nreduction of the critical current needed to switch\nthe magnetization together with the fast dynam-\nics of such switching, involving inter-sublattice ex-\nchange interactions. The possible exploitation of\nthis class of materials, however, depends on the\nability to produce high-frequency oscillations (in\nthe 100s of GHz range) and the depth of resis-\ntance modulation that junctions using these could\nsupport. Predictive computational guidance for\nthe magnitude and resilience of both the spin-\ntransfer torque (STT) and the tunneling magneto-\nresistance (TMR) e\u000bects, towards the di\u000ecult-to-\ncontrol experimentally barrier quality and inter-\nface reconstruction, can help to speed-up the de-\nvelopment of functional prototype devices.\nOne such candidate is Mn 3Ga, which in its\ntetragonal DO 22phase is a low-moment ferrimag-\nnet with a high spin-polarisation and anisotropy [1,\n2, 9], but also a low Gilbert damping and an estab-\nlished epitaxial relationship with MgO(001) [10].\nA large staggered long-ranged STT e\u000bect has been\nfound theoretically in MTJs based on DO 22-phased\nMn3Ga [11], present both in Fe/MgO/Mn 3Ga and\nin Mn 3Ga/MgO/Mn 3Ga tri-layers, and related to\n\u0003Contact email address: stamenom@tcd.iethe mismatch of the Fermi wavevectors of the ma-\njority and minority \u0001 1symmetry band in Mn 3Ga\nin the direction of transport. Theoretically, the\nTMR e\u000bect in these stacks reaches a few tens or\npercent and exhibits a sign change below 1V, which\nis in accordance with experimental observations for\nsimilar ferrimagnetic MTJs [12].\nAnother Mn-based Heusler Mn 3Al has been\nshown to exhibit half-metallicity and almost ide-\nally fully compensated moment in its cubic DO 3\nphase [13], similarly to Mn 3Ga in this phase[3],\nand proposed MTJs with GaAs have shown large\ntheoretical TMR ratios [14]. A GGA-PBE geome-\ntry optimisation of Mn 3Al reveals a stable tetrag-\nonal DO 22solution with almost fully compensated\nmoment and an in-plane lattice constant commen-\nsurate with MgO. As geometrically the Mn 3Ga and\nMn3Al stacks with MgO are very similar, but o\u000ber\ndi\u000berent placement of the Fermi level with respect\nto the main band dispersions, a comparison of the\nspin-transport in analogous MTJs could unveil fur-\nther insights about the Spintronic capacity of the\ntwo Heuslers. Here we \frst examine the electronic\nstructure properties of both materials in bulk (Sec-\ntion II) and then we compare the spin-dependent\ntransport properties of two pairs of MTJs, all with\n3-monolayers (ML) thick MgO spacers, but featur-\ning two di\u000berent terminations at one of the inter-\nfaces (Section III).\nII. BULK PROPERTIES\nIntermetallic Heusler alloys X 2YZ crystallize\nusually in the cubic L2 1structure, especially at\nhigh temperature, and at low temperature they\ncan develop a DO 22tetragonal structure with a\nratioc=aaroundp\n2. The larger departure of this\nratio fromp\n2 is a prelude for a higher magneto-arXiv:2301.08300v1  [cond-mat.mes-hall]  19 Jan 20232\nFIG. 1. Band structures and spin-polarised density of states of (a,c) Mn 3Al and (b,d) Mn 3Ga, respectively\nin their depicted unit cell, calculated using LDA-PW92 (for Mn 3Al) and LDA-CA (for Mn 3Ga) on the MgO-\nmatching (strained) geometries (black or red/blue \flled curves, see text and insets). In all panels a comparison\nis shown with a corresponding GGA-PBE calculation for the relaxed unit cell (green curves, see text/legends for\ndetails). In (a,b) the DO 22unit cells of both materials with the local spins of the Mn atoms shown as arrows.\ncrystalline anisotropy [1]. Our calculated optimum\nlattice parameters, using the GGA-PBE, of both\ncompounds in their antiferromagnetic con\fgura-\ntion area= 4:057,c= 5:911\u0017A anda= 3:78,\nc= 7:1\u0017A for Mn 3Al and Mn 3Ga, respectively.\nThe in-plane lattice constants are close to that\nof bulk MgO ( aMgO = 4:21\u0017A), which makes\ntheir integration in conventional magnetic tunnel\njunctions feasible. The DO 22Mn3Al(Ga) struc-\ntures are constructed by alternating planes of Mn-\nAl(Ga) (Mn in 2 bWycko\u000b position: Mn I) and Mn-\nMn (Mn in 4 dWycko\u000b position: Mn II) coupled\nanti-ferromagnetically, along the z-axis (see unit\ncell schematics in Fig. 1). In such a lattice struc-\nture, Mn 3Ga is metallic and both spins contribute\nto the conductivity, while Mn 3Al is a half-metal\nand only spin-up states contribute to the conduc-\ntion. Consequently, Mn 3Al has a 100% spin po-\nlarization at the Fermi level and therefore one can\nexpect a high TMR for junctions based on Mn 3Al.\nMn3Ga also shows a high spin polarization of 88%\nin other GGA-based calculations [2]. Similar re-\nsults for the Fermi level spin-polarisation are ob-\ntained also when the transverse lattice constants\narea=b= 4:1\u0017A in both materials (approaching\nthat of MgO thin \flms), as shown in the density\nof states presented in Fig. 1(c,d).\nThe corresponding band structure of both com-\npounds is presented in Fig. 1(a,b) and one can see\nthat for Mn 3Al the spin-down channel exhibits a\ngap of the order of 0.4 eV when computed with\nGGA-PBE on the relaxed unit cell (green points)\nand similarly about 0.25 eV, when calculated with\nLDA-PW92 (the black points), as implementedin the Siesta code [15]. This gap can be tuned\nby changing the lattice parameters. In the LDA\ncase, we apply a longitudinal tensile strain of 4%\n(c= 6:027\u0017A) to open the gap and approximate\nthe GGA result (where c= 5:795\u0017A). The impact\nof that on the layer-resolved magnetic moments\n(calculated by Mulliken population analysis) is a\nsmall decrease by about 7 % for both Mn sublat-\ntices compared to GGA. We have additionally es-\ntablished that Mn 3Al keeps its fully-compensated\nferrimagnetic character for applied strains ranging\nbetween -4% to 8% (so that the 100% polarization\nis kept for such a strain). Note that the orbitals\nbelow the Fermi level are mainly a hybridisation\nbetween d-t2gorbitals of the Mn atoms, while the\nempty bands above the Fermi level are egbands of\nthe Mn occupying site 4 d.\nFor Mn 3Ga we are using a longitudinal lat-\ntice constant c= 6:6\u0017A, which has been found\nto reproduce more reasonably the experimen-\ntal values of the Mn spins (as extracted from\nneutron-di\u000braction results [1]), within the LDA\n[see Fig. 2(b,c) for the computed layer-resolved\nmagnetic moments], compared to the GGA-relaxed\nor the experimental lattice parameter values c=\n7:1\u0017A (larger by about 7 %), for which the LDA-\ncomputed magnetic moments are larger by over\n15 % with respect to the c= 6:6\u0017A case and\noutside the experimental range. We assume that,\nwith these structural amendments (consistent also\nwith Ref. 11 for Mn 3Ga-based MTJs), the LDA,\nwhich is not currently replaceable by GGA in our\nnon-collinear-spin method for spin-transfer torque\n(STT)[11], captures the essential Fermi-surface3\nFIG. 2. (a) Schematic of the four MTJs considered, including the directions of the spin quantisation axes in the\ntwo leads at the 90 °alignment for the STT calculations. Note, that the corresponding 'asymmetric' MTJs have\none layer of Mn removed from the right interface. (b,c) Self-consistently calculated layer-resolved spin-components\n(xandz) for the four MTJs at equilibrium (see legend for the color code). (d,e) Corresponding layer-resolved in-\nplane STT components calculated at the Fermi level in linear response regime (so-called, torkance \u001c=dT(V)=dV\nat the limit V= 0, whereT(V) is the STT, see Ref. 16 for precise de\fnition) in the four di\u000berent MTJs (same\ncolour code). A=a2= 16:81\u0017A2is the cross-sectional area of the junction (it is the same for all; note the periodic\nboundary conditions in the x\u0000yplane).\nproperties of the two materials relevant for the\nlinear-response regime investigated here and we\ncontinue exclusively with the LDA and the de-\nscribed above lattice parameters of both materials\nin Section III.\nConsequently, for the just described geometries,\nboth compounds are ferrimagnetic with Mn 3Al be-\ning fully compensated (total magnetic moment '0\n), while Mn 3Ga having a 2.6 \u0016Bper cell (1.3 \u0016B\nper formula unit). The advantage of having a very\nlow-spin-moment lead is to reduce the demagnetiz-\ning \feld, but more importantly, to reduce the crit-\nical current for a spin-transfer torque switching,\nwhich we will discuss it in the next Section III. We\nshould mention here that the magneto-crystalline\nanisotropy calculated for Mn 3Ga is much bigger\nthan that of Mn 3Al and this is because of the much\nstronger spin-orbit interaction at the Ga site com-\npared to Al. The local spins (extracted by Mul-\nliken population analysis) on the two Mn sublat-\ntices in Mn 3Al are nearly fully compensated with\n3.31\u0016Bon the Mn Isite and -1.58 \u0016Bon the Mn II\nsite, respectively. In Mn 3Ga the corresponding val-\nues are 3.53 and -2.46 \u0016Bfor Mn Iand Mn II, re-\nspectively, which are consistent with the measured\nmoments[1].III. SPIN-TRANSPORT IN Mn 3Al AND\nMn3Ga JUNCTIONS WITH MgO\nBARRIERS\nFour di\u000berent junctions based on Mn 3Al and\nMn3Ga all sandwiching 3 MLs of MgO have been\ninvestigated [Fig. 2a]. In the junctions, which we\nrefer to as 'symmetric', the two interfaces are the\nsame, that is, in both cases the interface is be-\ntween the MgO and the Mn II-plane of the DO 22\nlattice and overall the junction is mirror-symmetric\nwith respect to the central plane in the MgO. In\nthe 'asymmetric' junctions we have removed one\nmonolayer of Mn from the right interface, but all\ndistances, including the interface spacing, are pre-\nserved. Note that these geometries have not been\nrelaxed { the interface distance we have chosen in\nthe Mn 3Al case is 2 \u0017A, while in the Mn 3Ga junc-\ntions we have chosen 2.2 \u0017A, as motivated in Ref.\n11.\nIn Fig. 2 we compare the linear-response STT\n[11, 16] for a 90 °misalignment of the spin po-\nlarisations in the two leads, computed within the\nnon-equilibrium Green's function (NEGF) open-\nboundaries method implemented in the Smeagol\ncode [17]. Panels (b) and (c) show the layer-\nresolved local moments in the scattering region {\nin (b) the mirror symmetry is readily observed be-4\ntween thexandzcomponents of the spins on both\nsides of the junction. In comparison, in (c) the\nasymmetry after the removal of one Mn plane at\nthe right interface is evident. It is worth noting\nthat the spins in the left lead across the MgO ap-\npear practically una\u000bected by this local structural\ndisturbance on the right interface, in both junc-\ntions.\nThe calculated layer-resolved in-plane STT for\nthe symmetric junctions [Fig.2(d)] displays a per-\nfect left-to-right symmetry as well { this time an in-\nversion symmetry with respect to the central layer\nof MgO. For both ferrimagnets the STT at the op-\nposite interfaces has an opposite sign. There is,\nhowever, a qualitative di\u000berence between the two\nferrimagnets { if the STT in the Mn 3Al case is lo-\ncalised at the interface, in Mn 3Ga it shows the fa-\nmiliar long-range oscillatory decay with periodicity\ndetermined by the di\u000berence of majority and mi-\nnority spin wave-vectors of the \u0001 1-symmetry band\nin Mn 3Ga, as described in Ref. 11. This ideal\ninversion symmetry of the STT across the bar-\nrier, however, will hinder the switching of the such\nideally symmetric junctions. For instance, a posi-\ntivex-component of STT will rotate anti-clockwise\nthe Mn IIspins at the left interface (aligned along\n\u0000z), and also anti-clockwise the Mn IIspins aligned\nalong\u0000xat the right-hand-side interface. Hence\nsuch an alignment of the STT cannot drive switch-\ning of one lead with respect to the other. It is\nworth noting, however, that the STT changes sign\nin the next bi-layer { this will act against the sub-\nlattice exchange coupling. This is would enable\nthe excitation of high frequency anti-phase modes\nand potentially lead to fasted switching dynamics.\nIt is likely that the ideal symmetry in our calcu-\nlations will be broken in real structures and there\nwill be an STT imbalance leading to the switch-\ning of one of the layers. In the Mn 3Al junction the\nSTT has analogous symmetry, which does not lead\nto switching. It is, however, much more localised\nat the interfacial layer and signi\fcantly lower in\namplitude.\nTo illustrate the e\u000bect of possible structural im-\nperfection we consider the 'asymmetric' versions of\nboth junctions { with removed Mn-Mn monolayer\nfrom the right interface, while keeping the inter-\nface spacing unchanged [Fig. 2(a)]. In this case\nthe STT pro\fles change substantially [Fig.2(d,e)].\nNow we see both in the Mn 3Al and Mn 3Ga junc-\ntions the STT becomes asymmetric on both sides\nof the barrier. This time it shows a tendency to ro-\ntate spins in opposite directions, especially in the\nMn3Al case, thus driving a switching of one of the\nlayers with respect to the other. In the asymmetric\nMn3Al junction the STT is signi\fcantly increased\nin magnitude from the symmetric case, especially\non the right-hand side, where we see a large STT\nalso beyond the interfacial layer.\nWe then examine the energy dependence of the\ntotal transmissions in the four junctions in their\nFIG. 3. Energy-resolved properties at 0 V equilibrium\nfor both the P and AP states of the junctions (see text).\n(a,b) transmission coe\u000ecients and (c,d) TMR ratio,\nde\fned as TMR = ( TP\u0000TAP)=(TP+TAP) for the two\nsymmetric and the two asymmetric MTJs, respectively.\ncollinear spin states { parallel (P) state in which\nthe Mn spin in each sublattice (Mn Ior Mn IItype)\non the two sides of the barrier are parallel, and\nAP state, when they are antiparallel. We clearly\nsee the half-metallicity of Mn 3Al manifesting it-\nself in the vanishing transmission in the AP state\naround the Fermi level. This in turn drives a large\nTMR (practically 100% from the conservative de\f-\nnition with the sum of the transmission coe\u000ecients\nin the denominator) for a wide range of energies\naroundEFin both the symmetric and the asym-\nmetric junctions [green curve in Fig. 3 (c,d)]. In\nthe case of Mn 3Ga we \fnd a TMR of about 60 %\nfor the symmetric junction, which drops to 30 %\nfor the asymmetric case. We \fnd a signi\fcant en-\nhancement of the TMR for energies higher than\n0.2 eV above EF, which is due to the \u0001 1symmetry\nband edge and the band gap opening above that\nenergy for majority spin in the \u0000 \u0000Z(transport)\ndirection [11] (note that the position of this band\nedge above EFis consistent between the LDA and\nGGA calculations in Fig.1). The role of the in-\nterfacial asymmetry appears to be in reducing the\ntransmission in the P state, where in the asymmet-\nric junctions the interfacial spins are pointing in\nopposite directions, giving rise to additional scat-\ntering at the interfaces.\nThis can also be seen in the 2D portraits of the\nFermi-level transmission coe\u000ecients in the trans-\nverse 2D Brillouin zone (2D-BZ) (Fig. 4). In the\nMn3Ga case the transmission in all spin-states and5\nFIG. 4. Contour portraits of the total transmission coe\u000ecient (both spin species) at the Fermi level decomposed\nover the transverse 2D-BZ [as a function of ( kx;ky) on each panel, where \u0000\u0019=a\u0014kx;y\u0014\u0019=a]. From left to right,\nthe four di\u000berent MTJs are depicted (as indicated above), while the two rows correspond to the parallel (P) and\nanti-parallel (AP) spin states of the junctions, respectively (see text). Color-code bars for each panel are for the\nT(kx;ky;E=EF), i.e. dimensionless transmission probabilities (see, e.g. Ref.16).\njunction geometries is predominantly around the\n\u0000 point. The P states show stronger transmis-\nsion in both junctions, while in the asymmetric\ncase both spin-polarised transmissions are some-\nwhat suppressed with respect to the correspond-\ning ones in the symmetric MTJ. In contrast to\nMn3Ga, the transmission around the \u0000 point is\nsuppressed in the two Mn 3Al junctions in the P\nstate, as in this material there is a band gap along\n\u0000\u0000Zdirection for majority (up) spins [see Fig.\n1(a)]. Interestingly, despite the small quantita-\ntive changes, which the asymmetric interface in-\ntroduces in some regions of the transverse 2D-BZ,\nfor the small range of wavevector angles relevant\nto realistic specular transport around the \u0000 point,\nthe transmission appears relatively una\u000bected by\nthe interfacial detail. This e\u000bect is expected to be\nenhanced further for thicker MgO barriers, where\nthe transmission is more strongly \fltered in the\nvicinity of \u0000 point and we expect the overall trans-\nmission in Mn 3Al-based MTJs to be even less sen-\nsitive to features of the interfaces. We also note\nthat in the AP state of the Mn 3Al junctions the\ntransmission is extremely small and at the verge\nof computational accuracy, as expected from its\nhalfmetallicity (see Fig. 1).\nIV. CONCLUSIONS\nWe have compared linear-response spin-\npolarised transport properties in ferrimagnetic\ntunnel junctions made from two similar Mn-\nbased binary Heuslers in their tetragonal phase\nsandwiching a MgO barrier. Junctions featuring\nMn3Al o\u000ber a di\u000berent compromise between the\nmagnitude of STT and TMR, when compared\nto otherwise equivalent Mn 3Ga-based ones. The\noverall transmission, and therefore the expecteddi\u000berential conductivity, is higher in the ferrimag-\nnetic Mn 3Ga case, together with the low-bias STT.\nThe absence of one type of spin states for Mn 3Al-\nbased junctions, leads to smaller overall absorbed\nmomentum (with STT acting only within a couple\nof lattice spacings away from the interfaces) and\na signi\fcantly suppressed low-bias di\u000berential\nconductance. It can be argued, therefore that at\nleast at low temperature and applied bias, when\ncompared on the basis of STT per unit dissipated\npower, the Mn 3Ga-based junctions would o\u000ber\nan edge over the otherwise better in their TMR\nperformance Mn 3Al counterparts. The relatively\nlow sensitivity of the calculated parallel-state\ntransmission (normal to the interface) and also\nof the STT, although to a lesser extent, on the\nexact surface reconstruction in both Mn 3Al and\nMn3Ga-based junctions, may further motivate the\nexperimental veri\fcation of spectroscopic TMR\nand STT features (at low temperatures), hence,\n\fnite-bias calculations of the same would be well-\nwarrantied. In view of the above, aluminium-rich\nsolid solutions of Mn 3Al-Mn 3Ga, with or without\nSn doping, could o\u000ber enhanced low-bias TMR\nperformance, while preserving tetragonality, at\nmanageable growth-induced strain, in practical\ndevice stacks.\nACKNOWLEDGMENTS\nWe gratefully acknowledge funding from the\nScience Foundation Ireland (SFI Grant No.\n18/SIRG/5515 and 18/NSFC/MANIAC). We\nthank the Irish Centre for High-End Comput-\ning (ICHEC) and the Trinity Research IT Centre\n(TCHPC) for the provision of computational facil-\nities and support.\nCopyright (2022) Authors. This article is dis-6\ntributed under a Creative Commons Attribution (CC BY) License.\n[1] K. Rode, N. Baadji, D. Betto, Y. C. Lau,\nH. Kurt, M. Venkatesan, P. Stamenov, S. Sanvito,\nJ. M. D. Coey, E. Fonda, E. Otero, F. Choueikani,\nP. Ohresser, F. Porcher, and G. Andr\u0013 e, Physical\nReview B 87, 184429 (2013).\n[2] J. Winterlik, B. Balke, G. H. Fecher, C. Felser,\nM. C. Alves, F. Bernardi, and J. Morais, Physical\nReview B 77, 1 (2008).\n[3] G. Y. Gao and K. L. Yao, Appl. Phys. Lett. 103,\n232409 (2013).\n[4] L. Fan, F. Chen, C.-m. Li, X. Hou, X. Zhu, J.-l.\nLuo, and Z.-Q. Chen, J. Magn. Magn. Mater. 497,\n166060 (2019).\n[5] S. Mizukami, S. Iihama, Y. Sasaki, A. Sugihara,\nR. Ranjbar, and K. Z. Suzuki, Journal of Applied\nPhysics 120, 10.1063/1.4961704 (2016).\n[6] S. Mizukami, A. 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Joseph Poon  and Chung Ting Ma   \nDepartment of Physics, University of Virginia, Charlottesville, Virginia 22904 USA  \n \nAbstract  \nRecently, magnetic skyrmion ha s emerged as  an active topic of fundamental study  and applications  in \nmagnetic materials research . Magnetic skyrmions  are vortex -like spin excitations with topological \nprotection and therefore are more robust to pinning compared with magnetic domain walls. We employ \natomistic simulations to  create  room -temperature ultra -small Néel skyrmions  in amorphous ferrimagnet. \nThe fast propagation and low -dissipation dynamics of ultra-small ferrimagnetic skyrmions make them \nattractive for utilization as an alternative to domain walls in spin-based memory and logic devices.  \n \nIntroduction  \nIn the 1960s, Professor Theodore (Ted) Geballe and his colleagues pioneered the study of the magnetic \nmoment  of 3D transition metal solutes in noble metals  and discovered the formation of giant moment in \nthese material systems  [1]. The seminal study by Professor Geballe  laid the foundation for the fundamental \nresearch on magnetic materials  and, equally important, also impacted the study of strong electronic \ncorrelation  and orbital effects.  To celebrat e his centennial birthday, it is fitting to discuss  a new type of \nmagnetic  phenomenon called magnetic skyrmion  (Figure 1 ), a topological excitation in a magnetic system  \nthat could potentially lead to better spin-based memory and logic devices . Originally, skyrmion  was named \nfor a certain type of topological soliton invented by T. Skyrme, also in the 1960s, in the study of particle \ncreation in quantum fields.  \nBeginning with some background, the d iscovery  of giant \nmagnetoresistance (GMR) led to the dawn of spintronic s. \nSpintronics utilize the magnetic moment instead of electric \ncharge to store data in magnetic memory , which potentially can \nhave a significant impact on the future of electronics  [2].  Non-\nvolatile magnetic memory is one of the essential approaches in \novercoming the von Neumann bottleneck  in computer \narchit ecture. Current magnetic memory relies on the bubble \nand domain wall (DW) to encode information, using DW s in the \nrace-track architecture to encode bits and to drive  them with spin \ncurrent. However, there are limitations that impede the advance of \nDW based magnetic memory. Skyrmions  can be an alternative to \nDW race -track memory  [3]. Being topologically protected and \nmuch smaller  than DWs  [4-5]., e.g. ~10 nm or smaller in diameter, \nskyrmions  can be easier to unpin from lattice defects . Along with \nthe low driving threshold, magnetic skyrmions  can form the basis for scalable and high -speed -low-power \nspin-based logic devices . However, such devices  can only be possible if  the small skyrmions  of 10 nm or Fig. 1  Top view of the spin texture  \nin a Néel skyrmion  from an \natomi stic simulation. Different from \nthe domain wall  spins , the spins in a \nskyrmion  rotate by 360 degrees from \none side to the other.  \nbelow in diameter exist  sufficiently long at room temperature. In this article, we will employ atomistic \nsimulations to study a class of materials , namely amorphous ferr imagnetic alloy films t hat are promising in \nhosting ultra -small skyrmions  at room temperature . Our s imulation s tudy also showed stable ultra -small \nskyrmions  at room temperature in ferromagnetic heterostructures, however, the conditions required are \nmuch more stringent.          \nMethod  \nSkyrmions are stabilized via the Dzyaloshinskii Moriya \ninteraction (DMI)  [6-7], which originated from  the interplay of  \nspin-orbit effect and inherent chiral asymmetries or interfacial \nsymmetry breaking. Figure 2 illustrates the chiral nature of the \ninterfacial DMI. Intrinsic DMI arises in non -centrosymmetric \ncrystals such as B20 alloy  where Bloch skyrmions have been \nfound  at low temperature  [8-9] Interfacial DMI originates from \ninversion symmetry breaking by a  heavy metal interfacial layer \nsuch as Pt and Ir with strong spin -orbit coupling  in multilayer \nstacks  that contain ferromagnetic Fe and Co. The latter was \nfound to host >40 nm  Néel skyrmion s at room temperature  [10-\n12].  \nWe have employed the classical atomistic Hamiltonian H to \nperform the simulation of magnetic textures , as shown below:   \n \n𝐻=−1\n2∑ 𝐽𝑖𝑗𝑠𝑖∙𝑠𝑗\n<𝑖,𝑗>−1\n2∑ 𝐷𝑖𝑗∙(𝑠𝑖×𝑠𝑗)\n<𝑖,𝑗>−𝐾𝑖(𝑠𝑖∙𝐾𝑖̂)2 \n−𝜇0𝜇𝑖𝐻𝑒𝑥𝑡∙𝑠𝑖−𝜇0𝜇𝑖𝐻𝑑𝑒𝑚𝑎𝑔 ∙𝑠𝑖                            (1) \nwhere 𝑠𝑖,𝑠𝑗 are the normalized spins and 𝜇𝑖,𝜇𝑗 are the atomic moments at sites i, and j respectively. The \natomic moment is absorbed into the exchange constant, 𝐽𝑖𝑗=𝜇𝑖𝜇𝑗𝑗𝑖𝑗, and the DMI interaction 𝐷𝑖𝑗=\n𝜇𝑖𝜇𝑗𝑑𝑖𝑗, which is proportional to r i x rj, the position vector between the atoms i, and j and the interface, and \nthe effective anisotropy 𝐾𝑖=𝜇𝑖𝑘𝑖. 𝐻𝑒𝑥𝑡 and 𝐻𝑑𝑒𝑚𝑎𝑔  are the external field and demagnetization  field \nrespectively. Only the nearest neighbor interactions are considered.  \nThe effective field H eff is calculated using  the atomistic Hamiltonian in Eq. (1), and  the ground state  of the \nmagnetic system  is obtained by evolv ing the spins  under the following stochastic Landau -Lifshitz -Gilbert \n(LLG) equation,  \n𝑑𝑀\n𝑑𝑡=−𝛾\n1+𝛼2𝑀×(𝐻𝑒𝑓𝑓+𝜉)−𝛾𝛼\n(1+𝛼2)𝑀𝑠𝑀×[𝑀×(𝐻𝑒𝑓𝑓+𝜉)]    (2) \nwhere 𝛾 is the gyromagnetic ratio, 𝛼 is the Gilbert damping constant,  𝐻𝑒𝑓𝑓 is the effective field, 𝜉 is the \nGaussian white noise term for thermal fluctuations and 𝑀𝑠 is the saturation magnetization.  \nAmorphous Ferrimagnet  \nAmorphous rare earth  transitional  metal (RE -TM) ferrimagnets  (FiM) is found to provide a favorable \nenvironment to host small skyrmions at room temperature. The magnetic structure of RE -TM FiM consists \nof two sublattices, one occupied by the RE atoms and the other occupied by the TM atoms.  The atoms \ncouple ferromagnetically w ithin eac h sublattice  and antiferromagnetically between the sublattice s. The \nFig. 2 Schemat ic illustration of \ninterfacial  Dzyaloshinskii Moriya \ninteraction . The DMI exchange \ncoupling is given by Dij·(SiSj), \nwhich favors spin canting that \nfacilitates the formation of Néel \nskyrmion .  amorphous structure helps to reduce defect pinning, while their intrinsic perpendicular magnetic anisotropy \n(PMA)  allows the formation of Néel skyrmion s in thicker  films ( e.g. up to 10  nm). Furthermore, a \ndistinctive feature of the ferrimagnet is that the magnetization of RE -TM alloys vanishes at the \nmagnetization compensation temperature  due to the cancelation  of the magnetization of the two sublattice s. \nFigure 3 shows the simulated ma gnetization of an amorphous Gadolinium -Cobalt ferrimagnet. With near \nzero magnetization , the skyrmion  velocity can reach  a high speed near ~1,000 m/s [13]. , while near the \nangular -momentum compensat ion temperature, the skyrmion Hall effect is vastly reduced  [14]. These \nmaterial advantages make amorphous ferrimagnet a n ideal material for spin -based memory and logic \ndevices.   \n \n \n \n \nResults and Discussion  \nUsing atomistic LLG simulations, we will now explore  the equilibrium  state and size of skyrmion in \namorphous Gd 25Co75 film at room \ntemperature  by varying the DMI, \nmagnetic anisotropy , and thickness . \nTo capture the unique short -range \norder in amorphous materials, an \namorphous structure of RE -TM \nwas obtained from ab initio \nmolecular dynamics simulation by \nProfessor Howard Sheng using the \nmethod  described in ref. 15 [1 5]. \nThe sample size is 50.7 nm x 50.7 \nnm x 5 nm  comprising 768000  \natoms .  Since the interfacial DMI in \nthese heterostructures  originates \nfrom the heavy metal interface , we \nwill use an exponential -decay law to describe \nthe DMI inside the magnetic layer. Indeed, s uch \nrapid decay of DMI has been found in both \ncalculations  [16] and experiments  [17].  \nFor a 5-nm thick Gd 25Co75 layer, interfacial \nDMI ranges from 0 to 2 mJ/m2 and anisotropy \nranges from 0.05 x 104 J/m3 to 4 x 105 J/m3 are \ninvestigated. From experiment s, the anisotropy \nFig. 3 Simulated saturation magnetization vs. \ntemperature of amorphous Gd 25Co75. The \ncompensation temperature is near 250 K , and \nthe magnetization is small  at room \ntemperature.  \nFig. 4 Simulated magnetic anisotropy vs. \ninterfacial DMI phase diagram of 5 nm \namorphous Gd 25Co75 at 300 K. Inserted figure \ncorresponds to a 13 nm skyrmions simulated at K \n= 3 x 104 J/m3. The color maps correspond to Co \nsublattice (top) and Gd sublattice (bottom).  \nof GdCo was found to be ~ 3 x 104 J/m3 [18] Figure 4  shows the simulated magnetic anisotropy versus \ninterfacial DMI  (K-DMI) phase diagram for the  5-nm thick amorphous  Gd 25Co75 at 300  K. For a given \nanisotropy, as DMI increases from 0 to 2 mJ/m2, the transition from FiM phase to skyrmions, followed by \nstripe  phase occurs . For a given DMI, as the magnetic anisotropy increases, the size of skyrmions decreases , \nand the skyrmions  finally collapse into the FiM state with large enough anisotropy . Using experimental \nvalue of K~3 x 104 J/m3, we found skyrmions as small as 13 nm. Such small skyrmion is stabilized  with \ninterfacial DMI of  ~0.6 mJ/m2. The 2D color map of the out-of-plane reduced magnetization (mz) for the \n13 nm skyrmion is inserted in Figure 4 . In an experiment , Caretta et al.[13] found skyrmion  size in the \nrange of 10 nm to 30 nm in \nPt/GdCo/TaO x with an average \nDMI of 0.12 mJ/m2, which \ncorresponds to an interfacial DMI \nof about 0. 9 mJ/m2. With such \ninterfacial DMI, our simulation \nshows a skyrmion size of ~20 nm \nat room temperature , which  is in \ngood  agree ment with experiment.    \nWe further investigate the stability \nof skyrmions  at room temperature  \nby increas ing the thickness of the \nGdCo  layer . Figure 5 shows the \nthickness -DMI phase diagram of \nGd 25Co75 at 300 K.  For all \nthicknesses, increase in DMI results in an \nincrease in  skyrmion size. The latter can \nbe understood in terms of the \neffectiveness of DMI in spin canting. \nDue to the exponential decay of DMI, a s \nthickness increases , the strength  of interfacial DMI required to stabilize skyrmions also increases.   Even \nthough the interfacial DMI is less effective  in the thicker films , smaller skyrmions, as small as 8 nm, are \nfound.  In the 10-nm thick GdCo  layer , such sub 10 nm skyrmions  are stabilized in the DMI rang e of 1.0 to \n1.2 mJ/m2, which is in the range of measured  interfacial DMI in Co/Pt films [17] . A color map of reduced \nmagnetization of a sub 10 nm skyrmion is shown in Figure 5 . Such skyrmion appears to be robust and \ncontain s a well -defined core at  the center .  \n \nTo further demonstrate the robustness of sub 10 nm skyrmions in GdCo,  a numerical  tomography  plot is \nutilized to reveal the spin texture of the skyrmion in three dimension s. The result is shown  in Figure 6 . This \nultra-small skyrmion is found in the 10-nm GdCo film with a n interfacial DMI of 1.1 mJ/m2, The color map \nof m Z is deliberately set to be brighter in order to more clearly show the skyrmions structure.  For the Co \nsublattice (left of Figure 6 ), the majority  of the spins are pointing down. Near the center, the stripe of green \nand blue corresponds to the center of the skyrmion. One can conclude from the columnar distribution of \ngreen and blue color that this skyrmion is distributed uniformly from top t o bottom. Such columnar \ndistribution of skyrmion is also found in the Gd sublattice (right of Figure 6 ). Such columnar growth of \nskyrmion is favorable for designing skyrmion -based devices. Spin-based logic devices using these \ncolumnar skyrmions will be robust and reliable.  Fig. 5 Simulated DMI -thickness phase diagram of \namorphous Gd 25Co75 at 300 K with K = 0.3 x 105 J/m3. The \ninserted figure corresponds to a sub 10 nm skyrmion \nrevealed in 10 nm Gd 25Co75. The c olor maps correspond to \nCo sublattice (top) and Gd sublattice (bottom).  \n \nConclusions  \nUsing atomistic simulations, we have explored the phase diagram of Néel skyrmion s at room temperature \nin amorphous ferrimagnetic  Gd 25Co75 films with interfacial Dzyaloshinskii Moriya interaction  (DMI) . Sub-\n10 nm skyrmions are found to form in thick  (10 nm)  films in the range of DMI values similar to that obtained \nin experi ment . Furthermore, despite the exponential decay  of DMI away from the interface, 3D spin texture \nexhibit s a uniform  columnar  distribution across the film thickness . The present study has revealed the \nrobustness of skyrmions , thus adding to the promise of these topological magnetic entities in spin -based \nnanoelectronics. .   \n \nAcknowledgement:  \nThis work was supported by the DARPA Topological Excitations in Electronics (TEE) program \n(grant D18AP00009). The content of the information does not necessarily reflect the position or \nthe policy of the Government, and no official endorsement should be inferred. Approved for public \nrelease; distribution is unlimited.  \n \n \n \nFig. 6 Tomograph of a sub 10 nm skyrmion in 10 nm amorphous Gd 25Co75 at 300 K. Co sublattice \nis on the left, and Gd sublattice on the right. A robust, columnar  distribution of a sub 10 nm \nskyrmion is revealed in both sublattices .  References  \n1. 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Mater. 339, 51-55 (2013)  "    },    {        "title": "1705.01335v1.Synthetic_ferrimagnet_spin_transfer_torque_oscillator__model_and_non_linear_properties.pdf",        "content": "Synthetic ferrimagnet spin transfer torque oscillator: model and non-linear properties\nB. Lacoste,1M. Romera,2,∗U. Ebels,2and L. D. Buda-Prejbeanu2\n1International Iberian Nanotechnology Laboratory, Braga, Portugal\n2Univ. Grenoble Alpes, CEA, INAC-SPINTEC, CNRS, SPINTEC F-38000 Grenoble, France\n(Dated: Date of submission October 12, 2018)\nThe non-linear parameters of spin-torque oscillators based on a synthetic ferrimagnet free layer\n(two coupled layers) are computed. The analytical expressions are compared to macrospin simula-\ntions in the case of a synthetic ferrimagnet excited by a current spin-polarized by an external fixed\nlayer. It is shown that, of the two linear modes, acoustic and optical, only one is excited at a time,\nand therefore the self-sustained oscillations are similar to the dynamics of a single layer. However,\nthe non-linear parameters values can be controlled by the parameters of the synthetic ferrimagnet.\nWith a strong coupling between the two layers and asymmetric layers (different thicknesses), it is\ndemonstrated that the non-linear frequency shift can be reduced, which results in the reduction of\nthe linewidth of the power spectral density. For a particular applied field, the non-linear parameter\ncan even vanish; this corresponds to a transition between a red-shift and a blue-shift frequency\ndependence on the current and a linewidth reduction to the linearlinewidth value.\nKeywords: spin transfer torque, synthetic ferrimagnet, non-linear auto-oscillator\nSpin transfer torque oscillators (STOs) have promis-\ning applications as high frequency microwave generators.\nA typical STO nano-pillar is composed of two magnetic\nlayers separated by a metallic spacer or an isolating bar-\nrier. The magnetization of the first magnetic layer re-\nmains fixed in-plane or out-of-plane. It acts as a spin\npolarizer for the current flowing through the nano-pillar.\nThe magnetization of the second layer can be driven into\nself-sustained oscillations by an applied DC current due\nto spin transfer torque (STT)1–3. The oscillation of the\nfree layer magnetization gives rise to a variation of the\nresistance of the pillar, so that an alternative voltage ap-\npears at its boundaries. For a single domain free layer,\nthe generated microwave signal is typically in the GHz\nrange. However, the large linewidth, in the order of tens\nof MHz, is an obstacle for functional devices.\nIn order to improve the STO characteristics, an accu-\nrate and simple model describing the dynamics is funda-\nmental. For a single-layer (SL) STO, the general frame-\nwork of non-linear auto-oscillators (NLAO), proved to\nbe a particularly well adapted model4–7. Indeed, most\nof the features exhibited by experimental devices could\nbe explained within this framework, such as the field\nand current dependence of the frequency, the broadened\nlinewidth8, but also synchronization to an external signal\nor to other STOs9. More importantly, this model defines\na key parameter for understanding the STO behavior:\nthe non-linear amplitude-phase coupling parameter. By\nevaluating this non-linear parameter from the magnetic\nproperties of the layer, it was found that the linewidth of\nthe STO was reduced when applying a transverse field,\nfor instance5. However this model is confined to a SL free\nlayer and some recent works studied STO devices where\nthe free layer is composed of two coupled layers consti-\ntuting a synthetic ferromagnet (SyF)10,11. Allegedly, the\nadditional coupling energy would increase the magnetic\nstiffness and reduce the fluctuations. However, coupled\nsystems are also more complicated to understand and ageneral analytical model is necessary to explain and de-\nfinetheimportantparametersofitsdynamics. Typically,\nit would be useful to be able to calculate the non-linear\namplitude-phase coupling parameter of a SyF-STO.\nToanswerthisquestion, weproposetoextendthisframe-\nwork, the NLAO model, to describe the dynamics of two\ncoupled layers subjected to spin transfer torque. In order\nto treat the most general case, different coupling are in-\ncluded : the Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction, the dipolar coupling and the mutual STT.\nThe non-adiabatic STT (or field-like torque) is also in-\ncluded, although its effect was found to be negligible in\ntheparticularconfigurationsexaminedinthispaper. Itis\nfundamental in the dynamics of self-polarized STO12,13,\nthough.\nThe NLAO theory is based on a change of coordinates\nto complex variables to represent the magnetization dy-\nnamics of the layers. The phase and amplitude of the\ncomplexvariablesdescribethenon-lineardynamicsofthe\nauto-oscillator. The validity of this approach is limited\nto quasi-conservative trajectories, for which the energy is\nalmost constant, and to small oscillation amplitudes. Us-\ning common diagonalization techniques, the conservative\npart of the magnetization equation of motion is simpli-\nfied to two terms : a linear and a non-linear contribution.\nThe dissipative part, which is supposed to be small com-\npared to the conservative part, defines the equilibrium\nenergy of the auto-oscillator by balancing the negative\nGilbert damping and the positive STT.\nThe first and second parts of this work describe the steps\nto extract the auto-oscillator equation for two coupled\nSyF layers in the macrospin approximation. In the third\nand fourth part, we describe the dynamics of the SyF-\nSTO defined by two coupled equations, so the STO can\nbe described by two modes. However, only one of them\nis usually excited into steady-state at a time, so the SyF-\nSTO is equivalent to a single-layer (SL) STO. This is an\nimportant result of this paper. The parameters of thearXiv:1705.01335v1  [cond-mat.other]  3 May 20172\nsingle-mode SyF-STO are computed, especially the non-\nlinear parameter, which is responsible for the frequency\ntunability, the large linewidth and the synchronization\nbandwidth. Another important result of this paper is\nthe link between the vanishing of the non-linear parame-\nter and the transition between a redshift and a blueshift\nregime. Finally in the fifth part, we study how to de-\ncrease the linewidth of a SyF-STO by changing the cou-\npling strength and the thickness of the layers.\nI. DESCRIPTION OF THE SYSTEM\nA. Landau-Lifshitz-Gilbert-Slonczewski equation\nWe consider the system in Figure 1 of two magnetic\nlayers, labeled 1 and 2 constituting a synthetic ferromag-\nnet (SyF). The total free energy Eholds the demagnetiz-\ning energy, the uniaxial anisotropy energy and the Zee-\nman energy (including an exchange energy) of both lay-\ners, plus a conservative coupling term between the two\nlayers, consisting of an RKKY interaction coupling and\nthe dipolar coupling :\nE=µ0\n2V1M1Hd1(m1·uz)2−µ0\n2V1M1Hk1(m1·ux)2\n+µ0\n2V2M2Hd2(m2·uz)2−µ0\n2V2M2Hk2(m2·ux)2\n−˜Dxm1xm2x−˜Dym1ym2y−˜Dzm1zm2z\n−µ0V1M1(Hx+Hex 1)(m1·ux)\n−µ0V2M2(Hx+Hex 2)(m2·ux) (1)\nHereµ0is the permeability of free space. V1=t1S\nandV2=t2Sare the volumes of the layers, with\nthicknesses t1andt2and surface S.M1andM2are\nthe saturation magnetizations of the layers, Hd1,Hd2\ntheir demagnetizing fields (supposed positive), Hk1,Hk2\nthe uniaxial anisotropy fields. For each layer labeled\nbyi= (1,2), we define the demagnetizing coefficients\n(Ni\nxx,Ni\nyy,Ni\nzz), and the interface anisotropy constant\nKSi, so thatHdi= (Ni\nzz−Ni\nyy)Mi−2KSi/(µ0Miti)\nandHki= (Ni\nyy−Ni\nxx)Mi.Hex1,Hex2are the exchange\nfields acting on each layer (for instance from a coupling\nwith a fixed anti-ferromagnet), and Hxthe applied field\nalong the easy axis.\nThe coefficients ˜Dx,˜Dyand ˜Dzaccount for the (conser-\nvative) coupling between the two layers. They include\nRKKY interaction term and the dipolar coupling, such\nthat, fori= (x,y,z ),˜Di=SJRKKY +Di, where\ntheDiare the dipolar coupling energy coefficients in\nmacrospin. Note that a negative JRKKYcorresponds to\nan anti-ferromagnetic coupling between the layers. Two\nadjacent layers give rise to negative DxandDy, and\npositiveDz.\nThe layers are also subject to a spin transfer torque\n(STT) due to a current flowing perpendicular to the lay-\ners. A positive current corresponds to electrons flowing\nFigure 1. Schematics of the synthetic ferrimagnet (SyF) as a\nfree layer with a fixed in-plane magnetized reference layer.\nfrom layer 2 towards layer 1, and then to the reference\nlayer. Thus, layer 1 is subjected to the STT from the\nreference layer and to the STT from layer 2 (with a neg-\native factor because layer 1 receives reflected electrons\nfrom layer 2). Layer 2 is subjected to the STT from the\nreferencelayerandfromlayer1(becauseofelectronsthat\nwere spin polarized after passing through layer 1). These\nspin torques acting on the two layers are modeled by two\nspin torque potentials, for layer 1 and 2 respectively, P1\nandP214:\nP1=−~\n2|e|Iη1m1·ux+~\n2|e|Iη21m1·m2\nP2=−~\n2|e|Iη2m2·ux−~\n2|e|Iη12m1·m2\nIis the current flowing through the layers. η1(resp.η2)\nis the effective spin-polarization of the current in layer\n1 (2) due to the fixed in-plane polarizer positioned be-\nfore layer 1 according to the direction of the current. η12\n(resp.η21) is the effective spin-polarization of the current\nin layer 2 (1) due to layer 1 (2).\nMoreover, the two layers are subjected to perpendicu-\nlar (or field-like) spin transfer torque (pSTT), from the\nreference layer and from the other layer. The pSTT is\nmodeled by two potentials, similar to the spin torque po-\ntentials defined above :\n˜P1=−~\n2|e|Iβ1m1·ux+~\n2|e|Iβ21m1·m2\n˜P2=−~\n2|e|Iβ2m2·ux−~\n2|e|Iβ12m1·m2\nThe equation of motion is given by the Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equation. In this\nform, the damping is defined with respect to the time-\nderivative of the magnetization vector; after moving all\nthe time-derivatives on the left-hand-side, the LLGS3\nwrites :\nµ0V1M1dm1\ndt=γ0m1×∂E\n∂m1+γ0m1×∂˜P1\n∂m1\n+γ0m1×/parenleftbigg\nm1×∂\n∂m1(P1−α1E)/parenrightbigg\nµ0V2M2dm2\ndt=γ0m2×∂E\n∂m2+γ0m2×∂˜P2\n∂m2\n+γ0m2×/parenleftbigg\nm2×∂\n∂m2(P2−α2E)/parenrightbigg\nThe Gilbert damping coefficients of the two layers are\ngiven byα1andα2. The correction to the gyromagnetic\nratio due to the damping coefficient has been neglected,\nsoγ0=µ0γwhereγis the gyro-magnetic ratio.\nAccording to the form of the LLGS equation used in\nthis paper, the coefficients βj(j= (1,2,12,21)) of the\nfield-like torques can contain a term proportional to the\ncoefficients ηjfrom the damping-like STT and to the\nGilbert damping constants of the two layers, that we call\npseudo-field-like torque. Namely β1=α1η1,β2=α2η2,\nβ12=α2η12andβ21=α1η21. Such additional terms\nwould be coming from the transformation of the STT\nfrom the Gilbert-form of the LLGS equation to the\nLandau-form.\nBy writing the LLGS equation in this form, the\nfree energy part, which is common to both layers, is\nseparated from the rest. This will allow to use a similar\nformalism as for the description of a single layer in\nprevious publications4,5.\nIn order to simplify the notations, we introduce the\nlayer asymmetry β, the geometrical mean magnetic vol-\numeMand the following normalized hamiltonian and\npotentials :\nβ=/radicalbigg\nM2t2\nM1t1M=µ0S/radicalbig\nM1t1M2t2\nH=γ0E\n2M∆1=γ0˜P1\n2M∆2=γ0˜P2\n2M\nΓ1=α1H−γ0P1\n2MΓ2=α2H−γ0P2\n2M\nIn the following, dotted variables represent their time\nderivative. Therefore the LLGS equation rewrites :\n1\n2β˙m1=m1×∂H\n∂m1+m1×∂∆1\n∂m1\n+m1×/parenleftbigg\nm1×∂Γ1\n∂m1/parenrightbigg\n(2)\nβ\n2˙m2=m2×∂H\n∂m2+m2×∂∆2\n∂m2\n+m2×/parenleftbigg\nm2×∂Γ2\n∂m2/parenrightbiggWe decompose the right-hand-side of the LLGS equation\nin two parts that will be treated separately: (i) the con-\nservative hamiltonian terms (simply called conservative\nin the following) that are composed of the first terms\non the right-hand-side and depend only on H. (ii) the\nconservative non-hamiltonian terms and the dissipative\nterms (simply called dissipative in the following because\nthe dissipative terms play a more important role) that\nare composed of the other two terms (respectively) on\nthe right-hand-side.\nIn general, the damping constants α1,α2are considered\nto be small ( <0.1) and the applied current is reasonably\nsmall, so the conservative part is larger than the dissipa-\ntive part. The two different orders of magnitude further\nsupport the distinction made between the two parts.\nB. Numerical parameters\nThe results from the extended NLAO model will be\ncompared to macrospin LLGS simulations. The case of\nan asymmetric SyF shows interesting properties, espe-\ncially in terms of linewidth reduction. As all cases can-\nnot be reproduced here, we focus on a SyF with thickness\nasymmetry between the two layers. However, asymme-\ntry can also be introduced by submitting one layer to an\nexchange field or by reducing the effective demagnetiz-\ning field of one of the layers with perpendicular interface\nanisotropy.\nFor the inter-layer coupling, two regimes are considered,\nsmall coupling JRKKY =−2×10−4J/m2and large cou-\nplingJRKKY =−5×10−4J/m2. The dipolar coupling\nis neglected in the macrospin simulations. This is sup-\nported by the fact that, in the macrospin approximation\nandinnano-pillarswithcircularcross-section,thedipolar\ncoupling is an antiferromagnetic coupling in the in-plane\ndirections ( xandydirections in our convention) and a\nferromagnetic coupling in the normal direction ( zdirec-\ntion). Because of the high demagnetizing field in thin\nlayers, the trajectories have a small out-of-plane compo-\nnent, so contribution from the dipolar coupling is com-\nparable to a low RKKY antiferromagnetic coupling. For\nthe layer thicknesses considered, the dipolar field is lower\nthan the RKKY coupling field, so the dipolar coupling is\nsimply neglected.\nThe rest of the parameters are defined in Table I.\nAccording to the value of the area Sof the pillars, cur-\nrents expressed in mA correspond to current densities of\n1011A/m2.\nThe current is considered to be unpolarized after going\nthrough the first layer, so η2= 0. However the same\nqualitative results were obtained15if we suppose that\nη2=±η1.\nThe conservative part is the most important to de-\nscribe the self-sustained oscillations because the trajec-\ntories of the self-sustained oscillations are close to the\nconstant energy trajectories. For this reason, a change of\nvariables that describes accurately the conservative part4\nIdentical properties Value\nMs1,Ms2 1×106A/m\nHd1,Hd2 0.9×106A/m\nHk1,Hk2 10×103A/m\nHex1,Hex2 0\nα1,α2 0.02\nS 10−14m2\nη21,η12 0\nβ1,β2,β12,β21 0\nDifferent properties Values\nt1andt21.8 and 2.2 nm\nη1andη2 0.5 and 0\nTable I. Properties of the magnetic layers.\nand treats the dissipative part as a small perturbation is\nadapted to describe the dynamics of the STO. This will\nbe developed in the next part.\nII. TRANSFORMATION TO COMPLEX\nVARIABLES\nA. Complex variables: conservative part\nWe intend to rewrite the LLGS equation in complex\nform representing the evolution of two modes a1anda2.\nLeta= (a1,a2)be a 2-dimensional complex vector. The\ngoal is to write the conservative part of the LLGS equa-\ntion in the form :\n˙a=−i∂H\n∂a†(3)\nThe elements of the basis, a1anda2, represent uni-\nform modes around the equilibrium position, with com-\nplex conjugates a†= (a†\n1,a†\n2). In the following, we focus\nonly on the modes around the parallel equilibrium state\n(or antiparallel depending on the sign of the RKKY cou-\npling constant and the dipolar coupling), i.e. the syn-\nthetic ferrimagnet (SyF) is in the plateau region. The\nequilibrium position is represented by :\nmeq\n1x=mux meq\n2x=mnux\nHeren= sign( ˜Dx)reflects the ferromagnetic or anti-\nferromagnetic type of coupling between the two lay-\ners:n= +1ferromagnetic coupling, n=−1anti-ferromagnetic coupling. The direction of layer 1 rela-\ntively to the fixed reference layer is given by m:m= +1\nfor a parallel (P) orientation, m=−1for an antiparallel\n(AP) orientation. The initial state is then defined by a P\nor AP configuration (with respect to the reference layer)\nand a ferromagnetic or anti-ferromagnetic coupling be-\ntween layer 1 and layer 2.\nWe proceed to a change of coordinate system so that\nthe equilibrium magnetizations have the same definitions\nfor all the layers. They are defined by meq\ni=ui\nζfor\ni= (1,2):\nu1\nζ=mux u2\nζ=mnux\nu1\nξ=muy u2\nξ=mnuy\nu1\nη=uz u2\nη=uz\nThe expressions of a1anda2with respect to the local\nmagnetization coordinates have to be chosen adequately\nso that the conservative part of LLGS in this new system\nof coordinates take the hamiltonian form of Eq. (3). For\nthat we set :\na1=1√βm1ξ−im1η/radicalbig\n2(1 +m1ζ)(4)\na2=/radicalbig\nβm2ξ−im2η/radicalbig\n2(1 +m2ζ)(5)\nNotice that there are other choices of (H,a1,a2)that al-\nlows to rewrite the LLGS equation in the hamiltonian\nform of Eq. (3), notably by multiplying a1anda2by the\nsame constant term CandHbyC2. The quadratic part\n(as it will be defined later) of the Hamiltonian would re-\nmain unchanged by changing this factor, but the quartic\n(and the other orders) part would be affected. Hence, it\nis not possible to compare coefficients of quartic or higher\norder for different geometries, as their definition depends\non the arbitrary choice of the constant C. Instead, nor-\nmalized coefficients should be compared.\nThe expression of Hwith respect to the new variables\n(a1,a2)and their complex conjugates (a†\n1,a†\n2)can be di-\nvided asH=H2+H4by dropping the constant term\nand neglecting higher order hamiltonian terms. In terms\nof the complex variables, H2is the quadratic part and\nH4is the quartic part.\nH2=A1a1a†\n1+A2a2a†\n2+1\n2/parenleftbig\nB1a2\n1+B2a2\n2+c.c./parenrightbig\n+/parenleftbig\nC12a1a2+D12a1a†\n2+c.c./parenrightbig\nH4=U1a2\n1a†2\n1+U2a2\n2a†2\n2+W12a1a2a†\n1a†\n2\n+/parenleftbig\nV1a3\n1a†\n1+V2a3\n2a†\n2+c.c./parenrightbig\n+/parenleftbig\nY12a2\n1a†\n1a2+Y21a1a2\n2a†\n2+c.c./parenrightbig\n+/parenleftbig\nZ12a1a†2\n1a2+Z21a†\n1a2\n2a†\n2+c.c./parenrightbig\nWe introduce new parameters that correspond to the5\ncharacteristic frequencies :\nω1\nk=γ0Hk1,ω2\nk=γ0Hk2,ω1\nd=γ0Hd1,ω2\nd=γ0Hd2,\nω1\na=γ0m(Hx+Hex1),ω2\na=γ0nm(Hx+Hex2),\nω0\nc=γ0\nMn˜Dx,ω−\nc=γ0\nMn˜Dy−˜Dz\n2,ω+\nc=γ0\nMn˜Dy+˜Dz\n2.\nUsing these notations, the coefficients of the hamiltonian\nare given by :\nA1=ω1\nk+ω1\nd\n2+ω1\na+βω0\nc,A2=ω2\nk+ω2\nd\n2+ω2\na+ω0\nc\nβ,\nB1=−ω1\nd\n2,B2=−ω2\nd\n2,C12=−ω−\nc,D12=−ω+\nc,\nU1=−βω1\nk−β\n2ω1\nd,U2=−ω2\nk\nβ−ω2\nd\n2β,\nW12=−2ω0\nc,V1=β\n4ω1\nd,V2=ω2\nd\n4β,\nY12=β\n2ω−\nc,Y21=ω−\nc\n2β,Z12=β\n2ω+\nc,Z21=ω+\nc\n2β.\nIn matrix form,H2rewrites :\nH2=1\n2/parenleftbig\na†\n1a†\n2a1a2/parenrightbig\nA1D12B1C12\nD12A2C12B2\nB1C12A1D12\nC12B2D12A2\n\na1\na2\na†\n1\na†\n2\n\nThe notation xis used for the complex conjugate of\nx, to distinguish scalar coefficients from the magnetiza-\ntion complex variables (a1,a2)with complex conjugates\n(a†\n1,a†\n2). As for a SL oscillator, it is possible to diago-\nnalize the quadratic part H2of the hamiltonian16. In\nfact it is possible to do so for any number of layers, al-\nthough it becomes difficult to find analytical expressions\nformorethantwolayers. Thenewcomplexbasisiscalled\n(bop,bac), with :\n\na1\na2\na†\n1\na†\n2\n=Tab\nbop\nbac\nb†\nop\nb†\nac\n(6)\nTab=\nuop\n1uac\n1vop\n1vac\n1\nuop\n2uac\n2vop\n2vac\n2\nvop\n1vac\n1uop\n1uac\n1\nvop\n2vac\n2uop\n2uac\n2\n\nWe note/hatwideIthe 4x4 block matrix /hatwideI=/parenleftbigg\nI20\n0−I2/parenrightbigg\nwithI2\nthe2×2unity matrix. T†\nabis the transpose conjugate of\nTab. It verifies :\nT−1\nab=/hatwideIT†\nab/hatwideIIn the new basis, H2takes the simple form :\nH2=ωopbopb†\nop+ωacbacb†\nac\nThe complex variables (bop,bac)are eigenvectors of the\nlinear hamiltonian. They correspond to the two linear\nmodes of the SyF-STO: optical and acoustic. This base\nof eigenvectors is then used to express the non-linear part\nof the hamiltonian.\nThe expressions of ωop,ωacand of the coefficients of\nthe matrix Tabcome from diagonalizing the matrix ˜H2:\n˜H2=\nA1D12B1C12\nD12A2C12B2\n−B1−C12−A1−D12\n−C12−B2−D12−A2\n\nThe expression of ˜H2is for the general case, for any di-\nrection of the equilibrium magnetizations. In the con-\nfiguration studied here, with equilibrium configurations\nand applied fields along the easy axis, all the coefficients\nare real. We take this assumption in the following.\nFrom computing the eigenvalues of ˜H2, the following val-\nues are obtained for ωop/ac:\nω2\nav= (A2\n1+A2\n2)/2.−(B2\n1+B2\n2)/2.+D2\n12−C2\n12\n∆ =/parenleftBig\nA2\n1−A2\n2−B2\n1+B2\n2/parenrightBig2\n+ 4/parenleftBig\nC12(B1+B2)−D 12(A1+A2)/parenrightBig2\n−4/parenleftBig\nC12(A1−A 2)−D 12(B1−B2)/parenrightBig2\nω2\nop/ac =ω2\nav±√\n∆\n2(7)\nThefrequencies ωopandωaccorrespondtothetwomodes\noptical and acoustic that are observed in ferromagnetic\nresonance (FMR) experiments with SyFs. By definition,\nthe optical mode corresponds to the mode with the high-\nest frequency. The expressions of the two mode frequen-\ncies are in agreement with the expressions found in the\nliterature14,17.\nThe eigenvectors of ˜H2, which correspond to the\ncolumns of the matrix Tab, have complicated expressions.\nHowever, due to normalization conditions, they can be\nexpressed by 6 angles. For the two labels j= (op,ac),\nthe elements of the matrix Tabare given by :\nuj\n1= coshθjcosφj\nuj\n2=−coshθjsinφj\nvj\n1=−sinhθjcosψj\nvj\n2= sinhθjsinψj\nThe details about the coefficients are given in Ap-\npendix A.\nThe angles φjandψjare related to the coupling6\nbetween the two layers. In fact, if the coupling vanishes\n(C12=D12= 0), these angles vanish for one mode,\nsay the acoustic mode, φac=ψac= 0, whereas for the\nother mode, φop=ψop=π/2. So the optical mode bop\ndepends only on the layer 2 complex variable a2and the\nacoustic mode bacon the layer 1 and a1.\nThe angles θjcorrespond to the mixing between the\ndiagonal termsA1,A2and the off-diagonal terms B1,\nB2, by analogy to the transformation coefficients for a\nsingle layer.\nHowever, it is not possible to obtain an exact diago-\nnalization of the quartic part H4of the hamiltonian but\nnon-canonical transformations provide good approxima-\ntions. We distinguish the resonant terms, for which the\noverall phase vanishes, like bopb†\nop, from the non-resonant\n(or off-diagonal) terms, for which the overall phase varies\nwith time, like bopb†\nac.\nBecause in this configuration, all along the easy axis,\nthere is no cubic term in the Hamiltonian, the (non-\ncanonical) transformation to remove the conservative\nnon-resonant terms18does not affect the value of the di-\nagonal quartic terms. Equivalently, we then assume that\nthe off-diagonal terms of the quartic term are negligi-\nble. However this assumption is valid only if the mode\nfrequencies ωopandωacare large compared to the off-\ndiagonal terms. Concretely, when applying an external\nfield that is comparable to the spin-flop field, the acous-\ntic mode frequency almost vanishes and the previous as-\nsumption is no longer valid. In this case the dynamics is\nmore complicated because the conservative non-resonant\nterms become important. We therefore limit the major\ndiscussion to the field range below the spin-flop field.\nNeglecting the non-resonant terms, the quartic part has\nthe simple expression :\nH4=Nop\n2b2\nopb†\nop2+Nac\n2b2\nacb†\nac2+Tbopb†\nopbacb†\nac\nWhereNac(resp.Nop) is the acoustic (optical) non-\nlinearfrequencyshiftcoefficientand Tisthemixed-mode\nnon-linear frequency shift coefficient. They are all real.\nAll these coefficients come from the conservative part of\nthe LLGS equation, they depend on the demagnetizing\nfields of the layers, applied field and coupling energy.\nHowever, they are independent of the damping coeffi-\ncients of the layers and of the applied current.\nB. Complex variables: dissipative part\nWenowfocusonthedissipativepartoftheLLGSequa-\ntion. After the transformation to the complex variables\na1,a2, the LLGS equation writes :\n˙a=−i∂H\n∂a†−Fa (8)\nWhere Fa= (Fa1,Fa2)is a vector with two complex\ncomponents. The two dissipative complex componentsFa1,Fa2are truncated to contain only linear and cubic\nterms ina1,a2,a†\n1anda†\n2. The polynomial coefficients\nare noted with 4 indices (k,l,m,n ), so that :\nFai=/summationdisplay\nk,l,m,nfk,l,m,n\naia1ka2la†\n1ma†\n2nfori= 1,2\nTheexpressionsofthecoefficientsofthedissipativeterms\nare given in the Appendix B. Using the linear transform\nwiththematrix Tab,similarcoefficientsforthe b-variables\nare obtained, with b= (bop,bac):\n˙b=−i∂H\n∂b†−Fb\nFb= (Fbop,Fbac)and fori=op, ac :\nFbi=/summationdisplay\nk,l,m,nfk,l,m,n\nbibopkbaclbop†mbac†n\nSo that :\n/parenleftbiggFb\nF†\nb/parenrightbigg\n=T−1\nab·/parenleftbiggFa\nF†\na/parenrightbigg\nWhere FaandF†\naare expressed in terms of b-variables\nusing the transform of equation (6).\nAll these fk,l,m,n\nbicoefficients in the b-coordinates\nare complex in general. However, if the coefficients\nof the conservative terms are real ( B1,B2,C12,D12,\nV1,V2, etc...), only the field-like torque contributes to\nthe imaginary part. For the magnetic configuration\nstudied in this paper, the applied field is aligned with\nthe magnetization, so the conservative coefficients are\nreal. The field-like torque is also set to zero, so the\ndissipative coefficients are real. In any case, the real part\nis the really important part, as it defines the power as it\nwill be shown in the next section. The imaginary part\nonly gives a contribution to the phase equation and it is\nnegligible compared to the contribution from the con-\nservative part in the configuration studied in this paper,\nwith an external polarizer. Without external polarizer,\nbut taking into account the mutual spin-torque in a\nself-polarizer structure, the contribution of the field-like\ntorque is non-negligible as shown in reference 19.\nOf all the dissipative terms, the most important are\nthe resonant terms, i.e. the terms that are similar to the\nresonant terms from the conservative part. Taking into\naccount only these resonant terms, the dissipative part\nreduces to :\nFbop=bop/parenleftBig\nγop+Qopbopb†\nop+Ropbacb†\nac/parenrightBig\nFbac=bac/parenleftBig\nγac+Qacbacb†\nac+Racbopb†\nop/parenrightBig\n(9)\nγac(resp.γop) is the acoustic (optical) linear relaxation\nrate.Qac(Qop) is the acoustic (optical) non-linear\nrelaxation rate coefficient. Rac(Rop) is the coefficient of\nthe acoustic (optical) non-linear mode mixing relaxation7\nrate.\nBecause of the linear dependence of the STT amplitude\nwith respect to the applied current hypothesized in this\npaper, these coefficients depend linearly on the applied\ncurrent. The linear coefficients, γopandγac, are positive\nfor zero current, in agreement with the fact that the\nGilbert damping is a relaxation to the minimum energy\nconfiguration. They decrease with the current if the\ncurrent is applied in the direction that destabilizes the\nmagnetization. The dissipative coefficients also depend\non the demagnetizing fields and coupling energy, like the\nconservative coefficients.\nThe analytical expressions of the coefficients are very\nlengthy and therefore they are not presented here in de-\ntail. Instead, for each value of field and current, the co-\nefficients are calculated numerically through the various\ntransformations, using the materials parameters given in\nsection IB. The variation of the different coefficients with\nfield and current are given in section III, where the cou-\npled complex equations are solved.\nIII. DYNAMICS WITH RESONANT TERMS\nONLY\nIn order to illustrate some of the basic features of the\ncoupled system, in a first approximation only the reso-\nnant terms, i.e.H2,H4and Eq. 9 are considered for the\ntime evolution of bopandbac:\n˙bop=−ibop(ωop+Noppop+Tpac)\n−bop(γop+Qoppop+Roppac)\n˙bac=−ibac(ωac+Nacpac+Tpop)\n−bac(γac+Qacpac+Racpop)(10)\nWherepop=bopb†\nopandpac=bacb†\nacare the powers of\nthe two modes. All the coefficients are supposed to be\nreal.\nWe notice that the dynamics of the coupled system does\nnot reduce to two independent oscillator equations. Even\nif the two modes are decoupled in the linear regime\n(pop,pac/lessmuch1), the acoustic and optical modes are cou-\npled through the non-linear coefficients.\nIntroducing the phases φop,φacof the two modes, let’s\ndefine :\nbop=√pope−iφop\nbac=√pace−iφac\nUsing the definitions of bopandbac, one can derive\nseparate equations for the power and the phase. These\nwill be discussed in the next sections. It is reminded\nthat for a single layer the equivalent analytical equations\nyield as a stationary solution a constant oscillation\npower (cancellation of the dissipative part). In the next\nsection it is shown that the coupled Eq. 10 can reduce\nto a single mode equation under specific conditions.\nFigure 2. Linear and non-linear dissipative coefficients versus\napplied current IforHx=−40kA/m and JRKKY =−5×\n10−4J/m2. (a) Optical coefficients, (b) acoustic coefficients.\nAll values are divided by 2πto be in units of Hz and not in\nrad/s.\nFor this we start discussing the solutions to the power\nequations.\nA. Power equations\nTheequationsoftimeevolutionofthepowerandphase\nare derived from the complex equations 10. The equa-\ntions of evolution of the powers of both modes are given\nby the generalized Lotka-Volterra (LV) equations20:\n˙pop=−2pop(γop+Qoppop+Roppac)\n˙pac=−2pac(γac+Qacpac+Racpop)(11)\nLotka-Volterra systems are well known for modeling the\nevolution of predator-prey populations. We define the\nsingle-mode equilibrium powers ¯popand¯pacas :\n¯pop=−γop\nQop¯pac=−γac\nQac(12)\nTheeffective linear coefficients are defined by :\ndop=γop+ ¯pacRopdac=γac+ ¯popRac\nAnd the inter-mode mixing coefficient ∆is defined by :\n∆ = 1−RopRac\nQopQac\nThe convergence to equilibrium for the LV system is\ndescribed in reference 21 and references 22, 23 provide a\nclassification with state diagrams. The two-modes sys-\ntem has four equilibriums, their conditions for existence\nand stability are defined by :8\n•P0= (0,0)ifγop>0andγac>0:\nNo mode is excited, this is the subcritical regime\nwith only damped modes.\n•Pop= (¯pop,0)ifγop<0,Qop>0anddac>0:\nOnly the optical mode is excited and the acoustic\nmode vanishes.\n•Pac= (0,¯pac)ifγac<0,Qac>0anddop>0:\nOnly the acoustic mode is excited and the optical\nmode vanishes.\n•P∗= (p∗\nop,p∗\nac)ifdopQac<0,dacQop<0,\nQopQac∆>0and(dop+dac)/∆<0:\nThe system converges to a mixed-mode equilibrium\nwhere both modes have a finite power given by:\np∗\nop=−dop\nQop∆p∗\nac=−dac\nQac∆(13)\nNotice thatP0andP∗are compatible, they can be\nstable local equilibriums at the same time, but they are\nincompatible with PopandPac. And reciprocally, Pop\nandPaccan be stable at the same time, but not at the\nsame time asP0andP∗.\nGiven the specific conditions are fulfilled, each equilib-\nrium is defined and locally stable. However, the global\nconvergence to this equilibrium depends on the initial\nconditions, if they are in the basin of convergence of this\nequilibrium. For instance, PopandPaccan be stable\nat the same time, it depends on the initial conditions if\nthe system converges to one or the other equilibrium,\nor even if it diverges (which corresponds to a switching\nof one or both layers). See Fig. 4 in reference 20 for a\nphase portrait of pacversuspop— notedn1andn2.\nThe coefficients of Eq. (11) are plotted in figure 2\nversus applied current Ifor the macrospin parame-\nters defined previously and for Hx=−40kA/m and\nJRKKY =−5×10−4J/m2. Two threshold currents\nfor the modes excitations, Iac\ncandIop\nc, are defined by\nthe vanishing of the linear coefficients γopandγac,\nrespectively. For this particular set of parameters, the\nacoustic threshold current Iac\ncis lower than the optical\nthreshold current Iop\nc. Therefore the critical current Ic\ncorresponds to the acoustic threshold current, which is\nIc= 3.4mA in this particular case. Above the critical\ncurrentIc, the acoustic mode is excited, and because Qac\nis positive (not shown in Figure 2 above 2mAQacin-\ncreases linearly), the power converges to the equilibrium\nacoustic power; the optical mode remains zero. Above\nthe optical threshold current, the equilibrium acoustic\npower still exists and it is stable, because dop>0(not\nshown on the figures). However, Qopis negative, so\nno equilibrium optical power is defined and the optical\nmode may diverge. Therefore, the final state depends\non the initial conditions : if the acoustic power is close\nto the equilibrium ¯pacand the optical power is close to\n0, the system converges to the powers {0; ¯pac}; if theoptical mode diverges faster than the acoustic mode\nconverges to its equilibrium value, the whole system\nwill diverge, which corresponds to a reversal of the layers.\nHaving defined the equilibrium powers, the oscillation\nfrequency is given by the phase equations that will be\nanalyzed in the next section.\nB. Phase equations\nThe corresponding phase equations of Eq. 10 including\nonly resonant terms are :\n˙φop=ωop+Noppop+Tpac\n˙φac=ωac+Nacpac+Tpop (14)\nWe notice that the phase velocities ˙φopand ˙φacof the\ntwo modes are constant if the powers are at equilibrium\n(˙pop= ˙pac= 0). Moreover, the phase of each mode\ndepends not only on its own power, but also on the power\nof the other mode through the non-linear phase mixing\nT. But both phases are independent of each other : each\nmode oscillates at its own constant frequency. Note that\nthis is true only if the non-resonant terms are excluded,\nas shown in section IV below.\nLet’s consider the case of a single-mode excitation of\nthe acoustic mode, as it is observed in the simulations\nshown in this paper. In this case, pop= 0andpac=\n¯pac=−γac\nQac>0. Therefore the magnetization oscillates\nat the frequency fof the excited acoustic mode, which\nis given by :\n2πf=ωstt= Ωac=ωac+Nac¯pac (15)\nThis equation is equivalent to the phase equation\nof an STO composed of a single-layer (SL) free layer\nas described in previous work5. The power increases\nwith the applied current, and the frequency decreases\nor increases depending on the sign of Nac. Figure 3\nshows the transition between the two regimes, red-shift\n(frequency decrease with the current) and blue-shift\n(frequency increase) with JRKKY =−5×10−4J/m2, by\nchanging the applied field. The frequency is computed\nfrom the extended NLAO model and compared to\nthe frequency obtained from macrospin simulations,\nboth show a transition between red-shift and blue-shift\nat around−75kA/m. The change of regime with\napplied field in an asymmetric SyF was already observed\nnumerically24and experimentally10.\nAs stated, this transition corresponds to Nacchanging\nsign. The value of the non-linear coefficients of Eq. (14)\nis plotted versus applied field Hxin Figure 4 (a). Nac\nchanges signs at around Hx=−75kA/m, which, indeed,\ncorresponds to the red-shift/blue-shift transition. The\nself-sustained oscillations frequency f=ωstt/(2π)versus9\nFigure 3. Self-sustained oscillations frequency versus applied\ncurrentIwithJRKKY =−5×10−4J/m2and for different\napplied fields, from top to bottom : −40kA/m to −90kA/m.\nThe frequency scale is identical in all the panels, from 0 to\n4 GHz. Solid red line : computed from the extended NLAO\nmodel. Dashed blue line : extracted from LLGS simulations.\nBeyond the spin-flop transition, for Hx=−90kA/m, the\nextended NLAO model is not applicable.\nfieldHxatI= 4mA is reported in Figure 4 (b)\nand compared to the acoustic FMR frequency ωac\nand the frequency obtained from the simulations. We\ndifferentiate four regions, from low to high fields : (i)\nbelow the spin-flop field, at −90kA/m, the extended\nNLAO model is not valid. (ii) for higher fields but below\n−75kA/m, the acoustic mode is excited in the blue-shift\nregime, so ωstt> ωac. The discrepancy between the\nfrequency obtained from the extended NLAO model and\nthe simulation is high, as expected because the model is\nnot valid anymore if ωacis small. (iii) above −75kA/m,\nthe acoustic mode is excited, in the red-shift regime,\nωstt< ωac. The frequency computed from the model\nagrees with the simulations. (iv) above −20kA/m,\nthe applied current is too low to excite a mode, the\noscillator is in sub-critical mode. Notice that in the\nvicinity of the field value at which Nacvanishes, the\noscillator frequency does not change much with the\napplied field, in agreement with the simulations. At this\nfunctioning point, the oscillator frequency is not very\nsensitive neither to the applied field, nor to the applied\ncurrent.\nWe showed that the frequency of the self-sustained os-\ncillationscanbepredictedbytheextendedNLAOmodel,\nin the next section the model will be compared to numer-\nical simulations to define its validity range.\nFigure4. (a)Non-linearand(b)linearfrequencytermsversus\napplied field HxforI= 4mA andJRKKY =−5×10−4J/m2.\n(a) Non-linear coefficients: optical Nop(green), acoustic Nac\n(red),inter-mode T(blue). (b)Linearcoefficients: dottedma-\ngenta line, linear ωac; red solid line, self-sustained oscillations\nfrequency from the model ωstt= Ω ac=ωac+pacNac; dashed\nblue line, self-sustained oscillations frequency from simula-\ntions. The field range is divided in four regions, from low\nto high fields : model non-applicable (NA), blue-shift regime\n(BS), red-shift regime (RS) and no excitation (NE). All values\nare divided by 2πto be in Hz units and not in rad/s.\nC. Single-mode description of the SyF-STO\nTwo sets of simulations are presented, showing the self-\nsustained oscillations frequency versus applied current\nand field for two coupling strengths: (i) Figure 5 in the\nsmallcouplingregime JRKKY =−2×10−4J/m2, (ii)Fig-\nure6the largecouplingregime JRKKY =−5×10−4J/m2.\nIn both figures, the frequency of the m1ycomponent of\nthe magnetization of layer 1 from macrospin simulations\nis plotted in the top panels (a). The frequency computed\nfrom the extended NLAO model is plotted in the bottom\npanels (b). State diagrams for these values of JRKKYare\ndisplayed in reference 25.\nWe observe a qualitative agreement between the model\nand the simulations, especially in the region close to\nthe critical current. First, above the acoustic critical\ncurrentIac\nc(region on the right of the red solid line), the\nmodel predicts self-sustained acoustic-like oscillations,\njust like the simulations (and other publications25).\nJust above the optical critical current Iop\nc(region on\nthe right of the green solid line and on the left of\nthe red solid line), there is no oscillation and the two\nlayersswitch, aspredictedbytheequationsofthepowers.\nThere are also several discrepancies, that will be dis-\ncussed in the following.10\nFigure 5. Frequency of the self-sustained oscillations versus\napplied current and field (a) from macrospin numerical sim-\nulations and (b) from the formulas for the power and phase\nfrom Eq. (15). The RKKY coupling is of −2×10−4J/m2.\nRed (green) solid lines represent Ic(Hx)the vanishing of the\nacoustic (optical) linear dissipative coefficient γac(op). Dotted\nlines correspond to the vanishing of the quadratic dissipative\ncoefficientQac(op).\nFirst, the out-of-plane precession (OPP) region is not\npredicted by the model. OPP are oscillations around\nthe energy maximum, which are not considered in this\nmodel. To describe the OPP, the projection base for the\ncomplexa-coordinates should be changed to the out-of-\nplane axes, instead of the equilibrium in-plane axes, and\nall the coefficients should be computed again.\nSecond, according to the simulations, self-sustained os-\ncillations are expected when the field is larger than the\nspin-flop field. However the extended NLAO model is\nnot valid in the spin-flop region. In fact it is not valid in\nthe vicinity of the spin-flop field either, as it was already\nmentioned. That is why for JRKKY =−2×10−4J/m2,\nFigure 5, the red-shift/blue-shift transition at around\nHx=−45kA/m, is not predicted by the extended\nNLAO model : it is too close to the spin-flop field\nvalue of−50kA/m. On the contrary, for JRKKY =\n−5×10−4J/m2, Figure 6, the red-shift/blue-shift tran-\nsition at around Hx=−70kA/m, with a spin-flop field\nat−90kA/m, is well predicted by the extended NLAO\nmodel.\nLast, the model predicts a much larger region of oscil-\nlations than the simulations. In the region on the right of\nthe optical critical current Iop\nc(green solid line), the dif-\nFigure 6. Same as Figure 5 with an RKKY coupling of −5×\n10−4J/m2.\nference between the model and the simulations becomes\nreally important. This was also shown in Figure 3. In\nthis region, the power is large, which is a known limit\nfor the validity of the NLAO model. But there could be\nanother explanation, because in this region, the model\npredicts a single-mode excitation with pop= 0, whereas\nthe simulations show that popdoes not vanish (not shown\nin the figures).\nTo explain the failure of the model in this region, we\npropose to study the influence of other terms that we\nfirstdiscardedinthemodel, namelythelinearcoefficients\nfrom the dissipative part that are non-resonant. The lin-\near terms are important corrections as they depend lin-\nearly in the powers, contrary to higher order terms. Also\nthey can be easily computed, which is not the case of\nhigher order terms.\nIV. CORRECTION DUE TO NON-RESONANT\nTERMS\nAs was shown in Section IIIC, Eq. 10 cannot capture\nall the features of the dynamics, in particular the fre-\nquency versus current. Therefore, in order to obtain a\nbetter description of the phase, we also include in Eq. 10\nnon-resonant, off-diagonal terms. This leads to the fol-11\nlowing equation:\n˙bop=−(iΩop+ Γop)bop−˜γopb†\nop−˜ϑopbac−ϑopb†\nac\n˙bac=−(iΩac+ Γac)bac−˜γacb†\nac−˜ϑacbop−ϑacb†\nop(16)\nHere Γop=γop+Qoppop+Roppacis the optical dissi-\npative part with only resonant terms from equation (9).\nIdentically, Γac=γac+Qacpac+Racpopis the acoustic\nresonant dissipative part. For the conservative part,\nΩop=ωop+Noppop+TpacandΩac=ωac+Nacpac+Tpop.\nThe coefficients of the non-resonant terms\n(˜γop,˜γac,ϑop,ϑac,˜ϑop,˜ϑac) are independent of the\npowers; for simplicity, we take the coefficients to be real,\nbut taking into account the imaginary part does not\nchange the general conclusions.\nThe equations for the amplitude and phase rewrite as :\n˙pop=−2(Γop+ ˜γopcos(2φop))pop\n−2√poppac/parenleftbig˜ϑopcos(φop−φac) +ϑopcos(φop+φac)/parenrightbig\n˙pac=−2(Γac+ ˜γaccos(2φac))pac\n−2√poppac/parenleftbig˜ϑaccos(φop−φac) +ϑaccos(φop+φac)/parenrightbig\n(17)\n˙φop= Ωop+ ˜γopsin(2φop)\n+/radicalbiggpac\npop/parenleftbig˜ϑopsin(φop−φac) +ϑopsin(φop+φac)/parenrightbig\n˙φac= Ωac+ ˜γacsin(2φac)\n+/radicalbiggpop\npac/parenleftbig˜ϑacsin(φac−φop) +ϑacsin(φop+φac)/parenrightbig\n(18)\nThe equations including the non-resonant terms are\nmore complicated, therefore each term will be treated\nseparately.\nWe first present a qualitative interpretation of each term\nand then evaluate its effect on the dynamics in Figure 7 :\nLLGS equation (Eq. (2)), extended NLAO model with\nonly resonant terms (Eq. (10)), with the addition of the\nlinear dissipative terms (Eq. (16)) and with all the terms.\nA. Inter-mode phase locking\nAn important disagreement between the LLGS sim-\nulation (Eq. (2)) and equation (10) is the phase of the\nnon-excited mode, as can be seen from the comparison\nof Fig. 7 (a) and 7 (c). In section IIIB, it was shown\nthat without the non-resonant terms the two modes have\ndifferent frequencies. However, in the LLGS simulations,\nFig. 7 (a), the two modes are locked, they have the same\nfrequency (although they can have an opposite sign19).\nThis discrepancy can be corrected by including the terms\nwiththe ˜ϑacand˜ϑopcoefficients, asisshowninFig.7(d).\nLet’s suppose that only an acoustic-like mode is ex-\ncited, buttheopticalmodedoesnotvanishtotally( pop≈0andpac= ¯pac). The powers are considered to be con-\nstant.\nThe differential equation for the phases of the two modes\nare :\n˙φop= Ωop+/radicalbiggpac\npop˜ϑopsin(φop−φac)(19)\n˙φac= Ωac+/radicalbiggpop\npac˜ϑacsin(φac−φop)(20)\nIn the acoustic phase equation (20), the second term\nis negligible compared to the constant frequency Ωac\nbecause of the powers ratio, so in the first order, the\nacoustic mode has a constant frequency ˙φac= Ωac, so\nφac= Ωact. However, in the optical phase equation (19),\nthe second term on the right-hand-side is dominant, also\nwith respect to the left-hand-side. This leads to the rela-\ntionsin(φop−φac) =/radicalbiggpop\npac/parenleftbigg˙φop−Ωop\n˜ϑop/parenrightbigg\n≈0, so in the\nfirst order, φop≈Ωact, orφop≈π+ Ωact. This means\nthat the frequency of the non-excited mode is locked to\nthe frequency of the excited mode in the supercritical\nregime.\nAt the second order, the phase difference is given approx-\nimately by :\nφop−φac=/radicalbiggpop\npac/parenleftbiggΩac−Ωop\n˜ϑop/parenrightbigg\n+kπwithk∈Z\n(21)\nSimilarly, the terms with the ϑacandϑopcoefficients\nare responsible for a locking with opposite frequency\n(same absolute frequency, but opposite phase sign), of\nthe form :φop+φac≈0, withφac(t) = Ωact.\nIf both ˜ϑopandϑopare included simultaneously, there\nis a competition between the two terms for the locking\nof the non-excited mode, to the same or the opposite\nfrequency as the excited mode. The resulting relation\nbetween the two phases is more complicated then. How-\never, regarding the time-average of the frequency, the\nnon-excited mode is locked to the frequency of the ex-\ncited mode if|˜ϑop|>|ϑop|, and to the opposite frequency\nif|˜ϑop|<|ϑop|. In other words, the coefficient with the\nhighest value (in norm) determines the type of locking,\ndirect or opposite. An example for opposite frequency\nlockingistheself-polarizedconfigurationdiscussedinref-\nerence 19.\nB. Power oscillations and second harmonics\nSecond, let’s focus on the term with the ˜γaccoefficient\n(itwillbesimilarforthetermin ˜γop). Weconsidera pure\nsingle-mode excitation of the acoustic mode, so pop= 0.\nNote that this analysis is valid for any single-mode non-\nlinear oscillator equation, including the SL case.12\nWithout the other non-resonant terms, the power and\nphase equations of the acoustic mode write :\n˙pac=−2Γacpac−2˜γaccos(2φac)pac\n˙φac= Ωac+ ˜γacsin(2φac)\nIn the assumption that the perturbation due to the ˜γac\nterm is small, one can use Lindstedt’s series to solve this\nsystem of equations26. If/epsilon1=˜γac\nΩacis small, then the\npowerpacand phaseφaccan be written as power series\nof/epsilon1:pac=p0+/epsilon1p1andφac=φ0+/epsilon1φ1. In the zeroth\norder,p0= ¯pacandφ0=¯Ωact, with ¯Ωac=ωac+Nac¯pac.\nIn the first order, the equation for the power deviation\np1and phase deviation φ1are :\n˙p1=−2¯pacQacp1−2¯pac¯Ωaccos(2 ¯Ωact)\n˙φ1=Nacp1+¯Ωacsin(2 ¯Ωact)\nWe use the fact that ¯pacQac=−γac/lessmuch¯Ωac, so the first\nterm on the right-hand side of the power equation is ne-\nglected. Therefore, in the first order and in the perma-\nnent regime, the power pacwrites :\npac(t) = ¯pac/parenleftbigg\n1−˜γac\nΩacsin(2 ¯Ωact)/parenrightbigg\nUp to the first order, the phase φacis given by :\nφac(t) =¯Ωact−˜γacωac\n2¯Ω2accos(2 ¯Ωact)\nTherefore the term ˜γacgives rise to oscillations of the\npower but also a second harmonics in the frequency spec-\ntrum. As a consequence, it also contributes to the STO\nsynchronization by an AC current on the second harmon-\nics. Notice that this term is also present in STO based on\na SL free layer but was omitted in previous descriptions4.\nC. Simulations and trajectories\nThe effect of the non-resonant terms on the dynamics\nis best seen by simulating the different equations.\nOn Fig. 7, we compare the simulations of different\nequations and performed in different coordinate sys-\ntems, and projected afterwards in the (pop,pac,φop,φac)-\ncoordinates for comparison. In Fig. 7 (a), the simulation\nis performed in the (m1x,m1y,m1z,m2x,m2y,m2z)-\ncoordinates, like the usual LLGS simulations, according\nto Eq. (2).\nIn Fig. 7 (b), the simulation is performed in the com-\nplexa-coordinates, using equation (8). The trajectory is\nvery similar to the LLGS trajectory. That is because the\nterms of order superior to 3 in (a1,a2)were dropped after\nthe canonical transformation (m1,m2)−→(a1,a2)and\nwith powers of the order of 10−2, this approximation isperfectly valid.\nIn Fig. 7 (c-e), the simulations are performed in the com-\nplexb-coordinates, from Equation (16). In Fig. 7 (c), all\nthe off-diagonal terms are omitted (which corresponds to\nEq.(10)). Thetrajectoryexhibitsaconstantfiniteacous-\ntic power, a vanishing optical power, and instantaneous\nfrequency for the acoustic and optical mode being con-\nstant but with different values. The constant power and\nfrequency of the acoustic mode are close to the averaged\nvalues computed from the LLGS equation.\nInFig.7(d), ϑop,ϑac,˜ϑopand˜ϑacaretakenintoaccount.\nThe powers are very similar to the powers obtained in\nFig. 7 (c), which justifies the approximation of constant\npowers used in the previous section. The frequency of\nthe non-excited mode, the optical mode, is locked to the\nacoustic frequency. The optical frequency is not constant\nthough, this is because of the competition between the\ntwo types of locking, direct and opposite. But its average\nvalue is close to the value of the acoustic frequency.\nIn Fig. 7 (e), ˜γopand˜γacare also included, so the sim-\nulated equation is exactly Eq. (16). The powers are not\nconstant anymore, but oscillate around the average value\ninstead. Although the average acoustic power is over-\nestimatedcomparedto theLLGSequation (0.043instead\nof 0.036, 20%over-estimated), the average frequencies\nmatch more accurately (-3.40 GHz instead of 3.46 GHz,\n2%under-estimated).\nIn conclusion, Eq. (10) with only resonant terms\npredicts accurately the excitation of the acoustic mode\nfor this set of parameters and it gives a good estimation\nfor the average values of the power and frequency of\nthe excited mode. In order to account for second order\nfeatures, like phase locking of the non-excited mode to\nthe excited mode and first harmonic oscillation, the\ncorrected equation (16) should be used. However, this\ncorrected model is not enough to explain the discrepancy\nwith the LLGS equation in the average power. When\nthe field becomes closer to the spin-flop field, this error\nbecomes so large that extended NLAO model is not\nvalid anymore. As stated in section IIA, the error is\nprobably due to higher order terms but this is out of\nthe scope of this paper. Similarly, the extended NLAO\nmodel fails at large applied currents and this cannot be\nexplained by the correction terms. It is also probably\ndue to higher order terms.\nWith the restrictions of the model of Eq. (10) in mind,\nin the next section we make predictions on how to reduce\nthe generation linewidth of the SyF-STO, which is a very\nimportant parameter for application. The value of the\nlinewidth given by the model were compared to LLGS\nsimulations.13\nFigure 7. Simulations for Hx=−40kA/m,I= 4mA andJRKKY =−5×10−4J/m2performed in the (a) m-variables, (b)\na-variables, (c) b-variables, only with resonant terms from Eq. (10), (d) b-variables with non-resonant terms from Eq. (16) but\n˜γop= ˜γac= 0and (e)b-variables with all non-resonant terms from Eq. (16). The results of the simulations are transformed to\ntheb-coordinates to compare them easily. Insets in (a) and (e) : zoom between 30 and 31 ns. Top panel figures : powers pac\n(red) andpop(green). Bottom panel figures : phase velocity or instantaneous frequency in GHz,∂φac\n∂t(red) and∂φop\n∂t(green).\nV. APPLICATION: REDUCE THE STO\nLINEWIDTH\nA. Thermal noise\nSo far, the system was supposed to be at zero tem-\nperature, however stochastic fluctuations arise at non-\nzero temperature. The effect of these fluctuations can\nbe estimated in regions where the single-mode approx-\nimation is valid. We consider single-mode acoustic-like\nself-sustained oscillations, but the same reasoning apply\nto any single-mode non-linear oscillator.\nWith finite temperature, the power and phase of the os-\ncillator are given by :\n˙pac=−2pac(γac+Qacpac) +/radicalbig\n4pacDacηp(22)\n˙φac=ωac+Nacpac+/radicalBigg\nDac\npacηφ (23)\nWhereηpandηφrepresent white Gaussian noise with\nnormalized variance and the diffusion coefficient Dacis\ndefined by :\nDac= Γ+\nacωT\nΩacwithωT=γ0kBT\n2Mwith Γ+\nacthe positive damping (without the contribution\nfrom the STT) computed at ¯pacandΩac=ωac+Nac¯pac.\nBecause of the thermal noise, the auto-oscillator\nexhibits a finite generation linewidth ∆ω, typical of a\nnon-linear single-mode oscillator6,7. The spectral density\ncan be Lorentzian or Gaussian depending on the value\nof the damping rate of the power fluctuations (or power\nrelaxation rate) Γp= ¯pacQac. The characterization of\na non-linear single-mode oscillator in the presence of\nthermal noise is detailed in Appendix C.\nIf the correlation time of the power fluctuations ( 1/Γp)\nissmallcomparedtothecharacteristicphasedecoherence\ntime(theinverseofthegenerationlinewidthbeingagood\nestimation), ∆ω/lessmuchΓp, the spectral density is Lorentzian\nand the full width at half-maximum (FWHM) ∆ωLis\ngiven by :\n∆ωL= ∆ω0/parenleftbig\n1 +ν2\nac/parenrightbig\n(24)\nwithνac=Nac/Qacand ∆ω0= Γ+\nacωT\n¯pacΩac(25)\nWhere ∆ω0is thelineargeneration linewidth and νacis\nthe normalized non-linear frequency shift coefficient.14\nOn the other hand, if the correlation time of the power\nfluctuations is much larger than the decoherence time,\n∆ω/greatermuchΓp, the spectral density is Gaussian with standard\ndeviation ∆ωGgiven by :\n∆ωG=|νac|/radicalbig\n∆ω0Γp (26)\nThe FWHM is given by√\n8 ln 2 ∆ωG.\nB. Key parameters to the linewidth\nThe expressions of Eq. 24 and 25 identify three pa-\nrameters that can be changed to reduce the value of\nthe linewidth to make functional devices : (i) increase\nthe power relaxation rate Γp, (ii) decrease the linear\nlinewidth ∆ω0and (iii) decrease the normalized non-\nlinear parameter νac.\n•The power relaxation rate Γp= ¯pacQac=|γac|, is\nproportional to the difference between the applied\ncurrent and the critical current Ic. An analytical\nexpression of γacis given in reference 14. In order\nto increase Γpwithout increasing Ic, the absolute\nvalue of the slope of |γac|versusIshould be in-\ncreased without increasing |γac|atI= 0.\n•The linewidth is proportional to the square of the\nnormalized non-linear parameter νac(if the nor-\nmalized non-linear parameter is large, which is the\ncase for STOs). Therefore, one way of reducing the\nlinewidth would be to reduce the non-linear param-\neterNacto zero. For the SyF structure discussed\nhere, this is the case at the transition from the red-\nshift to the blueshift regime. At the transition, the\nlinewidthisequaltothelinearlinewidthvalue ∆ω0.\nIn SL-STO, the vanishing of the non-linear param-\neter can be achieved by changing the equilibrium\nmagnetic state from in-plane along the easy axis to\nin-plane along the hard axis or out-of-plane4. This\nusually requires an external field. In SyF-STO, the\nvanishing of νaccan be achieved by applying an\nin-plane magnetic field along the easy axis. Such\na magnetic field can be generated by the dipolar\nfield from another magnetic layer with the same\neasy axis direction. Notice that a vanishing non-\nlinear parameter Nacmeans that the frequency of\nthe STO becomes independent of its power, and\nthen of the applied current; this loss of tunability\ncan be detrimental for applications. The synchro-\nnization bandwidth with an external signal is also\nproportional to the normalized non-linear parame-\nterνac5, so it should not be too small.\n•The linear linewidth is inversely proportional to\nthe geometrical mean magnetic volume M(see\nEq. (25)). With a SL, the critical current is pro-\nportional to the magnetic volume, so it is counter-\nproductive to increase it. For a SyF however, one\nFigure 8. Power relaxation rate Γpat constant super-\ncriticalityζ= 1andHx= 0versus RKKY coupling energy\nby area, plotted for different layer thicknesses (in nm). The\nother layer properties are the same as in Table I without ap-\nplied field,Hx= 0.Γpis divided by 2πto be expressed in Hz\ninstead of rad/s.\ncan think of a thin layer subjected to the spin-\ntransfer torque from the reference layer, coupled to\na thick layer not subjected to spin transfer torque.\nThus the critical current remains low, whereas the\nmean magnetic volume is increased.\nIn the next sections, we give some ideas about improving\nthese three parameters using a SyF-STO.\nC. Dependence of Γpon the coupling strength\nFirst, we study the variation of the power relaxation\nrate Γpwith some parameters of the SyF. However, be-\ncause Γpis related to the critical current Ic, we need\nto somehow normalize its value. To start, the super-\ncriticalityζis used instead of the current :\nζ=I−Ic\nIc\nUsing this normalized quantity, one can compare the\nvalues of Γpat twice the critical current value, which\ncorresponds to ζ= 1.\nThe applied field dependence of Γpis non-trivial but\nits value at zero field, Hx= 0, is interesting for ap-\nplications. The value of Γpat zero field, for the same\nsuper-criticality ζ= 1, is plotted in Figure 8 for different\nthicknesses of the two layers. It shows that Γpincreases\nwith the RKKY coupling strength, although it remains\nin the same order of magnitude as with a single layer\n(asymptotic value for JRKKY→0).\nD. Vanishing of the non-linear parameter Nac\nBecause of the quadratic dependence of the linewidth\non the normalized non-linear parameter νac, the most\neffective action to reduce the oscillator linewidth is to15\nFigure 9. Linewidth of m1y(yellow diamonds) from LLGS\nsimulations at 300 K, compared to the linewidth (solid red\nline), linearlinewidth (dotted red line) and Γp(dashed blue\nline) computed from the extended NLAO model, versus ap-\nplied field for a current of I= 4mA andJRKKY =−5×\n10−4J/m2.\ndecreaseNacby applying an in-plane field so the oscil-\nlator is excited close to the transition between red-shift\nand blue-shift.\nFigure 9 shows a comparison of the linewidth from LLGS\nsimulations at 300 K and from the extended NLAO\nmodel. The linewidth is plotted versus applied field,\natI= 4mA andJRKKY =−5×10−4J/m2. For\nthe simulations, the linewidth is calculated from a fit\nto a Lorentzian function. We observe a decrease of the\nlinewidth of almost two orders of magnitude between\nHx= 0andHx=−70kA/m. The linewidth decrease is\nassociated to the vanishing of the non-linear parameter\nNac. For small fields, |Hx|<50kA/m, the linewidth is\nmuch larger than the power relaxation rate, which cor-\nresponds to a Gaussian spectrum. On the other hand,\naroundHx=−70kA/m, the spectrum has a Lorentzian\nprofile. In the simulations, the spectrum appears to be\nindeed Lorentzian around Hx=−70kA/m. It is difficult\nto conclude about the line shape at lower absolute field\nvalue, though, because the noise is too large and both\nprofiles interpolate well the simulated spectrum.\nFigure 10 shows the linewidth versus field for a low cou-\npling,JRKKY =−2×10−4J/m2, and forI= 3mA.\nAs was shown above, the model does not predict a van-\nishing ofNac, therefore the predicted linewidth remains\nlarge in the whole field range. However, the macrospin\nsimulations show a redshift/blueshift transition at Hx=\n−45kA/m and a decrease of the linewidth to its linear\nvalueatthisfield. Infact, atthisfield, thefrequencydoes\nnot change with the applied current. In a single-mode\nmodel, it means that the phase does not depend on the\npower, so the linewidth is given by the linear linewidth\nalone. Therefore, in the low coupling regime, the os-\ncillation looks like it is single-mode, according to the\nmacrospin simulations, but the extended NLAO model\nis not sufficient to estimate the characteristic parameters\nof the oscillator.\nFigure 10. Same as Figure 9 with JRKKY =−2×10−4J/m2\nandI= 3mA.\nFigure11. Linewidthof m1yversusfield, comparisonbetween\na 2 nm thick single layer (blue) and a 2 nm layer coupled with\na 20 nm thick layer (red-orange), separated by (a) 1 nm and\n(b) 20 nm spacer. Symbols : LLGS simulations, solid lines :\nextended NLAO model, dotted lines : linearlinewidth from\nthe extended NLAO model.\nE. Coupling to a thick layer\nFinally, the last parameter that can be tuned to\nreduce the linewidth is the linear linewidth ∆ω0. The\nlinear linewidth does not depend much on the coupling\nstrength, but more on the magnetic volume, as stated\nbefore. In order to increase the total magnetic volume\nand keep a reasonable critical current, we can imagine a\nthinlayerof2nmcoupledtoathicklayerof20nm. With\nthis geometry, where the layers are very asymmetric,\nthe coupling strength plays an important role. Contrary16\nto the asymmetric case studied previously where the\nnon-linear parameter vanishes with the combination\nof asymmetric layers and strong coupling25, in the\nfollowing example, the non-linear parameter is reduced,\nso the linewidth is decreased, although not to the level of\nthelinearlinewidth. However, the linewidth reduction\nhappens at lower fields, more suitable for application,\nthan in the previous case.\nThe SyF of this example is compared to a nano-pillar\nbased on a single free layer. The SL-STO is composed\nof three layer : (1) a reference layer with in-plane\nfixed magnetization, a spin polarization of 0.3and\ncompensated dipolar fields (the total stray field is zero),\n(2) a tunnel barrier, and (3) a 2 nm thick free layer, with\nsaturation magnetization of 1×106A/m and damping\nconstant of 0.02. The nano-pillar has an elongated shape\nof150×100nm, giving a shape anisotropy to the free\nlayer along the x-axis. The SyF-STO of study comprises\nthe same SL nano-pillar, plus two additional layers : (4)\na spacer of variable thickness, 1 nm or 20 nm, and (5)\na 20 nm thick free layer with saturation magnetization\nof1×106A/m and damping constant of 0.02. The\nmagnetizations of the thick and the thin layers are\ncoupled through dipolar field, whose strength is lower or\nhigher depending on the thickness of the spacer. For the\ntwo cases, strong and weak coupling, the coefficients of\nequation (1) take the values :\nStrong coupling ( tMgO= 1nm) :\n(˜Dx,˜Dy,˜Dz)/S= (−1.9,−2.9,4.8)×10−4J/m2\nWeak coupling ( tMgO= 20nm) :\n(˜Dx,˜Dy,˜Dz)/S= (−0.9,−1.4,2.3)×10−4J/m2\nDue to the shape anisotropy, the magnetization of the\nthick layer is more stable than that of the thin layer, but\nit is still free to move. Like for the other stacks stud-\nied in this paper, the current is spin polarized between\nthe reference layer and the 2 nm thin layer, but it is\nconsidered unpolarized at any other point, including be-\ntween the thin and thick layers. The simulations were\nperformed at 300 K and the linewidth is computed do-\ning a Lorentzian fit of the power spectral density of m1y,\nthe magnetization of the thin layer along the y-axis. The\nlinewidth computed from the extended NLAO model and\nextracted from the simulation are showed in Figure 11.\nWhen the thin and thick layers are separated by 20 nm\nso they are weakly coupled, Figure 11 (a), the linewidth\nis of the same order of magnitude with or without the\nthick layer, in the hundreds of MHz range. Around\n10kA/m, which is the coupling field (at which the thick\nand thin layers have the same FMR frequency), the ex-\ntendedNLAOmodelpredictsanincreaseofthelinewidth\nabove the SL value, that is not observed in the simula-\ntions. On the contrary, the simulations show a decreaseof the linewidth around 10kA/m that we cannot explain.\nOverall, the value of the SyF linewidth is essentially com-\nparable to the value of the SL linewidth.\nIn the strongly coupled case, with a 1 nm spacer, Fig-\nure 11 (b), below the coupling field (around -10 kA/m),\nthe model predicts a reduction of the linewidth of one or-\nderofmagnitudebetweentheSyFandtheSLcase; above\nthe coupling field, an increase of the linewidth is pre-\ndicted. The simulations show a decrease of the linewidth\nofalmostoneorderofmagnitudefortheSyFcomparedto\nthe SL case for fields smaller than the coupling field, with\na minimum of 5 MHz at -25 kA/m, in agreement with\nthe model. Notice that the decreased linewidth is still\none order of magnitude larger than the linearlinewidth.\nAround the coupling field, the SyF and SL linewidths are\nequivalent, around 100 MHz. Above the coupling field,\nthe linewidth of the SyF is half the linewidth of the SL,\nin disagreement with the model.\nIn conclusion, we observe a reduction of the linewidth\nwhen a thin layer is strongly coupled to a thick layer.\nThe linewidth reduction occurs for all the fields except\nfor the coupling field, at which the linewidth value is as\nhigh as for a single layer.\nVI. CONCLUSION\nWe presented an extension of the NLAO model to de-\nscribe the self-sustained oscillations of a SyF composed\nof two layers coupled with RKKY coupling, dipolar cou-\npling and mutual STT. The analysis was restricted to\nthe plateau region of the SyF, where the two layers are\naligned along the same direction at equilibrium, paral-\nlel or anti-parallel. However, nothing prevents one from\napplying the same analysis to arbitrary initial configura-\ntions (and an arbitrary number of layers), by taking into\naccount a transverse field for instance, although the di-\nagonalization of the hamiltonian matrix would be more\ncomplicated and only numerical solution would be avail-\nable.\nIn the extended model, the SyF dynamics is described\nby two coupled complex non-linear equations, which cor-\nrespond, in the linear regime, to the acoustic and optical\nmode. Inthispaper, wefocusedonSyFswithfixedexter-\nnal polarizer and, for the set of parameters that we chose,\nonly one mode is excited at a time, the acoustic-like self-\noscillation. Therefore, the dynamics can be described by\na single mode power and a phase equation, as in the case\nof a single layer. It means that the self-sustained oscil-\nlations are defined by a constant power, resulting from\nthe balance between natural damping and STT. The fre-\nquency consists of a linear part and a non-linear part,\nproportionaltothepowerandtothenon-linearfrequency\nshiftNac. Identically, the linewidth of the power spectral\ndensity consists of a linear part and a non-linear part.\nIt was found that with a strong coupling and if the\ntwo layers are asymmetric, for instance if they have dif-\nferent thicknesses, the non-linear frequency shift Naccan17\nbe reduced strongly, so the linewidth is also strongly re-\nduced of one order of magnitude. In particular cases, Nac\ncan even vanish at a given field, which corresponds to a\ntransition between a red-shift and a blue-shift frequency\nversus current dependency. At this field, the linewidth is\nreduced to its linearlinewidth value, which is a reduction\nof almost two orders of magnitude. The power relaxation\nrateΓpwas not found to change much compared to the\nvalues found for a single layer STO.\nThis work confirms the robustness of the NLAO model\ntodescribesmalloscillationsofthemagnetizationaround\nthe equilibrium and it shows that it can be extended to\nseverallayers. Italsopresentedarelativelysimplesystem\nto study the interaction between oscillating modes and\nwe hope it can be extended to more general cases.\nACKNOWLEDGMENTS\nThisworkwassupportedbytheEuropeanCommission\nundertheFP7programNo.316657SpinIcurandtheFP7\nprogram No. 317950 MOSAIC.18\nAppendix A: Hamiltonian diagonalization :\ntransformation a-b\nThe expression of the coefficients of the transformation\nmatrixTabare given by the 6 angles : φj,ψjandθjfor\ni= (op,ac).\nFirst the angles φj(forj= (op,ac)) are computed :\nR+\nj= (A1+ωj)(A2+ωj)−D2\n12\nusj=A1−ωj+D12+C12D12−B1(A2+ωj)\nR+\nj(B1+C12)\n+B1D12−C12(A1+ωj)\nR+\nj(B2+C12)\nucj=A2−ωj+D12+B2D12−C12(A2+ωj)\nR+\nj(B1+C12)\n+C12D12−B2(A1+ωj)\nR+\nj(B2+C12)\nunj=/radicalBig\nus2\nj+ uc2\nj\nsinφj=usj\nunjcosφj=ucj\nunj\nNext the angles ψj(forj= (op,ac)) :\nR−\nj= (A1−ωj)(A2−ωj)−D2\n12\nvsj=A1+ωj+D12+C12D12−B1(A2−ωj)\nR−\nj(B1+C12)\n+B1D12−C12(A1−ωj)\nR−\nj(B2+C12)\nvcj=A2+ωj+D12+B2D12−C12(A2−ωj)\nR−\nj(B1+C12)\n+C12D12−B2(A1−ωj)\nR−\nj(B2+C12)\nvnj=/radicalBig\nvs2\nj+ vc2\nj\nsinψj=vsj\nvnjcosψj=vcj\nvnj\nAnd finally, the angles θj(forj= (op,ac)) are com-\nputed :\nF1=A1−B1−C12F2=A2−B2−C12tanhθj=−cosφj(F1−ωj)−sinφj(F2−ωj)\ncosψj(F1+ωj)−sinψj(F2+ωj)\nAppendix B: Coefficients of the dissipative part\nThe dissipative part is expressed as a power series in\nthea-coordinates, truncated after the cubic term :\nFai=/summationdisplay\np,q,r,sfp,q,r,s\naia1pa2qa†\n1ra†\n2sfori= 1,2\nWe use the following notations :\nν1=−mγ0\n2M~\n2|e|Iη1ν2=−mnγ0\n2M~\n2|e|Iη2\nν21= +γ0\n2M~\n2|e|Iη21ν12=−γ0\n2M~\n2|e|Iη12\nκ1=−mγ0\n2M~\n2|e|Iβ1κ2=−mnγ0\n2M~\n2|e|Iβ2\nκ21= +γ0\n2M~\n2|e|Iβ21κ12=−γ0\n2M~\n2|e|Iβ12\nHence the non-vanishing coefficients of Fa1andFa2with\nindices (p,q,r,s )are given by ( iis the imaginary unit,\ni2=−1) :\nFa1:\n(1,0,0,0) :α1A1+ 2nβν 21+ 2βν1−2inβκ 21−2iβκ1\n(0,1,0,0) :α1D12−(1 +n)ν21+i(1 +n)κ21\n(0,0,1,0) :α1B1\n(0,0,0,1) :α1C12+ (1−n)ν21−i(1−n)κ21\n(2,0,1,0) :−α1βA1+ 2α1U1−2nβ2ν21−2β2ν1\n(1,1,0,1) :α1W12−4nν21+ 4inκ21\n(0,2,0,1) :α1Z21+1 +n\n2βν21−i1 +n\n2βκ21\n(0,1,0,2) :α1Y21−1−n\n2βν21+i1−n\n2βκ21\n(1,1,1,0) : 2α1Z12+ (1 +n)βν21−iβ(1 +n)κ21\n(1,0,1,1) : 2α1Y12−(1−n)βν21+iβ(1−n)κ21\n(2,0,0,1) : 3α1Z12+ 31 +n\n2βν21−i1 +n\n2βκ21\n(2,1,0,0) : 3α1Y12−31−n\n2βν21+i1−n\n2βκ21\n(1,0,2,0) : 3α1V1\n(3,0,0,0) : 3α1V119\nFa2:\n(1,0,0,0) :α2D12−(1 +n)ν12+i(1 +n)κ12\n(0,1,0,0) :α2A2+2n\nβν12+2\nβν2−i2n\nβκ12−i2\nβκ2\n(0,0,1,0) :α2C12+ (1−n)ν12−i(1−n)κ12\n(0,0,0,1) :α2B2\n(0,2,0,1) :−α2\nβA2+ 2α2U2−2n\nβ2ν12−2\nβ2ν2\n(1,1,1,0) :α2W12−4nν12+ 4inκ12\n(2,0,1,0) :α2Z12+1 +n\n2βν12−i1 +n\n2βκ12\n(1,0,2,0) :α2Y12−1−n\n2βν12+i1−n\n2βκ12\n(1,1,0,1) : 2α2Z21+1 +n\nβν12−i1 +n\nβκ12\n(0,1,1,1) : 2α2Y21−1−n\nβν12+i1−n\nβκ12\n(0,2,1,0) : 3α2Z21+ 31 +n\n2βν12−i1 +n\n2βκ12\n(1,2,0,0) : 3α2Y21−31−n\n2βν12+i1−n\n2βκ12\n(0,3,0,0) : 3α2V2\n(0,1,0,2) : 3α2V2\nAppendix C: Thermal noise and Fokker-Planck\nequation\nThermal noise is introduced in Eq. (10) in the form :\n˙bop+bop(iΩop+ Γop) =/radicalbig\n2Dopηop\n˙bac+bac(iΩac+ Γac) =/radicalbig\n2Dacηac(C1)\nThe noise amplitudes DopandDac, also called diffusion\ncoefficients, are not constant and depend on the mode\npowers :Dop(bop,bac)andDac(bop,bac), but this depen-\ndence is omitted for clarity. They will be determined\nlater. Ωop,Ωac,ΓopandΓacare the conservative (for op-\ntical and acoustic modes) and the dissipative determin-\nistic coefficients. They also depend on the mode powers.\nηopandηacare two independent white noise sources with\nzero mean and correlators given by :\n/angbracketleftηi(t)/angbracketright= 0, for i∈(op, ac)\n/angbracketleftηi(t)ηj(t/prime)/angbracketright= 0, for i,j∈(op, ac)2\n/angbracketleftηi(t)¯ηj(t/prime)/angbracketright=δijδ(t−t/prime), fori,j∈(op, ac)2\nThe expressions of the diffusion coefficients are de-\ntermined by insuring that the equilibrium probability\ndensity function (PDF) for the powers and phase re-\nduces to the Boltzmann distribution without applied cur-\nrent5. Considering the Stratonovich stochastic differen-\ntial equation (SDE) (C1), the time evolution of the PDFP(pop,pac,φop,φac,t)is given by the following Fokker-\nPlanck (FP) equation :\n∂P\n∂t−∂\n∂pop(2popΓopP)−∂\n∂pac(2pacΓacP)\n+∂\n∂φop(ΩopP) +∂\n∂φac(ΩacP)\n=∂\n∂pop/parenleftbigg\n2popDop∂P\n∂pop/parenrightbigg\n+∂\n∂pop/parenleftbigg\nP∂\n∂pop(popDop)/parenrightbigg\n+∂\n∂pac/parenleftbigg\n2pacDac∂P\n∂pac/parenrightbigg\n+∂\n∂pac/parenleftbigg\nP∂\n∂pac(pacDac)/parenrightbigg\n+Dop\n2pop∂2P\n∂φ2op+Dac\n2pac∂2P\n∂φ2ac\nHere, we considered that the diffusion coefficients depend\nonlyonthemodepowers. Thetermsintheleft-hand-side\ncome from the deterministic equation, or drift, whereas\nthe terms in the right-hand-side represent the thermal\ndiffusion. At equilibrium/parenleftbigg∂P\n∂t= 0/parenrightbigg\n, the PDFP0is a\nuniform distribution for the phases, so we can remove the\nlast two drift terms of the left-hand-side. Moreover, the\nsecond and fourth diffusion terms of the right-hand side\nshould be compensated by two terms of drift that are\nusually neglected. They arise from the renormalization\nof the multiplicative noise terms27(see reference28where\nthese extra drift terms are included for a SL free layer).\nThe extra drift terms can be incorporated in (C1) to give\nthe correct equation in Stratonovich form :\n˙bop+bop(iΩop+ Γop) +fopbop=/radicalbig\n2Dopηop\n˙bac+bac(iΩac+ Γac) +facbac=/radicalbig\n2Dacηac\n(C2)\nWith :\nfop=−1\n2pop∂(popDop)\n∂pop\nfac=−1\n2pac∂(pacDac)\n∂pac\nInterestingly, these extra drift terms contribute only to\nthe power equations. In particular, they are responsible\nfor the non-zero average power below threshold (when\nsolving ˙p= 0,p= 0is not a solution anymore).\nAfter eliminating the extra drift terms, the FP equa-\ntion at equilibrium reduces to :\n0 =∂\n∂pop/parenleftbigg\n2popΓ+\nopP0+ 2popDop∂P0\n∂pop/parenrightbigg\n+∂\n∂pac/parenleftbigg\n2pacΓ+\nacP0+ 2pacDac∂P0\n∂pac/parenrightbigg\nWhere Γ+\nopandΓ+\nacare the dissipative terms at zero ap-\nplied current, i.e. the natural damping.20\nA solutionP0(pop,pac)of the former equation is :\nP0=Z−1exp/parenleftbigg\n−/integraldisplaypop\n0Γ+\nop\nDopdpop−/integraldisplaypac\n0Γ+\nac\nDacdpac/parenrightbigg\nWhereZis a normalization constant. The equilibrium\nPDF should correspond to the Boltzmann distribution,\nwhichisequalto Z/prime−1exp/parenleftbigg\n−E\nkBT/parenrightbigg\n, whereZ/primeisanother\nnormalization constant, Eis the energy of the system as\ndefined in Eq. (1) and Tis the temperature. Then the\ndiffusion coefficients are given by :\nDop= Γ+\nopkBT/parenleftbigg∂E\n∂pop/parenrightbigg−1\n= Γ+\nopωT\nΩop(C3)\nDac= Γ+\nackBT/parenleftbigg∂E\n∂pac/parenrightbigg−1\n= Γ+\nacωT\nΩac(C4)\nWe now consider the self-oscillation regime with a\nsingle-mode excitation of the acoustic mode, with ther-\nmal noise. The stochastic differential equation of the\npower and phase is expressed in the It¯ o form, which is\npreferred when solving analytically stochastic equations\nbecause the solutions are martingales. For clarity, the ac\nindex is dropped on the power pand phaseφ:\n˙p=−2p/parenleftBig\nγac+Qacp+˜fac(p)/parenrightBig\n+/radicalbig\n4pDacηp(C5)\n˙φ=ωac+Nacp+/radicalBigg\nDac\npηφ (C6)\nWhereηp=Re(√\n2ηaceiφac)andηφ=Im(√\n2ηaceiφac)\nare real stochastic variables with zero average and\n/angbracketleftηi(t)ηj(t/prime)/angbracketright=δijδ(t−t/prime), fori,j∈(p,φ).˜fac= 2fac\nis the extra drift term in the It¯ o form, computed from its\nStratonovich form and the diffusion coefficients.\nDue to the extra ˜facterm, the stationary power is dif-\nferent from the power p0without temperature. How-\never, above the threshold, we suppose that the stationary\npower ˜p0is close to the zero-temperature value:\n˜p0=p0(1 +δp0) withδp0/lessmuch1\nIt can be shown that δp0is given by:\nδp0=−˜fac(p0)\np0Qac=∆ω0\nΓp−ν∆ω0\nΩ0Γ+(p∞)\nΓ+(p0)Where ∆ω0=Dac(p0)\np0is thelineargeneration\nlinewidth, Γp=p0Qacis the power relaxation\nrate, Ω0=ωac+p0Nacis the stationary frequency,\nν=Nac/Qacis the normalized non-linear frequency\nshift coefficient and p∞=−ωac\nNac, with positive or\nnegative value. If Nacis negative, p∞corresponds to the\nmaximum oscillation power, for which ωac+Nacp∞= 0.\nAs long as ∆ω0/lessmuchΓpand the oscillation frequency Ω0is\nhigh enough ( Ω0/greatermuchν∆ω0), the effect of the extra drift\nterm can be neglected and ˜p0≈p0.\nThen, weconsiderfluctuationsofthepoweraroundthe\nequilibrium power p0and of the phase around φ0(t) =\nΩ0t:δp=p−p0(withδp/lessmuchp0) andδφ=φ−φ0:\n˙δp=−2p0˜Qδp+/radicalbig\n4p0Dacηp (C7)\n˙δφ=Nacδp+/radicalBigg\nDac\np0ηφ (C8)\nWhere the effective non-linear relaxation rate coefficient\nis˜Q=Qac+∂˜fac\n∂pac/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np=p0.\nThe correction due to the temperature-dependent term\non the non-linear relaxation rate writes as :\n˜Q\nQac−1 =∆ω0\nΓp+ν∆ω0/parenleftbigg2ωac−Ω0\nΩ2\n0/parenrightbiggΓ+(p∞)\nΓ+(p0)\nThe same conditions that assured that δp0/lessmuch1lead to\n˜Q≈Qac.\nBecause the stochastic equations are linear, the power\nand phase fluctuations are Gaussian processes with zero\nmean. There are contributions to the linewidth from the\nphase noise ( ηφ) and from the amplitude noise ( Nacδp).\nNotethattheothermode, theopticalmode, isconsidered\nto be subcritical, so its power is almost zero, and in any\ncase much smaller than the power of the acoustic mode.\nTherefore its contribution to the power spectral density\nis neglected.\nThe power is a weakly stationary process but the phase\nis a non-stationary Gaussian random walk. We obtain\nthe expression of the power variance ∆p2=/angbracketleftδp2/angbracketrightand\nthe phase variance ∆φ2=/angbracketleftδφ2/angbracketright7:\n∆p2=p2\n0∆ω0\nΓp\n∆φ2= ∆ω0/bracketleftbigg\n(1 +ν2)|t|−ν21−e−2Γp|t|\n2Γp/bracketrightbigg\n∗M. Romera is now working at Unité Mixte de Physique,\nCNRS, Thales, Univ. Paris-Sud, Université Paris-Saclay,\n91767 Palaiseau, France1S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. 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This process adds two extra drift terms\nthat do not balance each other.\n28T. Taniguchi, Applied Physics Express 7, 053004\n(2014), ISSN 1882-0778, URL http://stacks.\niop.org/1882-0786/7/i=5/a=053004?key=crossref.\n0d01d0a8539d0d8a726bd42873186e86 ."    },    {        "title": "2009.12073v2.Temperature_dependence_of_the_damping_parameter_in_the_ferrimagnet_Gd__3_Fe__5_O___12__.pdf",        "content": " Temperature dependence of the damping parameter in the ferrimagnet \nGd 3Fe5O12 \nIsaac Ng,1,2 a) Ruizi Liu1,3 a), Zheyu Ren1,3, Se Kwon Kim,4 and Qiming Shao 1,2,3 b) \n1Department of Electronic and Computer Engineering Department, Hong Kong University of \nScience and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China  \n2Department of Physics, Hong Kong University of Science and Technology , Clear Water Bay, \nKowloon, Hong Kong SAR, China  \n3Guangdong -Hong Kong -Macao Joint Laboratory for  Intelligent Micro -Nano Optoelectronic \nTechnology, The Hong Kong University of Science and Technology, Hong Kong, China  \n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, \nRepublic of Korea  \na) Contributed equally b) Email: eeqshao@ust.hk  \n \nAbstract  \nThe damping parameter 𝛼FM in ferrimagnets defined according to  the conventional practice for \nferromagnets is known to be strongly temperature dependent and diverge at the angular \nmomentum compensation temperature, where the net angular momentum vanishes. However, \nrecent theoretical and experimental developments  on ferrimagnetic metals  suggest that the \ndamping parameter can be defined in such a way, which we denote by 𝛼FiM, that it is free of \nthe diverging anomaly at the angular momentum compensation point and is little dependent o n \ntemperature. To further understand the temperature dependence of the damping parameter in \nferrimagnets, we analyze several data sets from literature for a ferrimagnetic insulator, \ngadolinium iron garnet , by using the two different definitions of the damping parameter. Using \ntwo methods to e stimate the individual sublattice magnetization s, which yield results consistent \nwith each other, we found that in all the used data sets, the damping parameter 𝛼FiM does not \nincrease at the angular compensation temperature and shows no anomaly whereas th e \nconventionally defined 𝛼FM is strongly dependent on the temperature.  \n \n \n \n Antiferromagnets have been one important focus in spintronics due to their properties distinct from \nmore conventional ferromagnets including the zero stray field, ultrafast dynamics, and immunity \nto external field1,2. Recently, antiferromagnetically coupled ferrimagnets have emerged as a new \nmaterial platform to study antiferromagnetic dynamics as suggested by the recent discoveries of \ncurrent -driven magnetization switching near magnetization compensation point3,4, where the net \nmagnetization vanishes, and fas t domain -wall dynamics at the angular momentum compensation \ntemperature4,5,6,7, where the net angular momentum vanishes. However, magnetic resonance and \ndynamics of ferrimagnets have not been fully understood par tly due to the involvement of multiple \nmagnetic sublattices and the resultant internal complexity. The dissipation rate of angular \nmomentum in magnetic material is manifested as the linewidth  in resonance spectrum . One \nquantity of particular importance in the dissipative dynamics of ferrimagnets is the damping \nparameter, which is a characteristic of the magnetic material that determines the Gilber t-like \ndamping  of angular momentum and is usually denoted by the dimensionless number 𝛼. Early \nliterature sugge sted that the effective damping parameter αFM for ferrimagnets defined by a value \nthat is proportional to the line width of the resonance response is strongly temperature -dependent \nand increases anomalously near the angular momentum compensation temperature ( TA).8  Recent \nstudies have provided a new interpretation: the damping parameter can be defined in such a way \nthat it is independent of temperature near the TA while the temperature depend ence of the \nferromagnetic resonance (FMR) is attributed to the temperature  dependence of the net angular  \nmomentum .9,10,11  A. Kamra et al. have theoretically demonstrated this new perspective by \naccounting the Rayleigh dissipation function in a two -sublattice magnetic system, and the resu ltant \nGilbert damping parameter is independent of temperature near the TA.11 This damping parameter \ndenoted by αFiM is defined as follows:  \n αFiM=|snet\nstotal|αFM, (1) \nwhere the snet  and stotal are the net and total angular momentum, respectively . The snet is calculated \nby the differenc e of the angular momentum between two sublattice s (snet=|𝑠1−𝑠2 |) and stotal is \ncalculated by the total magnitude of the angular momentum  (stotal=|𝑠1|+|𝑠2|). D.-H. Kim et al.  \nhave experimentally studied the current -driven domain wall motion in ferrimagnetic metal alloy \nGdFeCo  and revealed that the damping parameter αFiM is indeed independent of temperature near \nthe TA.10 Furthermore, T. Okuno et al.  has reported that αFiM of the GdFeCo  is temperature independent when the FMR measurement temperature is approaching the TA.6 The FMR  of \nferrimagnetic thin films below the TA is difficult to achieve because of much enhanced \nperpendicular magnetic anisotropy at lower temperatures . It would be desirable that a full \ntemperature range o f FMR can be investigated for ferrimagnets.  \nThe divergence of the conventionally defined damping parameter αFM at TA can be understood \neasily by considering the energy dissipation rate given by 𝑃=αFM𝑠net 𝒎̇2 (which is  twice the \nRayleigh dissipation function) , where 𝒎 is the unit magnetization vector. For the given power 𝑃 \nthat is pumped into the ferrimagnet by e.g.,  applying microwave for FMR , as the temperature \napproaches TA, the net spin density 𝑠net decreases and thus αFM increases. Exactly at TA, the net \nspin density vanishes, making αFM diverge and thus ill -defined. Note that the divergence of αFM at \nTA is due to the appearance of the net spin density 𝑠net in the dissipation rate and should not be \ninterpreted to indicate the divergence of the dissipation rate, which is always finite. In terms of the \nalternative damping parameter αFiM, the energy dissipation rate is given by 𝑃=αFiM𝑠tot 𝒎̇2. The \ntotal spin density 𝑠tot is always finite and has weak temperature dependence, and thus αFiM is well -\ndefined at all temperatures with possibly we ak temperature dependence.  This suggests that αFiM, \nwhich is well -defined at all temperatures, might be more useful to describe the damping of \nferrimagnetic dynamics, particular ly in the vicinity of TA, than the more conventional αFM which \ndiverges and thus ill -defined at TA. One way to appreciate the physical meaning of αFiM is to \nconsider a special model, where the energy dissipation of a ferrimagnet occurs independently \nthrough the dynamics of each sublattice and all the sublattice s have the same damping parameter . \nIn this case, αFiM is nothing but the damping parameter of the sublattices.  So far, the discussion of \nferrimagnetic damping is limited to ferrimagnetic metals, while ferrimagnetic insulators have \nshown the potential for u ltralow -power spintronics .12,13,14,15,16 \nIn this paper, we investigate the temperature dependence of damp ing parameters in  ferrimagnetic \ninsulator,  gadolinium iron garnet ( Gd3Fe5O12, GdIG) , by surveying the literature of studies on the \ntemperature dependence of FMR. Since the stotal is usually not given in the literature, we adopt \ntwo different methods to cal culate the individual sublattice magnetization ( MFe and MGd) and then \nevaluate stotal. The first method is to use the magnetization of yttrium iron garnet ( Y3Fe5O12, YIG) \nas the MFe as done in Ref.17, where nuclear magnetic resonance experiments show  that the \nmagnetization contribution from iron is similar in YIG and GdIG since yttrium does not cont ribute \nthe magnetization in YIG,  and then obtai n MGd from the net magnetization and MFe. The second method uses Brillouin -like function to simulate the temperature dependence of GdIG \nmagnetization , the angular momentum of each individual sublattice can be calculated with the  \nBrillouin  function . We found consistent results between  these two different methods that the \ndamping parameter αFiM is almost temperature -independent near the TA, unlike the conventionally \ndefined αFM which is strongly temperature -dependent and diverge at TA. \nThe FMR linewidth ( ΔH) of GdIG is utilized to find the conventional damping parameter αFM: \n ΔH=αFM\ngeffμB/ℏfres+ΔH0 , (2) \nwhere geff is the effective Landé  g -factor, μB is the Bohr magneton, ℏ is the reduced Planck \nconstant , ΔH0 is the frequency -independent inhomogeneous broadening linewidth, and fres is the \nresonance frequency. Then, to convert the αFM to the αFiM, we need to find the ratio snet\nstotal. Note that \nαFM  diverges as the temperature approaches TA, meaning that Eq. (2) can be used only when it is \nsufficiently far away from the TA. Therefore, we will only employ data sufficiently far away from \nTA in this perspective. The net spin density snet is calculated from  the difference between the \nangular momentum of Fe and Gd:  \n sFe=MFe\ngFeμB/ℏ , \nsGd=MGd\ngGdμB/ℏ , \nsnet=|sFe-sGd|=Mnet\ngeffμB/ℏ , (3) \nwhere the Mnet is the net magnetization, gFe and gGd is the Landé  g-factor of the iron and \ngadolinium sublattice, respectively. The net magnetization is given by  \n                   Mnet=|MFe-MGd| , (4) \nwhich is normally measured by a superconducting quantum interference device  or a vibrating -\nsample magnetometer  and provided in the literature.18,19,20 \n \nMETHOD 1 \nWe can use the magnetization of YIG as an approx imation for the MFe to calculat e the MFe and \nMGd from GdIG net magnetization,  as yttrium does not contribute to the magnetization of YIG , \nwhich we refer to as Method 1. Experimentally, Boyd et al.17  used the nuclear ferromagnetic resonance technique to determine temperature -dependent MFe in YIG and GdIG and found that \nthey are very similar. Thi s approximation has been used in previous literature and has produced \nreasonable results.21 The magnetization of YIG is obtained from Ref.18. With MFe and MGd \nknown, we can determine the angular momentum of each sublattice with it s respective g -factor . \nThe g factors of Fe and Gd are very similar , the g -factor of iron in measured from YIG and  is \ndetermined as 𝑔𝐹𝑒=2.0047 .22 The g -factor of Gd sublattice is 𝑔𝐺𝑑=1.994  and is determined by \nmeasurement of GdIG  23. The TM and TA will be very close to each other, with TA slightly higher \nthan TM. We can calculate the total spin density stotal  using  \nstotal=sFe+sGd . (5) \nThe net spin density snet can be calculated using Eq. (3) and we can obtain effective g -factor \nmeanwhile.  Finally, we can calculate the αFM using Eq. (2) and the αFiM using Eq. (1).  \n \nMETHOD 2 \nThe second  method  is to use the Brillouin -like function to simulate the temperature dependence of \nmagnetization.24 Due to the weak coupling of the Gd -Gd interaction, the gadolinium magnetic \nmoments follow a paramagnetic behavior  and increase drastically at low temperatures. The net \nmagnetization in GdIG can be describe d by the sum of the three sublattices with a and d sublattice s \ncorrespond ing to Fe an d c sublattice correspond ing to Gd:   \nMnet=|Ma+Mc−Md|. (6) \nThe individual magnetization component can be simulate d by the Brillouin function 𝐵𝑆𝑖(𝑥𝑖) \nMi(T)=Mi(0)BSi(xi) . (7) \nThe 𝑀𝑖(0) is the individual m agnetization at 0  K.  \nMd(0)=3nmFe=3ngdSdμB , \nMa(0)=2nmFe=2ngaSaμB , \nMc(0)=3nmGd=3ngcScμB , (8) \n𝑛 is the number of GdIG  formula unit per unit volume, it can be calculated using 𝑁𝐴/(𝜌𝑀𝑟), where \n𝑁𝐴 is the Avogadro’s number, 𝜌 and 𝑀𝑟 are the density ( 6.45 𝑔𝑐𝑚−3 25) and molar mass (942.97) of GdIG respectively . 𝑆𝑖 is the electron spin of the  respective sublattice . For GdIG,   𝑆𝑑 and 𝑆𝑎 are \n5/2 and 𝑆𝑐 is 7/2.  𝑔𝑖 is the individual g factor and  𝑥𝑖 is defined as:  \nxd=(μ0SdgdμB\nkBT)(nddMd+ndaMa+ndcMc) , \nxa=(μ0SagaμB\nkBT)(nadMd+naaMa+nacMc) , \nxc=(μ0ScgcμB\nkBT)(ncdMd+ncaMa+nccMc) , (9) \n𝑛𝑖𝑗 are the Weiss coefficients between two sublattice s, which account for the intersublattice \nmolecular field coupling ( 𝑖≠𝑗) or intrasublattice molecular field interactions ( 𝑖=𝑗).24 𝜇0 is \npermeability of vacuum . \n \nTo determine  the snet and stotal from the magnetization fitting will require the sublattice g -factor \ngGd and gFe.  gFe in a and d sublattice can be experimentally measured  from YIG  and is \ndetermined as  𝑔𝐹𝑒,𝑑=2.0047 ,𝑔𝐹𝑒,𝑎=2.003.22 The g -factor of Gd c sublattice has the same value \nas the one in Method 1, 𝑔𝐺𝑑=1.994.23 With the value of the individual sublattice g -factor, the \nangular momentum of each sublattice can be calculate d from Eq. (3). Then we can calculate the \neffective gyromagnetic ratio and effective g -factor with the sublattice magnetization and angular \nmomentum . \nγeff=MFe,d−MFe,a−MGd,c \nsFe,d−sFe,a−sGd,c ,  \n(11) \ngeff=γeffℏ\nμB ,  \nThe ratio snet/stotal can be calculated where snet=|sFe,d−sFe,a−sGd,c| and stotal =sFe,d+\nsFe,a+sGd,c, with both the 𝑔eff and angular momentum known . Eventually, the value of αFiM is \nobtained from Eq. (1). \n   \nFigure 1. The a nalysis of  GdIG data from Rodrigue et al.23  and Dionne et al.18 (a) Calculated \nindividual magnetization as a function of temperature  using Method 1. (b) The Magnetization \ncurve of GdIG using Brillouin fitting method (Method 2) compare d to the magnetization from \nDionne  et al.18 (c) The geff  of GdIG calculated from  Method 1 as the green cross  and from  Method \n2 as the red line compared to the grey dot geff from Rodrigue  et al. ([100] direction) .23 (d) (e) (f) \nComparing the damping parameter  αFM (red dot) to αFiM based on Method 1 (green cross ) and αFiM \nbased on Method 2 (grey dot) for three directions  ([100], [110] and [111]) . \n \nFigure 2. The a nalysis of  GdIG  data from  Flaig et al.26 (a) Calculated individual magnetization as \na function of temperature using Method 1.  (b) The Magnetization curve of GdIG using Brillouin \nfitting method (Method 2) compare d to the magnet ization from Flaig et al.26 (c) The geff  of GdIG \ncalculated from Method 1  as the red cross  and from Method 2 as the green  line compared to the \nblue dot geff from Flaig  et al.26 (d) Comparing the damping parameter αFM (red dot) to αFiM based \non Method 1 (green cross ) and αFiM based on Method 2  (blue dot). \n \nFigure 3. The a nalysis of  GdIG data from  Calhoun et al19,20. (a) Calculated individual \nmagnetization as a function of temperature using Method 1.   (b) The Magnetization curv e of GdIG \nusing Brillouin fitting method (Method 2) compare d to the magnetization from Calhoun et al.20 (c) \nThe 𝑔eff  of GdIG calculated from Method 1 as the green  cross  and from Method 2 as the grey line \ncompared to the yellow  dot 𝑔eff from  Calhoun et al.19 (d) Comparing the damping parameter αFM \n(red dot) to αFiM based on Method 1 ( green  cross  and αFiM based on Method 2 ( blue dot). \n \nRESULTS AND DISCUSSION S \nWe analyze three datasets and evaluate the validity of Method 1 and M ethod 2 using the formula \nprovided above.  The first dataset is from Rodrigue et al.23, where the ΔH and geff in three \ndirections [100], [110], [111] are provided. Note that  the value of  geff is calculated using the Kittel \nequation in Rodrigue ’s paper . The Mnet is obtained from  Dionne  et al.18 where the GdIG has a \nsimilar compensation temperature to Rodrigue et al.23. fres=9.165GHz and we assume that ΔH0 is \nzero since the GdIG is a polished sphere . We analyze t he data using Method 1 and Method 2 and  \nplot the results in Fig. 1. For Method 1, w e can observe that the calculat ed temperature dependence \nof the MGd (see Fig. 1a) and the obtained g -factor (see Fig. 1c) are  reasonable . Using Method 2, \nwe get the fitting curves for magnetization from each sublattice and g -factor, which fit accurately \nto the experimental data.  \n \nThe second dataset of the temperature dependence of FMR below the TA is from Maier -Flaig et \nal.,26 where the g -factor, ΔH, and Mnet are also provided.  Again, we can see that the magnetization  \nas a function of temperature from two  methods  are in accordanc e with Flaig’s data  (see Fig. 2). \ngeff calculated dots from Method 1 and fitting curves from Method 2 are highly consistent with \nthe data, which illustrates that both two methods  are well established.  \n \nThe third set of data is from B. A. Calhoun et al. ,19,20 where fres= 9.479GHz. Similar results to the \nabove two datasets are obtained as shown in Fig. 3. \n \nTo directly compare the above two methods, the ferrimagnetic damping parameter αFiM calculated \nfrom the se two methods are plotted against each other in Fig. 1, 2 and 3, using the data from \nRodrigue et al.23, Flaig et al.26 and Calhoun et al.19. For all datasets , two different methods all  give \nconsistent results and have similar values: the newly defined damping parameter αFiM of a \nferrimagnetic material is not divergent near the TA and has much lower value than αFM. The αFiM \nin all three datasets is at low value, revealing the achievability  of fast domain -wall dynamics  in \nferrimagnetic insulator at the angular momentum compensation temperature . \n \nCONCLUSION  \nIn this work, we survey the literature dataset of FMR studies on the fe rrimagnet ic insulator  GdIG \nand find that the ferrimagnetic damping parameter αFiM does not increase when the temperature \napproaches the TA, differing from the conventionally defined αFM that shows divergence near the \nTA. This validates the recently developed theory about damping in the ferrimagnetic systems and \nreveals that the damping parameter, when it is appropriately defined with no divergence at all \ntemperatures, is not  as high as previously thought. Our work suggests that analyzing the dynamics  \nof ferrimagnets needs extra caution, that is not required for ferromagnets, in particular in the vicinity of the TA to avoid unphysical divergences . Besides, potentially lower damping in \ninsulators suggests that ferrimagnetic insulators are promising for future ultrafast and ultralow -\npower spintronic applications.  \n \nACKNOWLEDGEMENT  \nThe authors at HKUST were  supported by the Hong Kong Research Grants Council -Early Career \nScheme  (Grant No. 26200520 ) and the Research Fund of Guangdong -Hong Kong -Macao Joint \nLaboratory for Intelligent Micro -Nano Optoelectronic Technology (Grant No. \n2020B1212030010) . S.K.K. was supported by Brain Pool Plus Program through the National \nResearch Foundation of Korea funded by the Ministry of Science and ICT (Grant No. NRF -\n2020H1D3A2A03099291) and by the National Research Foundation of Korea funded by the \nKorea Government via the SRC Center for Quantum Coherence in Condensed Matter (Grant No. \nNRF -2016R1A5A1008184).  \n \nREFERENCES  \n1. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. 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Persson,4, 8and Ram Seshadri6, 7, 9\n1Department of Chemistry and Biochemistry, California State University, Fullerton, California 92834, United States\n2Department of Physics, Carleton College, Northfield, Minnesota 55057, United States\n3Current Affiliation: Applied Physics Graduate Program,\nNorthwestern University, Evanston, Illinois 60208, United States\n4Department of Materials Science and Engineering, University of California Berkeley, Berkeley, California 94720, United States\n5Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States\n6Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States\n7Materials Department, University of California, Santa Barbara, California 93106, United States\n8Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States\n9Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States\nCo1−xMnxCr2O4crystallizes as a normal spinel in the cubic Fd3mspace group, and the end mem-\nbers have been reported to display a region of collinear ferrimagnetism as well as a low-temperature\nspin-spiral state with variable coherence lengths from 3 nm to 10 nm in polycrystalline samples. Here,\nwe present the synthesis of the entire solid solution, and data showing that the ferrimagnetic or-\ndering temperature as well as the spin-spiral lock-in temperature are tunable with the Co/Mn ratio.\nThe peak magnetocaloric entropy change was determined to be ∆SM=−5.63 J kg−1K−1in an ap-\nplied magnetic field change of ∆H= 0 T to 5 T for the Mn end-member at the ferrimagnetic ordering\ntemperature. Using density functional theory (DFT), we explore the shortcomings of the magnetic de-\nformation proxy to identify trends in ∆SMacross composition in this spinel system, and explore future\nextensions of theory to address these discrepancies.\nINTRODUCTION\nIn the search for alternative refrigeration technologies,\nmagnetic refrigeration has emerged as an environmen-\ntally friendly and more efficient alternative to conven-\ntional vapor-compression refrigeration [1]. Magnetic re-\nfrigeration uses the magnetocaloric effect (MCE) by al-\nternating between states of high and low entropy (low\nand high magnetization, respectively) to impart a re-\nversible temperature change. Since the discovery of the\ngiant magnetocaloric effect in Gd 5Si2Ge2[2], research\nhas focused on methods of understanding which mate-\nrials may exhibit a large magnetocaloric effect. Satura-\ntion magnetization ( Msat) is typically used as an indica-\ntor for the magnitude of the magnetocaloric effect, and\nhigh magnetization compounds have been thought to be\nwell-performing magnetocalorics.\nRecent analysis has shown Msatis not always the most\neffective predictor of which materials may exhibit fa-\nvorable magnetocaloric properties, and Msatshows poor\ncorrelation with the MCE as quantified by the magnetic\nentropy change, ∆SM[3]. As such, there has been an\nincrease in interest in systems whose lattice and spin\ndegrees of freedom are strongly coupled, especially in\nmagnetocaloric materials. It has been proposed that this\ncoupling of spin and lattice – termed magnetostructural\ncoupling – is important to understanding, and even pre-\ndicting, the magnitude of the magnetocaloric effect in\nferromagnets [3].\n∗jcooley@fullerton.eduFor many decades, the MCE has been used to reach\ncryogenic (mK and µK) temperatures [4–6]. Recently,\nthere has been great interest in magnetocalorics for\nroom-temperature magnetic refrigeration [7]. This ne-\ncessitates magnetic ordering temperatures near room\ntemperature such that the maximum ∆SM(and, there-\nfore, MCE)t will be near room temperature. However,\nother technologies exist that would require excellent\nmagnetocaloric performance in more intermediate tem-\nperature ranges below the boiling point of liquid nitro-\ngen ( e.g.2 K to 70 K). This includes staged/cascaded cy-\ncles necessary to cover the wide temperature range re-\nquired for hydrogen liquefaction [8], which boils near\n20 K. While some effort has been made toward study-\ning inexpensive materials like chalcogenide spinels [9],\ncurrently much of the research into magnetocalorics vi-\nable for hydrogen liquefaction involves expensive rare-\nearth elements [10–12], stemming partially from the fact\nthat these often exhibit high Msatvalues. Additionally, a\nsegmented series of materials with gradually changing\ntransition temperatures would be beneficial to reach hy-\ndrogen liquefaction temperatures [13]. Here, we look to\nan earth-abundant transition metal oxide solid solution –\nCo1−xMnxCr2O4– with gradually varying transition tem-\nperatures magnets as a more cost-effective alternative to\nrare-earth magnetic materials.\nWe investigate how compositional changes and slight\nstructural changes affect the magnetism and MCE as\nquantified by the magnetic entropy change ∆SMin the\nspinel solid solution Co 1−xMnxCr2O4. This series of com-\npounds is geometrically frustrated stemming partially\nfrom the 3D-pyrochlore sublattice of the Cr3+and par-\ntially from the diamond sublattice formed by cations inarXiv:2309.16168v1  [cond-mat.mtrl-sci]  28 Sep 20232\nthe Mn/Co site. Frustrated spin systems often possess in-\nteresting and exotic ground states [14, 15], such as the\nspiral spin textures found in the Mn [16] and Co [17, 18]\nchromite spinels, and have been linked to good magne-\ntocaloric performance [19]. According to a 2014 study\nby Dey et al., MnCr 2O4exhibits some degree of magne-\ntostructural coupling and shows magnetoelastic transi-\ntions at both the N ´eel temperature and the spin spiral\nlock-in temperature [20], and its geometric frustration\nlikely plays a part in coupling the spin and lattice [21].\nAs the first study of the magnetocaloric properties of this\nsolid solution, we find that the Mn-rich members exhibit\na high ∆SMof up to −5.67 J kg−1K−1for a field change\nof 0 T to 5 T.\nI. MATERIALS AND METHODS\nA. Synthesis\nPolycrystalline powders of the solid solution series\nCo1−xMnxCr2O4were prepared using solid-state syn-\nthesis. Precursor materials CoC 2O4•2H2O, MnO, and\nCr2O3were used as received. For clarity, herein samples\nwill be referred to according to their nominal xvalues\n(i.e.x= 0.00, 0.25, 0.50, 0.75, and 1.00). CoCr 2O4was\nsynthesized following literature procedure [17] but re-\ngrinding and annealing was not found to improve phase\npurity so samples were heated at 800 °C for 24 hours,\nthen at 1000 °C for 24 hours without removing from the\nfurnace, then the furnace was allowed to cool. Samples\nx= 0.25 and 0.50 were also synthesized using this proce-\ndure. However, due to initial phase separation into two\ndifferent spinel phases with slightly different unit cell\nparameters, samples x= 0.25 and 0.50 were quenched\nfrom 1000 °C. MnCr 2O4was also synthesized according\nto literature [22], and x= 0.75 also followed this proce-\ndure.\nB. Structural characterization\nHigh-resolution synchrotron powder X-ray diffraction\n(XRD) data were collected at room temperature on all\nsamples using beamline 11-BM at the Advanced Pho-\nton Source (APS), Argonne National Laboratory. For\nroom temperature scans, powderized samples were\nloaded into 0.8 mm diameter Kapton capillaries with\neach end sealed with clay and measured for 6.7 minute\nscans. x= 0.00, 0.75, and 1.00 were measured with\nλ= 0.457850 ˚A and x= 0.25 and 0.50 were measured\nat another time with λ= 0.457845 ˚A. Rietveld refine-\nments of data were performed using TOPAS [23]. All\nsamples were fit with size and strain parameters, except\nthe complex peak shape of x= 0.25 which required a 2 θ-\ndependent split Pearson VII peak shape to describe varia-\ntion in peak asymmetry fully. Structures were visualized\nusing VESTA-3 [24].C. Magnetic measurements\nMagnetic properties were measured on 5 mg to 10 mg\nof powder loaded into capillaries and measured a Quan-\ntum Design MPMS3 equipped with a vibrating sample\nmagnetometer (VSM). Zero field– and field–cooled mag-\nnetization ( M) vs temperature ( T) measurements were\ntaken upon warming at a rate of 5 K min−1. In order to\ndetermine ∆SM,Mversus Tmeasurements were taking\non cooling (using a rate of 5 K min−1) at various fields\nfrom H= 0.1 T to H= 5 T. Temperature derivatives of\ntheM(T)s were calculated using Tikhonov regulariza-\ntion [25], and then integrals with field were calculated\nusing the trapezoid method to obtain ∆SM. Data were\nanalyzed using the magentro.py code, and more details\nof this procedure have previously been reported [26].\nD. First-principles calculations\nThe crystal structures corresponding to the x= 0and\nx= 1 endpoints of the spinel system Mn xCo1−xCr2O4\n(CoCr 2O4and MnCr 2O4, respectively) were obtained\nfrom the Materials Project database[27]. To identify the\nenergetically stable collinear magnetic ordering of these\nspinel structures, an enumeration of the possible mag-\nnetic orderings and evaluation of their respective ener-\ngies was performed using the atomate workflow [28].\nDespite the experimental results in support of both\nCoCr 2O4and MnCr 2O4exhibiting a spin-spiral ground\nstate [17, 20, 29–32], we performed collinear ferrimag-\nnetic calculations. The justification for this approach is\nbased on experiments that show that the spin-spiral tran-\nsition temperature is significantly lower than the N ´eel\ntemperature, TN. Furthermore, we are interested in the\nferrimagnetic (FiM) to paramagnetic (PM) phase transi-\ntion, which yields the greatest change in entropy. Based\non the experimentally reported values of the spin-spiral\ntransition temperatures for CoCr 2O4and MnCr 2O4, the\norder-disorder phase transition has been experimentally\ndetermined to involve a predominantly collinear FiM to\nPM transition.\nUsing the endpoint spinel structures, a set of possi-\nble structures of each composition Mn xCo1−xCr2O4were\nenumerated based on their crystallographic symmetries,\nup to a supercell size of two times the formula unit using\nthepymatgen interface with enumlib [33–36].\nFollowing the generation of intermediate crystal struc-\ntures for x=0.25, 0.5, and 0.75, we generalized the\ncollinear FiM ordering that would be expected for the\nspinel system endpoints, with a net spin-up moment on\nthe Cr atoms, and spin-down moment on the Mn and\nCo atoms. This collinear FiM ordering was used in\nspin-polarized calculations for computing the deforma-\ntion proxy, ΣM, and saturation magnetization, M0\nnet.3\nFIG. 1. A view of one unit cell of the spinel structure type,\nACr2O4(A= Co, Mn) where the Asite is tetrahedrally coor-\ndinated by O and corner shared, and the Cr site is octahedrally\ncoordinated and edge- and corner shared.\nII. RESULTS AND DISCUSSION\nA. Structure and Phase Purity\nThe spinel structure type is pictured in Figure 1 where\nCo/Mn atoms are in green corner-shared A-site tetrahe-\ndra, Cr atoms are in deep blue edge-and corner-shared\nB-site octahedra, and O atoms are in tangerine at the\nvertices of the tetrahedra and octahedra. The normal\nspinel structure type exists such that divalent cations are\nin tetrahedral A-sites and trivalent cations are in octa-\nhedral B-sites. Many spinel compositions have the abil-\nity to form partially- or fully-inverted spinels such that\nA-site atoms oxidize to trivalent and occupy the B-site,\nandB-site atoms reduce to divalent and occupy the A-\nsite. Chromite spinels, however, are a unique case. Since\ntrivalent Cr is d3, it is more stable in an octahedral crys-\ntal field splitting arrangement, where all spins are in the\nlowest energy, triply-degenerate t2gorbitals. Trivalent\nCr in a tetrahedral crystal field would necessarily have\ntwo low-energy spins in the doubly degenerate egorbital\nand one high energy spin in the t2gorbitals, destabiliz-\ning this arrangement relative to the octahedral arrange-\nment. Thus, in chromite spinels we can be relatively cer-\ntain that Co and Mn only occupy the tetrahedral A-sites\nwhile Cr only occupies the octahedral B-site.\nRietveld refinement results of synchrotron powder x-\nray diffraction for samples in this study are shown in Fig-\nure 2. The left panel shows that each sample fits to a sin-\ngle spinel phase at the resolution of the instrument. The\nright panel shows a close view of the evolution in peak\nposition of the (311) peak as a function of x: a mono-TABLE I. Crystallographic Data for Co 1−xMnxCr2O4.\nx a(˚A) u R wp Cr2O3(%)\n0.00 8.33304(4) 0.26125(9) 14.46 –\n0.25 8.34771(4) 0.25949(1) 10.81 –\n0.50 8.3851(4) 0.26043(6) 12.149 0.81(1)\n0.75 8.41180(2) 0.26241(7) 13.72 0.54(1)\n1.00 8.43853(1) 0.26385(6) 12.82 1.40(2)\ntonic decrease in Qcorresponding to an increase in unit\ncell parameter. We note that samples with x= 0.25 and\nx= 0.50 show slightly more broadening than other sam-\nples. These samples did require quenching from max-\nimum temperature in order to yield single phase sam-\nples, and we note that there could be two spinel phases\nwith slightly different unit cell parameters, yet even the\nextremely high resolution of beamline 11-BM at the Ad-\nvanced Photon Source ( <1.4×10−4∆Q/Q) is not able\nto resolve two different phases, so we proceed as if each\nsample is single phase. There are no crystalline fer-\nromagnetic impurities present in the samples, however\nx= 0.50, 0.75, and 1.00 contain a small ( <1.5 wt%)\namount of unreacted Cr 2O3, which is antiferromagnetic\nat room temperature [37], that could not be eliminated\nby further reaction.\nFurther results of Rietveld refinements, including the\nweighted profile R-value Rwp, are tabulated Table I. The\ncell parameter a(also depicted in Figure 3) and oxygen\nposition uwere determined during Rietveld refinement.\nThe ionic radii of tetrahedrally coordinated Co2+and\nMn2+are 0.72 ˚A and 0.80 ˚A [38], respectively, so it fol-\nlows that increasing Mn composition should increase the\nlattice parameter throughout the solid solution. The an-\nion in spinel structures – here, O – is located on the crys-\ntallographic equipoint 32eand variation in this position,\nrepresented as the value u, reflects how the structure\nchanges when accommodating different sizes of cations.\nWhen u= 0.25, the anions are in ideal cubic closest-\npacking arrangement with perfect CrO 6octahedra; in-\ncreases in uabove the ideal value of 0.25 indicate the\nsize of the tetrahedron is increasing, and the octahedron\nis shrinking and undergoing a trigonal compression [39].\nTable I shows that the uparameter overall increases with\nincreasing Mn substitution, shifting it higher than the\nideal value, and implying the tetrahedral site size in-\ncreases across the solid solution, in good agreement with\nthe unit cell parameter increase and larger ionic radius\nof Mn2+.\nFigure 3 shows that the lattice parameter aof the se-\nries increases with increasing xand follows V ´egard’s law\nas indicated by the dashed line (created from the liter-\nature lattice parameters of the end members CoCr 2O4\n[40] and MnCr 2O4[41]). This is in good agreement with\nthe larger ionic radius of tetrahedral Mn2+as compared\nto Co2+. Each data point is also in good agreement with\nthe V ´egard line indicating that this series forms a com-4\nFIG. 2. The results of Rietveld refinement of synchrotron X-\nray diffraction data measured for each sample show the spinel\nstructure type. The data are black circles, the overlaying line\nrepresents the model based on the structure, and the line be-\nlow each pattern indicates the difference between the data and\nmodel. A close view of the (311) peak (right) indicates a com-\nplete solid solution at the resolution of the instrument.\nplete solid solution.\nB. Magnetic Properties\nBoth the Co and Mn end members of the solid solu-\ntion undergo a paramagnetic to ferrimagnetic transition\nand, after a region of collinear ferrimagnetic order, have\na complex spiral ground state at low temperatures. In\nFigure 4, which shows temperature-dependent magneti-\nzation for each composition, the transition from the para-\nmagnetic state to the collinear ferrimagnetic state can be\nseen to decrease as the Mn composition is increased. This\nis important from an engineering standpoint as tunabil-\nity and control of transition temperature, and thus peak\n∆SM(discussed later), in this range is especially valu-\nable to cascaded magnetic refrigeration for hydrogen liq-\nuefaction.\nThe high temperature (300 K to 390 K) inverse sus-\nceptibility of each sample was fit to a linear regres-\nsion and the equation of the resulting line was used\nto extract parameters from the Curie-Weiss equation,\nχ=C/(χ−ΘCW) – namely, the effective paramagnetic\nmoment µeff, Curie constant C, and Curie-Weiss inter-\nFIG. 3. a) Lattice parameters from Rietveld refinement of syn-\nchrotron X-ray diffraction data for each sample show that each\nfollows the V ´egard law (dashed line) as expected for a solid\nsolution.\nFIG. 4. Temperature-dependent magnetization data reveal an\nincrease in magnetization and decrease in ordering tempera-\nture as Mn content is increased.\ncept ΘCW. These values are presented in Table II. The\nspin-only formula for µeffis generally taken as valid for\n3dtransition metal ions, however it only strictly applies\nto specific cases. In reality, the orbital contribution is\nnot totally quenched and spin-orbit coupling plays a role\nin determining µeff.d1−d4transition metal ions have\na spin orbit coupling constant smaller than zero, and\ntheir µeffvalues tend to be smaller than what is calcu-\nlated by the spin-only formula. d6−d9transition met-5\nals have a spin-orbit coupling constant larger than zero\nand tend to have µefflarger than calculated by spin-only\nformula. The samples in this series show a stochastic\ntrend with composition concerning how they compare\nto the spin-only and unquenched µeffvalues. Co2+is\nd7and we would expect compounds rich in Co2+to de-\nviate higher than the spin-only value; conversely Mn2+\nisd4and we would expect that Mn-rich samples devi-\nate lower than the spin-only value. Values of µefffromthe Co- and Mn-rich ends of the solid solution are both\nfar above unquenched estimates and far below spin-only\nestimates with no real trend to speak of. This may be\ndue to not fitting completely in the paramagnetic regime\nfor each of these samples as, due to instrumentation and\nsample limits, only measurement to 390 K is practical.\nAsxincreases, the magnitude of ΘCWdecreases sug-\ngesting that the number of dominant antiferromagnetic\nCo/Mn interactions with Cr decreases as Mn content is\nincreased, and the decrease is overall monotonic.\nTABLE II. Results from fitting inverse magnetic susceptibility data of Co 1−xMnxCr2O4to a linear regression to extract parameters\nin the Curie-Weiss equation. TCis assumed to be the peak of the ∆SMcurve. Estimated moments (spin-only and unquenched) are\ncalculated using: µeff=p\n2µ2\nCr+ (1−x)(µCo)2+x(µMn)2. Individual moments, such as µMnare calculated as µeff=gp\nS(S+ 1)\nusing an isotropic Land ´e g factor of 2.\nµeff(µB)\nx C( [µB/ f.u.] T−1K ) measured spin-only unquenched ΘCW(K) TC(K)\n0.00 7.83 7.9 6.7 7.5 −613 97.9\n0.25 7.00 7.5 7.1 7.7 −471 88.7\n0.50 6.96 7.5 7.4 7.8 −369 74.2\n0.75 7.82 7.9 7.7 7.9 −337 58.8\n1.00 7.53 7.8 8.1 8.1 −271 43.7\nTABLE III. Results from fitting inverse magnetic susceptibility data of Co 1−xMnxCr2O4to Equation 4 using\nscipy.optimize.curve_fit .TCvalues are reproduced from Table II. We hypothesize that the “ A” sublattice corresponds\nto tetrahedral sites in the spinel structure occupied by Mn and Co species, and the “ B” sublattice corresponds to octahedral sites\noccupied by Cr atoms.\nx|ΘCW|TC CB CA CA+CB λBB λAA λAB\n(K) ( [µB/ f.u.] T−1K ) ( [µB/ f.u.]−1T )\n0.00 97.4 97.9 0.95±0.01 9.53 ±0.11 10.48 66.5±0.2 86.2 ±0.4 127.6 ±0.6\n0.25 89.6 88.7 0.89±0.01 7.85 ±0.07 8.74 67.1±0.3 72.0 ±0.4 118.4 ±0.6\n0.50 78.7 74.2 6.76±0.02 1.20 ±0.00 7.96 71.7±0.1 -37.7 ±0.0 48.2 ±0.0\n0.75 65.8 58.8 8.66±0.03 0.57 ±0.00 9.23 60.2±0.1 -113.0 ±0.0 15.1 ±0.1\n1.00 45.4 43.7 8.37±0.11 2.54 ±0.01 10.91 73.6±0.4 1.3 ±0.1 38.9 ±0.0\nA plot of C/(χ|ΘCW|) +sgn(ΘCW)vsT/ΘCWcol-\nlapses all high temperature susceptibility data as shown\nin Figure 5. The dashed straight line intersecting the ori-\ngin corresponds to ideal Curie-Weiss behavior; since the\nhigh temperature data fit well onto this line, this indi-\ncates that the fits in the high temperature regime are\nvalid. Plotting the susceptibility data in this manner al-\nlows us to understand the nature of dominant magnetic\nexchange interactions [42, 43]. Since all show negative\ndeviations from the ideal Curie-Weiss line, this indicates\nuncompensated antiferromagnetism manifesting as ferri-\nmagnetism.\nFrom inspecting Table II, it is clear that |ΘCW|severely\noverestimates TC. This is not entirely surprising, be-\ncause this disagreement is expected for more than oneanti-parallel sublattice [44]. We explore the extension\nof Weiss mean field theory to two magnetic sublattices in\nSection V A. By fitting Equation 4 to 1/χdata versus tem-\nperature, we were able to achieve agreement between\n|ΘCW|andTCto within a few Kelvin, as reported in Ta-\nble III. This Curie-Weiss temperature, ΘCW, is defined as\n|ΘCW|= max {ηi[−W]}, as explained in Section V A.\nThe effectiveness of this theoretical extension is ex-\nemplified in the agreement of the model to 1/χat all\ntemperatures greater than TC, as shown in Figure 6.\nThe parameters of Equation 4 are reported in Table\nIII, with their associated uncertainty values output by\nscipy.optimize.curve_fit . The “ A” and “ B” labels in-\ndicate a separate magnetic sublattice. Based on the an-\nticipated low temperature ferrimagnetic ordering of this6\nFIG. 5. Scaled inverse susceptibility data as a function of scaled\ntemperature as described by equation are shown. The dashed\nline indicates simplified (ferromagnetic) Curie-Weiss paramag-\nnetism. Negative deviations in all samples reflect uncompen-\nsated interactions and suggest ferrimagnetic interactions.\nFIG. 6. The inverse susceptibility is plotted versus temperature\nfor each composition, xin Co 1−xMnxCr2O4. The lighter mark-\ners indicate the model equation fit to the data, Equation 4, for\ntwo magnetic sublattices. The fitted parameters of this equa-\ntion are supplied in Table III.\nspinel system, these lattices should each correspond to\neither tetrahedral sites, occupied by Mn or Co atoms, or\noctahedral sites, occupied by Cr atoms. Observing that\nλBBremains relatively constant versus Mn composition\nxcompared to the other values of λ, we anticipate that\ntheB-sublattice corresponds to the octahedral (Cr) sub-\nlattice, and the A-sublattice to the tetrahedral (Mn, Co)\nFIG. 7. Field-dependent isothermal magnetization of\nCo1−xMnxCr2O4were taken at 30 K for all samples to avoid\nthe spiral magnetic ordering region and reflect the magnetiza-\ntion in the collinear ferrimagnetic regime. Increasing Mn con-\ntent increases the saturation magnetization.\nTABLE IV. Spontaneous ( M0) and Saturation Magnetization\n(Msat) Data for Co 1−xMnxCr2O4.\nx M 0(µB/f.u.)aMsat(µB/f.u., 2 T) Msat(µB/f.u., 5 T)\n0.00 – 0.14 0.21\n0.25 0.32 0.38 0.44\n0.50 0.60 0.68 0.72\n0.75 0.88 0.92 0.99\n1.00 0.97 1.06 1.17\naExtrapolated values are from 30 K isotherms for each sam-\nple\nsublattice.\nFigure 7 shows field-dependent magnetization data of\neach sample at 30 K. This temperature was chosen to\navoid the spiral magnetic ordering region and reflect the\nmagnetization of in the collinear ferrimagnetic region in\neach sample. The x= 0.00 end has considerable coerciv-\nity as compared to the other compositions, and this coer-\ncivity has been seen to increase with decreasing temper-\nature (from 75 K to 3 K) [45]. Since none of the samples\nfully saturates at the highest applied field of 7 T, Table IV\nshows spontaneous magnetization ( M0) as extrapolated\nfrom Arrott-Belov plots (which could not be determined\nforx= 0.00) as well as saturation magnetization ( Msat)\nvalues at fields of 2 T and 5 T. Overall, this table shows\nthat magnetization values increase as Mn content is in-\ncreased. Also, none of the individual magnetization val-\nues are very large, again indicating that Msatis not al-\nways a useful indicator of magnetocaloric viability, dis-\ncussed further below.7\nFIG. 8. (a) Calculated ∆SMcurves for each sample for field\nchanges of 0 T to 2 T (dashed lines) and 0 T to 5 T (solid lines)\nin the species shows an increasing peak ∆SMand decreasing\npeak temperature with increasing Mn content. (b) Increasing\nRC with increasing Mn content indicates that the overall per-\nformance of the magnetocaloric increases with Mn content.\nC. Magnetocaloric Properties\nThe magnitude of the magnetocaloric effect can be de-\nrived from temperature-dependent magnetization mea-\nsurements at varying fields which are then derived and\nintegrated over field to yield ∆SM, depicted in Fig-\nure 8(a). The absolute value of ∆SMincreases as Mn\ncontent increases, and the temperature occurrence of\nthe peak value decreases as expected from previously\ndiscussed temperature-dependent magnetization mea-\nsurements. This series spans a wide range of peak\n∆SMvalues, from −1.67 J kg−1K−1forx= 0.00 to\n−5.63 J kg−1K−1forx= 1.00. It is difficult to visu-\nally determine whether or not ∆SMpeaks broaden or\nsharpen throughout the series, and this can affect overall\nmagnetocaloric performance. Thus we employ a more\nmathematically rigorous method of understanding mag-\nnetocaloric performance, the refrigerant capacity. This\ninvolves integrating the area under the ∆SMcurve using\nthe full width at half maximum of each curve as temper-\nature limits. These results are shown in Figure 8(b) and\nindicate that the overall magnetocaloric performance in\nthe series does improve as Mn content is increased since\nthe refrigerant capacity increases nearly 5-fold through-\nout the series.\nOften, Msatis used as a proxy for understanding the\nmagnitude of the magnetocaloric effect, however in this\ncase it is clear that other factors – such as magne-\ntostructural coupling – are at play. For example, the\nsaturation magnetization (Figure 7) of the x= 0.75 andx= 1.00 samples near 7 T is approximately 1.12 µBand\n1.25µB, respectively. However, the difference in their\npeak ∆SMvalues nearly doubles from 3.51 J kg−1K−1to\n5.67 J kg−1K−1, respectively, indicating that in this case\nMsatis not an appropriate metric by which to understand\nmagnetocaloric performance as quantified by ∆SM.\nD. First-Principles Calculations\nThe experimental measurements of ∆SMare com-\npared to first-principles computational proxies for the\n∆SMfigure of merit for magnetocaloric materials. Bo-\ncarsly et al. [3] and others have demonstrated the pre-\ndictive power of computational magnetic deformation\nproxy, ΣM[46]. The “magnetic deformation proxy” pro-\nvides a computationally inexpensive indicator of a large\nchange in magnetocaloric isothermal change in entropy,\n∆SMabove a threshold of ΣM>1.5%.ΣMis a mea-\nsure of the deformation strain between the magnetic and\nnonmagnetic structures of the material. The reason for\nthe strong correlation between ∆SMandΣMcan be ex-\nplained, in a general sense, by the role that coupled mag-\nnetic and structural degrees of freedom play in promot-\ning phase transitions with a large change in entropy, and\neven latent heat in the case of a first-order magnetostruc-\ntural phase transition.\nIn this study, the magnetic deformation proxy ΣMwas\ncomputed for a representative set of structures for each\nMn composition xin Mn xCo1−xCr2O4. In addition to\nΣM, previous studies have identified that the total mag-\nnetization at T= 0 K, M0\nnet, calculated from DFT, also\ncorrelates well with ∆SM, although to a lesser degree\nthan ΣMfor the materials database analyzed in the orig-\ninal study that tested the correlation of ∆SMwith differ-\nent first-principles indicators [3]. In this context, M0\nnet\nis defined as the “net magnetization” within the unit\ncell and can be computed from the difference between\nspin-up and spin-down electron densities produced by\nDFT calculations. Bocarsly and others have shown that\nfor a representative sample of ferrimagnetic and anti-\nferromagnetic materials, the product of the deforma-\ntion proxy with the saturation magnetization, M0\nnetΣM,\nin many cases provides a greater indication of a large\nchange in entropy, ∆SM, compared to either M0\nnetorΣM\nindividually [47].\nFigure 9(a) includes the distribution of ΣMvalues at\neach composition. Due to the large spread of values as-\nsociated with a different crystal structure, we are unable\nto make a conclusion regarding the trend of the defor-\nmation values versus composition. The reported stan-\ndard deviation of energy values was found to be no more\nthan 0.4 meV, normalized by the number of atoms in the\nunit cell. The maximum difference in average energy\nper atom output by VASP was 1 meV for each ferrimag-\nnetic configuration for both CoCr 2O4and MnCr 2O4. This\nsmall difference in energy values may be indicative of the\nfrustrated nature of magnetism that has been confirmed8\nFIG. 9. (a) The deformation proxy, ΣM, versus Mn composi-\ntion. (b) The computed net magnetization M0\nnet(µBper f.u.)\nscaled by the deformation proxy, M0\nnetΣM, versus Mn com-\nposition. (c) The experimentally measured saturation magne-\ntization Msat(µBper f.u.) scaled by the deformation proxy,\nMsatΣM, versus Mn composition. Colored data-points indicate\nthe individual calculations for each enumerated structure. Each\nblack X indicates the mean deformation value at each compo-\nsition. Over x=0.25, 0.5, and 0.75, the maximum standard\ndeviation in energy per atom values at each composition com-\nputed by VASP was 0.4 meV.\nexperimentally in in CoCr 2O4and MnCr 2O4[17, 20, 29–\n32] and theoretically using the classical theory of mag-\nnetic ground-states [48, 49].\nIn addition to ΣM, Figure 9(b) shows the trend of the\nproduct of the computed net magnetization and the de-\nformation proxy M0\nnetΣMversus composition. M0\nnetis\nthe net magnetization of the simulation cell output by\nVASP. These M0\nnetvalues are in units of Bohr magnetons\n(µB), normalized by the formula unit (f.u.). In this case,\nthere is a clear downward trend, which would be ex-\npected for the collinear configuration that we chose to\ncompute. M0\nnetΣMdecreases with x, which is opposite\nto the trend that was observed from the experimental\nmeasurements.\nCompared to the DFT derived M0\nnetΣM, Figure 9(c)\nprovides a plot of the experimentally measured Msat\nscaled by ΣMat each composition. These Msatvalues are\nalso reported in Bohr magnetons, normalized by the for-\nmula unit. The upward trend with composition is due to\nthe dominant behavior of the experimentally measured\nMsatvalues, which increase with Mn composition.\nFrom the results of the computed ΣMandM0\nnetΣM\nversus Mn composition, it is clear that there is a large\ndisagreement with the experimentally observed behav-ior. This is possibly due to the fact that ΣMsimply quan-\ntifies the degree of magnetostructural coupling in a ma-\nterial, without treating the lattice and spin contributions\nto∆Sof the phase transition explicitly. The opposite\ntrend of M0\nnetwith the experimental saturation magneti-\nzation, Msat, is stark, but can be described by the incon-\nsistencies between MsatandM0\nnet. The two are strictly\ncomparable only at zero temperature. Even then, M0\nnet\nis calculated under zero applied field, where Msatis not,\nby definition. The saturation magnetization and effec-\ntive moment µeffare inherently temperature-dependent\nquantities. For example, the µeffis derived from the high\ntemperature paramagnetic decay of the susceptibility ac-\ncording to the Curie-Weiss law. For this reason, Monte\nCarlo methods or a mean-field description is necessary in\norder to connect the DFT derived parameters to the ex-\nperimentally measured thermodynamic quantities such\nasMsatandµeff. In addition, it has been shown that M0\nnet\ndoesn’t correlate well with ∆SMfor a nonzero applied\nmagnetic field [3].\nIn future studies of this spinel system, thermodynamic\nquantities versus temperature will be computed using a\nspin-lattice coupling Hamiltonian with parameters de-\nrived from density functional theory calculations. This\nmodel will allow for the quantification of the change in\nentropy due to structural, magnetic, and their coupling\nusing Monte Carlo methods that directly quantify spin\nand phonon contributions to ∆SM.\nIII. CONCLUSION\nWe have studied the solid solution Co 1−xMnxCr2O4as\na candidate for magnetocaloric applications by synthesiz-\ning using standard solid-state synthesis. Synchrotron X-\nray diffraction measurements revealed a complete solid\nsolution of spinel samples. Magnetic measurements be-\ntween 2 K and 390 K show the spiral spin-state at low\ntemperatures transitioning to collinear ferrimagnetism\nat moderate temperatures then to paramagnetism at\nhigh temperatures, and the transition temperatures for\neach phase decrease monotonically as Mn content is in-\ncreased. The maximum magnetic entropy change is also\nfound to increase monotonically, from −1.67 J kg−1K−1\nfor the Co end member to −5.63 J kg−1K−1for the Mn\nend member (for a field change of 0 T to 5 T). Overall,\nthe tunability of this series and robust peak ∆SMvalues\nin the range of 40 K to 75 K make this series attractive for\ncascaded hydrogen liquefaction systems relying on active\nmagnetic refrigeration.\nThe effect of variation in Mn/Co composition on max-\nimum magnetic entropy difference, ∆SM, cannot be ex-\nplained by DFT-computed magnetic deformation proxy\nvalues, ΣM[3], but are more closely related to the trends\ninMsatacross composition. This is notthe case for\nM0\nnetfrom DFT, therefore, we suggest that there are cru-\ncial thermodynamic mechanisms that underlie the wide\nrange of ∆SMvalues across the series. We argue that9\nthis finite-temperature behavior cannot be resolved at\nthe level of DFT alone [50], but requires finite tempera-\nture modeling approaches, such as Monte Carlo, or even\nWeiss mean-field models.\nIV. ACKNOWLEDGEMENTS\nThis work and facilities employed here were supported\nby the National Science Foundation through the MR-\nSEC Program NSF DMR 1720256 (IRG-1). The UCSB\nMRSEC is a member of the Materials Research Facil-\nities Network (www.mrfn.org). Use of the Advanced\nPhoton Source at Argonne National Laboratory was sup-\nported by the U. S. Department of Energy, Office of Sci-\nence, Office of Basic Energy Sciences, under Contract No.\nDE-AC02-06CH11357. G.M. acknowledges support from\nthe Department of Energy Computational Science Gradu-\nate Fellowship (DOE CSGF) under grant DE-SC0020347.\nM.K.H acknowledges support by the U.S. Department of\nEnergy, Office of Science, Office of Basic Energy Sciences,\nMaterials Sciences and Engineering Division under Con-\ntract No. DE-AC02-05-CH11231 (Materials Project pro-\ngram KC23MP).\nV. APPENDIX\nA. Curie-Weiss Law for a Ferrimagnet\nThe Weiss molecular field theory for magnetism [51]\nis based on a mean-field approximation, and thereforeneglects fluctuations that influence behavior below and\nnear to the critical temperature. However, Curie-Weiss\ntheory is useful for studying the behavior of magnets at\ntemperatures above their paramagnetic transition tem-\nperature. This theory was originally formulated for\nthe ferromagnetic to paramagnetic order-disorder phase\ntransition, however, it is possible to generalize this theory\nto the study of ferrimagnetism and antiferromagnetism,\nby treating the magnetic configuration as an antiparallel\ncoupled arrangement of parallel (ferromagnetic) lattices.\nThis first section follows the reasoning presented in the\ntext of Kittel [44].\nFor the sake of clarity, we will start from the system of\nequations for the Weiss molecular field coupling between\nthe magnetization of A and B sublattices in a ferrimag-\nnet, assuming a negative - AFM exchange between A and\nB sites,\nMA=CA\nT(B0−λAAMA−λABMB)\nMB=CB\nT(B0−λABMA−λBBMB) (1)\nwhere MA&MBandCA&CBare the net magneti-\nzation and Curie-Weiss constants of the A and B sublat-\ntices, respectively. B0represents the applied field, and\nλAA,λBB, &λABare the Weiss field constants. These\nWeiss field constants ( λ’s) capture the net exchange in-\nteraction within and between the two sublattices. The\nlatter of which is symmetric under both directions of the\nexchange (A →B and vice versa). Equations 1 can be\nstated in matrix-vector form as Equation 2.\n(T·I+W)\"\nMA\nMB#\n= \nT\"\n1 0\n0 1#\n+\"\nλAACAλABCA\nλABCBλBBCB#!\"\nMA\nMB#\n=B0\"\nCA\nCB#\n(2)\nBy inspecting Equation 2, we see that for B0= 0, a\nnonzero solution for MA&MBexists only if [44]\ndet{T·I+W}= 0 (3)\nTherefore, the critical temperature of the system will cor-\nrespond to the opposite sign of an eigenvalue of W,\nspecifically TC= max {ηi[−W]}, where ηi[A]are the\neigenvalues of a matrix A. From Equation 2, we arrive\nat a generalized Curie-Weiss law, for T > T C\nχ=∂(Pn\ni=1Mi)\n∂B0=1T[T·I+W]−1C (4)\nFor the two lattice case ( n= 2),Pn\ni=1Mi=MA+MB,\n1T= [1 1] , andCT= [CACB].1. Uncertainty quantification\nIn order to fit Equation 4 to experimental data,\nwe used scipy.optimize.curve_fit [52]. This op-\ntimization routine utilizes the Jacobian of the objec-\ntive function in the minimization procedure, as well as\nfor calculating the propagation of uncertainty. If one\ndoes not specify the jacobian argument explicitly, then\nscipy.optimize.curve_fit computes derivatives using\nfinite differences (FD). If we use the default FD scheme,\nthe uncertainty values of the parameters are estimated to\nbe at least 104. However, we find improvement if we sup-\nply the analytical derivative presented below in Equation10\n5,\n∂\n∂φχ=−1TW−1\nT∂W\n∂φW−1\nTC+1TW−1\nT∂C\n∂φ,\nwhere WT= [T·I+W]. (5)\nIn this expression, φrepresents a parameter of the model\nfunction, φ∈ {CA, CB, λAA, λBB, λAB}. If we provide\nthe analytical Jacobian, all uncertainties fell below 1.\nThis improvement is likely due to numerical artifacts that\narise from using the FD scheme near to the singularity at\nthe critical temperature.\n2. Simplified exchange\nWe can simplify the Equation 2 by setting λAA=\nλBB= 0, and letting λ=λAB. This simplification yields\nthe following result for susceptibility above the critical\ntemperature TC[44]:\nχ=∂(MA+MB)\n∂B0=(CA+CB)T−2λCACB\nT2−T2\nC(6)\nAnother consequence of this solution is that TC=\nλ√CACB[44].\n3. High temperature limit\nContinuing from Equation 6, we can obtain the following\nfactorization\nχ=∂(MA+MB)\n∂B0\n=CA+CB\nT+TC·T−2λCACB\nCA+CB\nT−TC(7)\n=CA+CB\nT+TC·T−ρ·TC\nT−TC(8)\nwhere ρis the ratio between the geometric and arith-\nmetic mean between CAandCB,\nρ=C1/2\nAC1/2\nB\n1\n2(CA+CB). (9)\nTherefore, ρ= 1ifCA=CB. If we examine the high-\ntemperature limit in which T >> T C, we can make the\nfollowing simplification\nlim\nT→∞T−ρ·TC\nT−TC= 1 (10)\nUnder approximation, we arrive at a more compact form\nof the CW law for a ferrimagnet\nχ≈CA+CB\nT+TC, (11)which is applicable for temperatures significantly larger\nthan the critical temperature. An illustration of this is\nshown in Figure 10.\nFIG. 10. Comparison between the Curie-Weiss (CW) expres-\nsions for χ, where the full expression (Equation 6) is plotted\nagainst the approximate high-temperature CW law (Equation\n11). The parameters, λ,CA, and CBare arbitrary dimension-\nless constants that are chosen to justify this approximation.\n4. Relation to effective moment\nThe effective moment can be expressed in terms of the\nCurie-Weiss law using Weiss mean-field theory [44]\nχ=Np2µ2\nb\n3kB(T−TC)=Nµ eff2\n3kB(T−TC)=C\nT−TC\nµeff2=3kB\nNC (12)\nNis the number of atoms per unit volume [44].\nCombining Equations 11 and 12, we arrive at the fol-\nlowing relationship between the “net” effective moment\nand the effective moment for each sublattice (A & B):\nC≈CA+CB=Naµ2\na\n3kB+Nbµ2\nb\n3kB\nµeff2≈Na\nNµ2\na+Nb\nNµ2\nb (13)\nThis relationship allows us to approximate, at T >\nTC, the effective magnetic moment of a sample from a\nweighted sum of the µeffof the constituent magnetic sub-\nlattices.11\n5. 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Bright, et al. , Nature methods 17, 261\n(2020)."    },    {        "title": "2211.05028v1.Magnetization_reversal_through_an_antiferromagnetic_state.pdf",        "content": "Magnetization reversal through an antiferromagnetic state\nS. Ghara,1,\u0003E. Barts,2K. Vasin,1, 3D. Kamenskyi,1L. Prodan,1\nV. Tsurkan,1, 4I. K\u0013 ezsm\u0013 arki,1M. Mostovoy,2and J. Deisenhofer1\n1Experimentalphysik V, Center for Electronic Correlations and Magnetism,\nInstitute for Physics, University of Augsburg, D-86135 Augsburg, Germany\n2Zernike Institute for Advanced Materials, University of Groningen,\nNijenborgh 4, 9747 AG Groningen, The Netherlands\n3Institute for Physics, Kazan (Volga region) Federal University, 420008 Kazan, Russia\n4Institute of Applied Physics, MD-2028 Chi\u0018 sin\u0015 au, Republic of Moldova\nMagnetization reversal in ferro- and ferrimagnets is a well-known archetype of non-equilibrium\nprocesses, where the volume fractions of the oppositely magnetized domains vary and perfectly\ncompensate each other at the coercive magnetic \feld. Here, we report on a fundamentally new\npathway for magnetization reversal that is mediated by an antiferromagnetic state. Consequently,\nan atomic-scale compensation of the magnetization is realized at the coercive \feld, instead of the\nmesoscopic or macroscopic domain cancellation in canonical reversal processes. We demonstrate\nthis unusual magnetization reversal on the Zn-doped polar magnet Fe 2Mo3O8. Hidden behind the\nconventional ferrimagnetic hysteresis loop, the surprising emergence of the antiferromagnetic phase\nat the coercive \felds is disclosed by a sharp peak in the \feld-dependence of the electric polarization.\nIn addition, at the magnetization reversal our THz spectroscopy studies reveal the reappearance\nof the magnon mode that is only present in the pristine antiferromagnetic state. According to\nour microscopic calculations, this unusual process is governed by the dominant intralayer coupling,\nstrong easy-axis anisotropy and spin \ructuations, which result in a complex interplay between the\nferrimagnetic and antiferromagnetic phases. Such antiferro-state-mediated reversal processes o\u000ber\nnovel concepts for magnetization control, and may also emerge for other ferroic orders.\nMagnetization reversal processes in ferro- and ferri-\nmagnets are essential in many of their applications. The\non-demand control of domains upon magnetization re-\nversal is a hotbed of new developments for spintronics1{4.\nThe mutual coupling of magnetization and polarization\nin multiferroic materials can be exploited to reverse mag-\nnetization by electric \feld and for electric read-out of\nthe magnetic state5,6. In this paper we report an uncon-\nventional magnetization reversal observed in the \feld-\ninduced ferrimagnetic (FiM) state of lightly Zn doped\nFe2Mo3O8. We \fnd that reversing the magnetization re-\nquires the resurrection of the antiferromagnetic (AFM)\nstate, which is evidenced by an increase of the static po-\nlarization and the reemergence of a THz excitation char-\nacteristic of the AFM state. Our theoretical calculations\nshow that the spin ordering in Fe 2Mo3O8is strongly af-\nfected by the presence of two magnetic Fe sites with dif-\nferent magnetic moments and magnitudes of magnetic\nanisotropy. In combination with a strong intralayer and\na weak interlayer exchange coupling of the honeycomb\nlayers the remarkable magnetization reversal through the\nAFM state in Fe 2\u0000xZnxMo3O8withx= 0:14 can be re-\nproduced by our simulations.\nRecently, the honeycomb antiferromagnets A2Mo3O8\n(A=Mn,Fe,Co,Ni,Zn) have emerged as a versatile mate-\nrial class with a hexagonal structure in the polar space\ngroupP63mc7{15. These compounds exhibit optical\nmagnetoelectric e\u000bects such as directional dichroism16,17\nand giant thermal Hall e\u000bects18,19, and the control of\n\u0003somnath.ghara@physik.uni-augsburg.demagnetization and polarization can be achieved in mag-\nnetic \felds of a few Tesla20,21or even by ultrafast mod-\nulation via laser pulses22. Particular attention has been\ndrawn to Fe 2Mo3O8with a N\u0013 eel temperature TN=60 K,\nbelow which a collinear AFM spin order of the magnetic\nFe2+ions coaligns with the polarization along the crys-\ntallographic c-axis as shown in Fig. 1a. The Fe2+ions\noccupy tetrahedrally coordinated sites (A-site) and octa-\nhedrally coordinated ones (B-site). Within each of the\nhexagonal layers (Fig. 1c) the inequality of the orbital\ncontributions to the magnetic moments of A- and B-site\nions with spin S= 2 results in \fnite magnetizations M1;2\nof each layer with opposite sign for adjacent layers (green\narrows in Fig. 1a). Such an AFM spin con\fguration can\nbe characterized by the AFM N\u0013 eel vectors for each sub-\nlattice of Fe2+, e.g. LA=1\n2(M1A\u0000M2A) with sub-\nlattice magnetizations M1AandM2Afor the two di\u000ber-\nent A-sites. The AFM order parameter is then given by\nL=LA+LB=1\n2(M1\u0000M2). Upon application of a mag-\nnetic \feld along the c-axis, a transition to a FiM phase\noccurs, where the two anti-aligned sublattices are formed\nby A-site and B-site ions, respectively, corresponding to\na \rip of the layer magnetization in every second layer, as\nshown in Fig. 1b20,21. This complies with a clearly dom-\ninating AFM intralayer exchange coupling Jk(Fig. 1c)\nin comparison with the weaker interlayer couplings JAA\nandJBB(Fig. 1d). The order parameter of the FiM state\ncan be expressed as the sum of the two sublattice con-\ntributions MA;Bor of the layer magnetizations M1;2as\nM=MA+MB=M1+M2.\nThe FiM state was shown to exhibit a linear magneto-\nelectric e\u000bect21and can be stabilized by substituting FearXiv:2211.05028v1  [cond-mat.str-el]  9 Nov 20222\n04008001200\nP\n (µC/m2)ab c d\neA\nBJ||S\n1AS 1BS2BS 2AJ\nAAJABJBBM1M\n2M2M10\n25507510001020χ'(emu/mol)T\n (K)TN ~ 53 KF\ne(B)F e(A)MoOz x\ny \nFIG. 1.jCrystal structure and magnetic order in Fe 2\u0000xZnxMo3O8. a, b , Unit cell of Fe 2Mo3O8in the AFM state and\nthe FiM state, with tetrahedral ( A) and octahedral ( B) sites shown in red and blue, respectively. The greeen arrows indicate\nthe overall magnetization of each layer. c, Hexagonal layer formed by the Fe sites with dominant AFM exchange coupling\nJk.d, Exchange couplings JAA,JBBandJABbetween the Fe ions at AandBsites of adjacent layers in the AFM state.\ne, Temperature dependence of the real part of the magnetic acsusceptibility \u001f0(solid line) and polarization P(open symbols)\nin Fe 1:86Zn0:14Mo3O8.\nby non-magnetic Zn, which preferably occupies the tetra-\nhedral A-sites8,9,21,23,24. The persistence of the \feld-\ninduced FiM state in Zn-doped Fe 2Mo3O8upon decreas-\ning and reversing the \feld was reported beforehand21,25,\nbut the origin of this metastable state has not been ad-\ndressed previously. The N\u0013 eel temperature in the system\nFe1:86Zn0:14Mo3O8is reduced to 53 K as indicated by the\npeak in the magnetic susceptibility shown in Fig. 1e,\nwhile the polarization shows an increase at the mag-\nnetic ordering and saturation-like behavior in the AFM\nphase comparable to pure Fe 2Mo3O821. The FiM state\nat this Zn concentration persists as a metastable state\nin zero magnetic \feld below about 40 K, i.e. the rema-\nnent magnetization during hysteretic cycling remains \f-\nnite, as seen in Figs. 2a and 2b25. By comparing the\nhysteretic cycle of magnetization and polarization mea-\nsurements in external magnetic \felds in this temperature\nrange, we \fnd a strong increase in polarization, whenever\nthe magnetization changes sign in the vicinity of the co-\nercive \felds. We ascribe this increase in polarization to\nthe reoccurrence of the AFM ground state, which has a\nlarger polarization than the FiM state20,21, during the\nmagnetization reversal of the FiM state. This assign-\nment is supported by the reoccurrence of the character-\nistic low-energy THz excitation of the AFM phase in the\nsame \feld range. Our microscopic theoretical approach\ncan explain the origin of the FiM metastable state and\nthe resurrection of the AFM phase upon magnetization\nreversal.\nAn unusual magnetization reversal\nIn this section, we discuss how the reoccurrence of the\nAFM state at the magnetization reversal is re\rected in\nthe polarization. In Figs. 2a and 2b we show the mag-\nnetizationM(H) (dashed lines) and polarization P(H)\n(solid lines) measured as a function of the external mag-\nnetic \feld along the c-axis atT= 20 K and T= 13 K,\nrespectively. The magnetization curves starting at zeromagnetic \feld (black dashed lines) in the pristine AFM\nstate show a transition to the FiM state in the shaded\n\feld range, and upon reversing the magnetic \feld, a typi-\ncal FiM-like hysteresis shows up with coercive \felds of the\norder of the critical \felds of the AFM-to-FiM transition.\nThis is in agreement with previously reported data25and\nsamples with similar Zn concentrations21. The polariza-\ntion shows a strong decrease upon the transition from the\nAFM to the FiM state, and a linear magnetoelectric ef-\nfect occurs in the FiM state (shown clearly in Supplemen-\ntary Fig. 1), both observations in line with literature21.\nIntriguingly, whenever the magnetization starts to devi-\nate from its saturation values \u0006MSduring the cycling, a\nstrong increase in polarization occurs, reaching its maxi-\nmum at the coercive \feld (see e.g. shaded area in Fig. 2a).\nSuch an unusual behavior has not been observed before-\nhand in Zn-doped Fe 2Mo3O8or other FiM multiferroics\nexhibiting a linear magnetoelectric e\u000bect26.\nIt is important to note that both the critical and co-\nercive \felds and the peak height of the polarization de-\npend on temperature, as can be seen by the compari-\nson of Figs. 2a and 2b. At 20 K the maximal value of\nthe polarization at the coercive \feld is about half of the\none in the pristine AFM phase (Fig. 2a), but at 13 K\nthe value is already considerably reduced. At 30 K the\nmaximal polarization value at the coercive \feld almost\nreaches the value of the pristine AFM phase (see Sup-\nplementary Fig. 2). We regard this as evidence that\nin the vicinity of the coercive \felds, when the mono-\ndomain FiM phase usually turns to an equal share of\nup and down domains with volume fractions x\fm\"(H;T)\nandx\fm#(H;T), respectively, a signi\fcant fraction of\nthe sample volume, denoted as xafm(H;T), exhibits the\nproperties of the pristine AFM state with high polariza-\ntion values. The AFM fraction assisting the magnetiza-\ntion reversal emerges now as a metastable state. Conse-\nquently, we analyze the magnetic-\feld dependent mag-\nnetization and polarization data assuming that the en-3\n-6- 30 3 6 -1.0-0.50.00.51.0M (µB/f.u.)µ\n0H (T)20 K a4\n008001200P\n (µC/m2)-\n6- 30 3 6 -1.0-0.50.00.51.0M (µB/f.u.)µ\n0H (T)13 KP\n0a\nfmP\n0f\nimMS4\n008001200P\n (µC/m2)-\n6- 30 3 6 0.00.51.0xafmµ\n0H (T)20 Kcd bx\nmaxa\nfm0\n1 02 03 04 00.00.51.0xmaxa\nfmT\n (K)\nFIG. 2.jSignatures of the resurrection of the AFM state in magnetization and polarization. a,b , Magnetic\n\feld-dependent magnetization M(dashed lines) and polarization P(solid lines) along the caxis at 20 K and 13 K, respectively.\nc, Magnetic \feld-dependent antiferromagnetic volume fraction ( xafm) at 20 K extracted from magnetization and polarization\ndata using Eq. (2). The vertical shading in aandcindicates the \feld range, where the pristine AFM state reappears upon\nmagnetization reversal. d, Temperature dependence of the maximum values of xafm. The solid line is a guide to the eye.\ntire sample volume is distributed between three fractions\nonly, i.e.xafm+x\fm\"+x\fm#= 1. The magnetization is\nthen solely determined by the FiM volume fractions as\nM(H) =MS(x\fm\"\u0000x\fm#), while the polarization bears\nthe contribution of all three magnetic volume fractions,\nP(H) =xafmP0\nafm+P0\n\fm(x\fm\"+x\fm#)+\u000bHM(H)\nMS:(1)\nHere,P0\nafmdenotes the polarization of the pristine anti-\nferromagnet in zero magnetic \feld, P0\n\fmthe correspond-\ning polarization of the FiM phase upon lowering the \feld\nto zero after reaching the mono-domain FiM states with\nM(H) =\u0006MS(see Fig. 2b). The only parameter, which\nhad to be determined by \ftting the P(H) curves, is the\nmagnetoelectric susceptibility coe\u000ecient \u000b, which was\nderived in the corresponding linear regimes (a possible\ncontribution/H2was found to be negligible, see Sup-\nplementary note 1). With this approach we extracted the\nAFM volume fraction given by,\nxafm(H) =1\nP0\nafm\u0000P0\n\fm\u0014\nP(H)\u0000P0\n\fm\u0000\u000bHM(H)\nMS\u0015\n:\n(2)\nThe result for T= 20 K is shown in Fig. 2c. The values\nofxafm(H) yield symmetric peaks centered at the coer-\ncive \felds. The corresponding maximum values xmax\nafmare\nshown in Fig. 2d for all investigated temperatures. It\ndecreases linearly with decreasing temperature and ex-\ntrapolates to zero at around 10 K. This is in agreement\nwith the absence of any macroscopic polarization peak at\nthe magnetization reversal below 10 K.To corroborate the above scenario derived from dc\nmagnetization and polarization measurements, we will\ndiscuss in the following the dynamic \fngerprints of the\nmagnetic volume fractions in the THz frequency regime.\nProbing magnetic phases via THz spectroscopy\nThe resurrection of the pristine AFM phase and its\ncoexistence with the FiM fraction upon magnetization\nreversal can also be revealed in the THz spectra by inves-\ntigating the \feld evolution of the characteristic elemen-\ntary excitations of the two magnetic phases, which were\npreviously identi\fed25. In Fig. 3b we show the THz ab-\nsorption spectra (red colors) at 25 K for light polarisation\nE!kafor several magnetic \felds during the virgin mag-\nnetization curve with the external magnetic \feld Hkc\nas indicated by the same colors in the M-H-diagram in\nFig. 3a. The spectrum of the AFM ground state at zero\n\feld and the one of the saturated FiM state in a \feld of\n2 T serve as benchmarks for the virgin AFM and mono-\ndomain FiM states with xafm= 1 andx\fm\"= 1, respec-\ntively. The absorption spectra shown in Fig. 3c corre-\nspond to reversed magnetic \felds of the hysteretic cycle\n(green curves) and show the evolution of the magnetic\nphases upon approaching the magnetization reversal at\nabout\u00001 T and reaching again a fully saturated FiM\nstate at\u00002 T withx\fm#= 1. The spectrum of the pris-\ntine AFM state is characterized by one distinct spectral\nfeature in this con\fguration - the narrow electric-dipole\nactive absorption mode at 44 cm\u00001(AFM mode) shown\nin Fig. 3b, which was identi\fed previously as a clear \fn-\ngerprint of the AFM state both in pure Fe 2Mo3O827and4\n-1\n-2 20\n01M   /f.u.)\nH(T)\n0 10 20 300.00 51.0\nT(K)P(H)\nAFM\nFiMxafmmode\nmode2030405060708003060901201501800\n21\nwavenumber(cm-1)\nabsorptioncoefficient(cm-1)H (T)\n203040506070800306090120150180-2-1-0.5\nwavenumber(cm-1)\nabsorptioncoefficient(cm-1)H (T)\nAFM \nmodea\ncb\nd\nFiM\nmodeAFM \nmode\nFiM\nmode\n0\nμ0μ0\nA\nA\nBBC\nD\nC\nDμ0(μB\nFIG. 3.jDemonstration of the reappearance of the AFM state upon magnetization reversal via THz spec-\ntroscopy. a , Magnetic \feld-dependent magnetization M(H) at 25 K. Selected segments (A-B and C-D) of the magnetization\ncurve are highlighted by colors to indicate the \feld regions of the THz data shown in bandc.b, THz absorption spectra at 25 K\nrecorded at di\u000berent magnetic \felds ( Hkc) ranging between the segment A-B of panel a.c, THz absorption spectra recorded\nat di\u000berent magnetic \felds ranging between the segment C-D of panel a. The THz spectrum indicated by the symbol \u0002is\nrecorded at the coercive \feld (see panel a) .d, Temperature dependence of the maximum values of xafmat the magnetization\nreversal obtained directly from the integrated intensity of the AFM mode and the FiM mode (via 1 \u0000x\fm). For comparison,\nxafmvalues obtained from the polarization analysis are reproduced in d. The solid line is a guide to the eye.\nin Fe 1:86Zn0:14Mo3O825. The mode at 44 cm\u00001is clearly\nabsent in the spectra of the saturated FiM states for\nH=\u00062 T, where only one excitation at about 83 cm\u00001\n(FiM mode) is observed as a characteristic \fngerprint of\nthe FiM state25.\nUpon lowering and reversing the external magnetic\n\feld from the saturated FiM state at 2 T no signi\fcant\nchanges in the absorption spectra occur as long as the\nconditionM(H) =MSholds (spectra not shown). In-\ntriguingly, on approaching magnetization reversal an ex-\ncitation at the eigenfrequency of the characteristic AFM\nmode at 44 cm\u00001emerges, reaches a maximum in inten-\nsity at around Hc=\u00001 T, and decreases in intensity\nagain forjHj> H cas shown in Fig. 3c. Finally, the\nmode disappears again when reaching magnetization sat-\nuration with M(H) =\u0000MSon approaching a \feld of\n\u00002 T. Based on the comparison with the properties of\nthe spectrum in the pristine AFM state, we identify this\nemerging mode with the AFM mode: eigenfrequency and\nlinewidth of the two modes coincide and they obey the\nsame electric-dipole selection rule E!ka(see Supple-\nmentary Fig 3). In addition, the intensity of the FiM\nmode clearly decreases with increasing intensity of AFM\nmode during magnetization reversal, showing that theTHz spectra provide direct information on the coexis-\ntence of the AFM and FiM states in this magnetic \feld\nrange.\nWe used the integrated intensities of AFM and FiM\nmodes for all spectra to extract the AFM volume fraction\nxafmand the FiM one x\fmby normalizing the intensity\nvalues to the intensity of the modes in the pristine AFM\nand the saturated FiM states, respectively (see Supple-\nmentary note 2 for details). Note that our THz probe\ndoes not distinguish up and down FiM states, therefore\nx\fm=x\fm\"+x\fm#. The values of xafmobtained di-\nrectly from the AFM mode and using xafm= 1\u0000x\fm\nfrom the FiM mode are shown in Fig. 3d for all measured\ntemperatures, together with the values of xafmfrom the\nabove analysis of the magnetization and polarization us-\ning Eq. (2). The agreement of the xafmvalues from the\ndynamic THz probe and the static polarization and mag-\nnetization values is very good and con\frms the unusual\nresurrection of the AFM ground state as a metastable\nstate during the magnetization reversal of the FiM state.\nHaving established the magnetization reversal through\nthe AFM state using our experimental observations, we\nwill now discuss a theoretical approach that explains this\nscenario from a microscopic point of view.5\n3.5 T 3.9 T 5.1 T 5.5 T 6.1 T 7.4 T 8.7 T\n20\n020246810cc\nba\nH (T) || cFiM, M<0 AFM, L<0 AFM, L>0 FiM, M>0a\nb\nFIG. 4.jSimulation of spin-state evolution upon magnetization reversal. a,b , Magnetic \feld dependence of\ncalculated magnetization M(dashed lines) and polarization P(solid circles) at 25 K and 35 K, respectively. c, Snapshots of\nspin con\fgurations at di\u000berent magnetic \felds at 25 K during a magnetization reversal process from a FiM down-domain state\nto a FiM up-domain state. Each plaquette corresponds to a unit cell of Fe 1:86Zn0:14Mo3O8, withM=\u00001;L= 0 (white),\nM= 0;L=\u00001 (blue),M= 0;L= +1 (red) and M= +1;L= 0 (black), where the AFM order parameter L= (S1\u0000S2)=2 and\nthe FiM order parameter M= (S1+S2)=2 are determined by the B-site spinsS1;2in the unit cell. Color coding of magnetic\norders is shown at the top of the panel. The planes are ablayers stacked along the caxis.\nMicroscopic theory of the magnetization reversal\nIn order to understand the puzzling appearance of the\nAFM phase with its high polarization at magnetization\nreversals, we consider the following spin model\nE=JkX\nhi;jisi\u001bj+J?X\ni2Asi\u0000\n\u001bi+c=2+\u001bi\u0000c=2\u0001\n\u0000H0\n@X\ni2AmAsi+X\nj2BmB\u001bj1\nA; (3)\nwheresi=Sz\ni=SwithSz\ni= 0;\u00061;\u00062 denoting the c-\naxis projection of the spin of a tetrahedrally coordinated\nFe2+ion (S= 2) on sublattice A, and \u001bj=\u00061 is an Ising\nvariable describing the strongly anisotropic spins on sub-\nlattice B of octahedrally coordinated Fe ions. The \frst\nterm in Eq. (3) describes the dominating AFM exchange\ninteraction Jk>0 between neighboring spins in the ab-\nlayers (see Fig. 1c) and the second term describes the\nweak e\u000bective ferromagnetic interaction J?<0 between\nspins in neighboring layers along the vertical AB bonds,\nresulting from the interplay between the three AFM in-\nterlayer interactions JAA,JBBandJAB28depicted in\nFig. 1d. Here, \u001bi\u0006c=2denotes the B-spins located above\nand below the spin on the A-site i. The last term is the\nZeeman energy for a magnetic \feld applied along the c\naxis with the magnetic moments of A and B site spins set\ntomB= 4:5\u0016BandmA= 4:2\u0016B. These values are inagreement with neutron di\u000braction measurements9and\nreproduce the experimentally observed saturation mag-\nnetizationMs= 0:86\u0016Bper f.u. for x= 0:14 at low\ntemperature assuming that Zn substitutes Fe on tetra-\nhedral sites8,9,21,24. The large deviation of mBfrom the\nspin-only value 4 \u0016Bresults from the unquenched orbital\nmoment of Fe2+ions on octahedral sites, which also leads\nto strong single-ion anisotropy along the caxis and allows\nus to describe B-spins by Ising variables. The relatively\nweak anisotropy of the A-site ions is neglected.\nWith this approach we \frst simulate the temperature\nand magnetic \feld dependence of the magnetization and\npolarization in pure Fe 2Mo3O8showing the \feld-induced\ntransition from the AFM to the FiM state (see Supple-\nmentary notes 3 & 4 for details of calculations). Sup-\nplementary Fig. 4a shows M(H) curves calculated for\nJk= 47 K,J?=\u00001:2 K, which are in quantitative agree-\nment with experiment20,21. The magnetically-induced\nelectric polarization (shown in Supplementary Fig. 4b) is\ncalculated using a microscopic magnetoelectric coupling.\nThe \feld-induced transition corresponds to a spin-\rip\ntransition instead of the spin-\rop transition expected for\nisotropic spins. The net magnetic moment of an ablayer\nthat governs this transition depends on temperature, be-\ncause the isotropic A-site spins show larger thermal \ruc-\ntuations than the Ising spins on B-sites. As a result, the\nnet magnetic moment of an ablayer nearTNis larger than\nat low temperatures21. Conversely, the critical \feld nec-6\nessary to \rip spins just below TNis signi\fcantly smaller\nthan that at low temperatures, which explains the dra-\nmatic increase of the spin-\rip \feld upon decreasing tem-\nperature in pure Fe 2Mo3O820,21. The expansion of the\nboundary of the AFM phase towards higher magnetic\n\felds at low temperatures plays an important role in the\nemergence of this phase at the magnetization reversals.\nIn Fig. 4 we show the main results of our cal-\nculations with regard to the experimentally observed\npolarization anomaly upon magnetization reversal in\nFe1:86Zn0:14Mo3O8. Our theory reproduces the hys-\nteretic magnetization and the polarization curves with a\npeak at the coercive \feld (Figs. 4a and 4b). As tem-\nperature decreases, the coercive \feld increases due to\nthe reduction of thermal \ructuations. The substitution\nof nonmagnetic Zn for Fe on tetrahedral sites increases\nthe net magnetic moment, which further stabilizes the\nFiM state reducing the critical \feld to 0 just below TN\natx= 0:1425. To reproduce the strong reduction of\nthe critical \feld upon doping, we use a much smaller\nvalue of the interlayer exchange constant, J?=\u00000:2 K.\nIn fact, the e\u000bective ferromagnetic interaction J?is ex-\npected to be x-dependent, as it results from the interplay\nbetween the three AFM interlayer interactions JAA,JBB\nandJAB(as depicted in Fig. 1d)28. This interplay sen-\nsitively depends on the removal of magnetic ions from\nA-sites. The much weaker interlayer exchange coupling\nJ?may be the reason why this phenomenon is so promi-\nnent in the present compound Fe 1:86Zn0:14Mo3O8. Even\nthough it has not been reported for other compositions\nin the Fe 2\u0000xZnxMo3O8series21, the necessary J?for this\nunusual magnetization reversal may be realized only in a\nlimited range of Zn concentrations.\nFigure 4c shows snapshots of the spin con\fgurations\nduring the simulated magnetization reversal that starts\nfrom a prepared FiM state with negative saturation mag-\nnetization, which remains negative up to +3.5 T. Each\nplane corresponds to two neighboring magnetic layers of\nFe1:86Zn0:14Mo3O8and the color denotes the local mag-\nnetic ordering in the B-site sublattice described by the\ncalculated local order parameters L= (SB1\u0000SB2)=2 =\n+1,M= (SB1+SB2)=2 = 0 (red), L=\u00001;M= 0\n(blue),L= 0;M= +1 (black), and L= 0;M=\u00001\n(white). The strong intralayer AFM coupling aligns the\nspins of the A and B sublattices opposite to each other.\nTherefore, we consider only the B-site sublattice order\nparameters here. Starting out from a FiM state with neg-\native saturated magnetization (white planes) on the left,\nplanes with non-zero AFM order parameter emerge with\nincreasing \feld in the vicinity of the coercive \feld (blue\nand red planes) before the FiM state with positive mag-\nnetization (black planes) is reached at the highest \felds.\nThe nearly uniform color of each plane re\rects strong\nspin correlations in the ablayers. These spin correla-\ntions suppress the growth of droplets with the opposite\nmagnetization, which are induced by the magnetic \feld\nreversal. As a result, the emergence of the AFM phase is\ndelayed up to the coercive \feld. The above calculations,supporting nicely the experimental \fndings, provide the\nmicroscopic understanding of how the strong strong easy-\naxis anisotropy together with the strong intralayer and\nweak interlayer couplings causes the unusual reappear-\nance of the AFM state at the magnetization reversal in\nFe1:86Zn0:14Mo3O8.\nSummary\nTo summarize, by independent measurements of polar-\nization, magnetization and THz absorption, we discov-\nered a highly unusual magnetization reversal process of\nthe FiM state in Zn-doped Fe 2Mo3O8, which involves the\nreappearance of the AFM ground state as a metastable\nstate during the reversal process. Our theoretical sim-\nulations nicely reproduce this \fnding, showing that an\nIsing-like anisotropy of the octahedrally coordinated Fe\nspins, small magnetic moment of FiM ablayers and weak\ninterlayer interactions are the main ingredients for the\nemergence of these metastable spin con\fgurations: The\npersistence of the FiM state in zero \feld and the reap-\npearance of the AFM state at the coercive \feld.\nAlthough Zn-doped Fe 2Mo3O8has a unique combi-\nnation of properties that slow down kinetics of tran-\nsitions between the FiM and AFM phases and allow\nfor electric detection of these states, the conditions for\nlayer-wise magnetization switching via metastable spin\nstates of distinct characters may be realized in other\n(quasi) 2D magnetic systems29{32. At this point we\nwant to stress the di\u000berence between the honeycomb an-\ntiferromagnets A2Mo3O8and the van der Waals mag-\nnets. In the latter systems, the coexistence of AFM\nand ferromagnetic states usually requires a complex ma-\nterial design using nanofabrication, such as mechanical\ntwisting in CrI 33, whereas the honeycomb antiferromag-\nnetsA2Mo3O8(A=Fe,Co,Mn,Ni,Zn) provide a versa-\ntile intrinsic toolbox for tuning and controlling di\u000berent\nspin con\fgurations through layer-by-layer switching. In\nfact, a special coexistence of collinear antiferromagnetism\nwith canted ferrimagnetism on adjacent honeycomb lay-\ners was recently observed in high-magnetic \felds33. We\nbelieve that similar exotic switching processes, involv-\ning metastable states of distinct magnetic order, can be\nachieved in these compounds by other external stimuli\nlike pressure, voltage or intense light-\felds, which will\nallow to control the growth and decay rates of the spin\nstates and open new pathways for spintronic applications.7\nMETHODS\nSynthesis\nPolycrystalline Fe 2\u0000xZnxMo3O8samples were prepared\nby repeated synthesis at 1000\u000eC of binary oxides FeO\n(99.999%), MoO 2(99%), and ZnO in evacuated quartz\nampoules aiming for a concentration with x= 0:2. Single\ncrystals were grown by the chemical transport reaction\nmethod at temperatures between 950 and 900\u000eC. TeCl 4\nwas used as the source of the transport agent. Large\nsingle crystals up to 5 mm were obtained after 4 weeks\nof transport. The X-ray di\u000braction of the crushed sin-\ngle crystals revealed a single-phase composition with a\nhexagonal symmetry using space group P63mcand a Zn\ncontent corresponding to x\u00190:14. The obtained lattice\nconstants are a=b= 5:773(2) \u0017A andc= 10:017(2) \u0017A25.\nTHz spectroscopy\nTemperature and magnetic \feld dependent time-domain\nTHz spectroscopy measurements were performed on a\nplane-parallel ac-cut single crystal of Fe 1:86Zn0:14Mo3O8.\nA Toptica TeraFlash time-domain THz spectrometer was\nused in combination with a superconducting magnet,\nwhich allows for measurements at temperatures down to\n2 K and in magnetic \felds up to \u00067 T. Measurements\nwere performed in Voigt transmission con\fguration with\nthe magnetic \feld parallel to the c-axis.\nMagnetization Measurements\ndc magnetization measurements were carried out with a\nMagnetic Property Measurement System (5 T MPMS-\nSQUID, Quantum Design). The magnetic \feld was ap-\nplied along the cdirection of a hexagonal-shaped single\ncrystal with clear top-bottom abplanes.\nPolarization Measurements\nElectric polarization was obtained by measuring pyro-\nelectric/magnetoelectric current with a Keysight elec-\ntrometer (model number B2987A). For accessing low\ntemperature and high magnetic \felds, a Physical Prop-\nerty Measurement System (9 T PPMS, Quantum Design)\nand an Oxford helium-\row cryostat (+14 T) were used.\nThe measurements were carried with a hexagonal-shaped\nsingle crystal, where electrical contacts were made on\ntop-bottom abplanes by silver paint and the electric po-\nlarization along the caxis was probed. The crystal was\nmounted in PPMS/cryostat in such a way that the mag-\nnetic \feld could be applied along the caxis and the \feld\nwas varied with a rate of 100 Oe/sec. The magnetic\n\feld-dependent electric polarization was obtained by in-\ntegrating the magnetoelectric current over measurement\ntime.\nNumerical Simulations\nThe sum over spin projections on tetrahedral sites, si, in\nEq.(3) can be performed analytically, which leaves an ef-\nfective model of Ising variables f\u001big(for more details see\nSupplementary materials). This e\u000bective model was used\nto calculate the magnetization and antiferromagnetic or-\nder parameter, as well as for simulations of hysteresisloops. We assume that the isotropic spins on A-sites\nquickly reach thermal equilibrium with the neighboring\nB-site spins, whereas the dynamics of the Ising spins oc-\ncurs on a longer time scale and can be described by the\nGlauber dynamics34. To simulate hysteresis curves, we\nemploy Glauber dynamics, a Markov chain Monte Carlo\nalgorithm used to study non-equilibrium physics of Ising\nmodels. For each curve, the initial state is prepared with\nsimulated annealing starting from a high-temperature\n(T= 90 K) random state. At each magnetic \feld value,\nwe performed 50 measurements with 107Glauber steps\nin-between after the waiting time of 5 \u0001108steps with no\nmeasurements. The total number of \feld points is 32.\nThe results were averaged over 20-30 disorder realiza-\ntions. Open boundary conditions in all three directions\nwere used to speed up the magnetization reversal. The\nnonmagnetic Zn impurities were modeled by Si= 0 on\nrandomly chosen A-sites. We simulated the lattice of\n20\u000220\u000220 Ising spins.\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nACKNOWLEDGEMENTS\nThis research was partly funded by Deutsche\nForschungsgemeinschaft DFG via the Transregional\nCollaborative Research Center TRR 80 \\From Elec-\ntronic correlations to functionality\" (Augsburg, Mu-\nnich, Stuttgart). The support via the project ANCD\n20.80009.5007.19 (Moldova) is also acknowledged. EB\nand MM acknowledge Vrije FOM-programma `Skyrmion-\nics' and the Peregrine high performance computing clus-\nter.\nAUTHOR CONTRIBUTIONS\nL.P. and V.T. synthetized and characterized the crys-\ntals; S.G. and L.P. performed magnetization and polari-\nsation measurements and analyzed the data; D.K., K.V.,\nand J.D. performed the THz measurements and analyzed\nthe data; E.B. and M.M. performed the theoretical sim-\nulations; S.G., E.B., M.M., I.K., and J.D. wrote the\nmanuscript; J.D. planned and coordinated the project.\nAll authors contributed to the discussion and interpre-\ntation of the experimental and theoretical results and to\nthe completion of the manuscript.8\nCOMPETING INTERESTS\nThe authors declare that there are no competing inter-\nests.\n[1] Ostler, T. et al. Ultrafast heating as a su\u000ecient stim-\nulus for magnetization reversal in a ferrimagnet. Nat.\nCommun. 3, 666 (2012).\n[2] Duine, R. A., Lee, K.-J., Parkin, S. S. P. & Stiles, M. 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Phys. 33, 549{564\n(1972).\n[9] Bertrand, D. & Kerner-Czeskleba, H. \u0013Etude structurale\net magn\u0013 etique de molybdates d'\u0013 el\u0013 ements de transition.\nJ. Phys. 36, 379{390 (1975).\n[10] Le Page, Y. & Strobel, P. Structure of iron(II) molybde-\nnum(IV) oxide Fe 2Mo3O8.Acta Cryst. B 38, 1265{1267\n(1982).\n[11] Strobel, P., Le Page, Y. & McAlister, S. P. Growth and\nphysical properties of single crystals of FeII\n2MoIV\n3O8.J.\nSolid State Chem. 42, 242{250 (1982).\n[12] McAlister, S. P. & Strobel, P. Magnetic order in\nM2Mo3O8single crystals (M = Mn, Fe, Co, Ni). J. Magn.\nMagn. Mater. 30, 340{348 (1983).\n[13] Abe, H., Sato, A., Tsujii, N., Furubayashi, T. & Shi-\nmoda, M. Structural re\fnement of T2Mo3O8(T=\nMg, Co, Zn and Mn) and anomalous valence of trinu-\nclear molybdenum clusters in Mn 2Mo3O8.J. Solid State\nChem. 183, 379{384 (2010).\n[14] Tang, Y. S. et al. 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Nano Lett. 20, 2741{2746 (2020).\n[20] Wang, Y. et al. Unveiling hidden ferrimagnetism and\ngiant magnetoelectricity in polar magnet Fe 2Mo3O8.Sci.\nRep.5, 12268 (2015).\n[21] Kurumaji, T., Ishiwata, S. & Tokura, Y. Doping-tunable\nferrimagnetic phase with large linear magnetoelectric ef-\nfect in a polar magnet Fe 2Mo3O8.Phys. Rev. X 5,\n031034 (2015).\n[22] Sheu, Y. M. et al. Picosecond Creation of Switchable\nOptomagnets from a Polar Antiferromagnet with Giant\nPhotoinduced Kerr Rotations. Phys. Rev. X 9, 031038\n(2019).\n[23] Streltsov, S. V., Huang, D.-J., Solovyev, I. V. & Khom-\nskii, D. I. Ordering of Fe and Zn Ions and the Magnetic\nProperties of FeZnMo 3O8.JETP Letters 109, 786{789\n(2019).\n[24] Ji, Y. et al. Direct observation of preferential occupation\nof zinc ions in (Fe 1\u0000xZnx)2Mo3O8.Solid State Commun.\n344, 114666 (2022).\n[25] Csizi, B. et al. Magnetic and vibronic terahertz excita-\ntions in Zn-doped Fe 2Mo3O8.Phys. Rev. B 102, 174407\n(2020).\n[26] Arima, T. et al. Structural and magnetoelectric proper-\nties of Ga 2\u0000xFexO3single crystals grown by a \roating-\nzone method. Phys. Rev. B 70, 064426 (2004).\n[27] Kurumaji, T. et al. Electromagnon resonance in\na collinear spin state of the polar antiferromagnet\nFe2Mo3O8.Phys. Rev. B 95, 020405(R) (2017).\n[28] Solovyev, I. V. & Streltsov, S. V. Microscopic toy model\nfor magnetoelectric e\u000bect in polar Fe 2Mo3O8.Phys. Rev.\nMater. 3, 114402 (2019).\n[29] Wu, H. C. et al. Anisotropic spin-\rip-induced multifer-\nroic behavior in kagome Cu 3Bi(SeO 3)2O2Cl.Phys. Rev.\nB95, 125121 (2017).\n[30] Zhang, T. et al. Magnetism and optical anisotropy in van\nder waals antiferromagnetic insulator CrOCl. ACS Nano\n13, 11353{11362 (2019).\n[31] Peng, Y. et al. Magnetic structure and metamagnetic\ntransitions in the van der waals antiferromagnet CrPS 4.\nAdv. Mater. 32, 2001200 (2020).\n[32] Zhang, M. et al. Metamagnetic Transitions in Few-\nLayer CrOCl Controlled by Magnetic Anisotropy Flip-\nping (2021).\n[33] Szaller, D. et al. Coexistence of antiferromagnetism\nand ferrimagnetism in adjacent honeycomb layers. arxiv\n(2022).\n[34] Glauber, R. J. Time-Dependent Statistics of the Ising\nModel. J. Math. Phys. 4, 294{307 (1963)."    },    {        "title": "1609.02048v1.Magnetodielectric_and_spin_lattice_coupling_in_quasi_1D_Ising_spin_chain_CoNb___2__O___6__.pdf",        "content": "Magnetodielectric and spin-lattice coupling in quasi 1D Ising spin chain\nCoNb 2O6\nM. Nandi,1D. Prabhakaran,2and P. Mandal1\n1)Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India\n2)Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU,\nUK\n(Dated: 19 November 2021)\nWe have studied magnetodielectric and spin-lattice coupling in CoNb 2O6single crystals. Magnetostriction\nand magnetodielectric experiments are performed at temperatures in and above anti\u000beromagnetic phase of\nquasi 1D Ising spin chain CoNb 2O6. Field induced magnetic transitions are clearly re\rected in magnetodi-\nelectric measurement as well as magnetostriction measurement also. Two sharp anomalies are found around\nthe critical \felds of antiferromagnetic to ferrimagnetic transition and ferrimagnetic to saturated paramag-\nnetic transition in both magnetodielectric and magnetostriction experiments. High \feld anomaly is more\npronounced for magnetodielectric response and magnetostriction also. So, in CoNb 2O6, spins are strongly\ncoupled with lattice as well as charges also.\nI. INTRODUCTION\nIn recent years geometrically frustrated triangular lat-\ntice systems have attracted immense interest due to\nits di\u000berent kind of magnetic phase transitions and de-\ngenerate ground states. Several triangular lattice sys-\ntems also exhibit multiferroic behavior.1Geometrical\nfrustration plays a key role to produce magnetodielec-\ntric coupling. Dielectric constant measurements in pres-\nence of magnetic \feld can probe the coupling between\ncharges and spins in insulating systems. Ising spin\nchain CoV 2O6,2Ca3Co2O63{5with triangular network\ndisplay magnetodielectric coupling at low temperature.\nQuasi-one-dimensional Ising spin chain CoNb 2O6is a\nvery good example of frustrated triangular lattice sys-\ntem which exhibits several interesting features like meta-\nmagnetic transition, quantum criticality behavior etc.\nRecently, quantum phase transition in transverse \feld\nhas been experimentally evidenced in CoNb 2O6.6A E 8\nsymmetry has been experimentally observed near the\nquantum critical point of Ising ferromagnet CoNb 2O6.\nAt low temperature, this system also exhibits var-\nious degenerate magnetically ordered states such as\nfourfold-degenerate antiferromagnetic (AF) phase, \feld-\ninduced threefold-degenerate ferrimagnetic (FR) phase,\nsinusoidally amplitude-modulated incommensurate (IC)\nphase, con\frmed by Neutron di\u000braction study.7{9In\nCoNb 2O6system, Co2+ions form zigzag chains along\nc-direction and they are arranged into isosceles trian-\ngular geometry in the a\u0000bplane. At low tempera-\ntures, Co spins orient along two di\u000berent easy axes in\nthe nearly a\u0000cplane with a 31\u000ecanting angle from\nthecaxis. Intrachain interaction is ferromagnetic in\nnature and chains are weakly coupled by antiferromag-\nnetic interaction. In this paper, we have performed meg-\nnetodielctric measurement to evidence the coupling be-\ntween electrical charges and magnetic moments. In ad-\ndition, we have done magnetostriction measurements to\nprobe coupling between spin and lattice. In some sys-\ntems, spins are simultaneously coupled with both latticeand charges. For example, in EuTiO 3, magnetostriction\nmeasurement exhibits several similarities with the \feld\ndependence of the dielectric constant.10The correlation\nbetween spin-phonon coupling and dielectric constant has\nbeen observed in TbFe 3(BO 3)4.11In this paper we have\nstudied and compared magnetic, magnetodielectric, mag-\nnetothermal properties of CoNb 2O6.\nII. EXPERIMENTAL DETAILS\nSingle crystal of CoNb 2O6was grown by the travel-\ning solvent \roating zone method.12Laue XRD was per-\nformed to determine crystal axes and crystal was cut\nalong di\u000berent crystallographic planes according to ex-\nperiment. Laue di\u000braction patterns are illustrated in\nFigure 1. A rectangular shaped piece of single crystal\nwas used for dielectric measurements. Two parallel faces\nof the crystal were covered with silver paint in order to\napply an electric \feld perpendicular to the chains. Here\nelectric \feld was applied along aaxis where as magnetic\n\feld was applied along the easy axis direction cso that\nthe~E?~Hcondition was always ful\flled. Magnetostric-\ntion measurements were done by capacitive method using\na miniature tilted-plates dilatometer with applied \feld\nparallel to caxis. The capacitance measurements were\nperformed using a commercial AH2700A ultra-precision\ncapacitance bridge. The magnetic measurements were\ndone in SQUID-VSM (Quantum Design). The speci\fc\nheat measurements were done using a physical property\nmeasurement system (Quantum Design) by conventional\nrelaxation time method.\nIII. MAGNETIZATION MEASUREMENTS\nTemperature dependence of magnetization along c-axis\nin zero \feld cool(ZFC) and \feld cool(FC) conditions is\nplotted in Figure 2(a). At low temperature, M vs. T\ncurve shows two successive transitions below 3 and 2 K.arXiv:1609.02048v1  [cond-mat.str-el]  7 Sep 20162\nFIG. 1. Laue di\u000braction patterns of (100), (010) and (001)\nplanes.\nZFC and FC of M(T) do not show any bifurcation down\nto 1.8 K. M(T) curve exhibits slope change around 3\nK (T 1) due to transition from paramagnetic to incom-\nmensurate phase. Another transition occurs below 2 K\n(T2) where a sharp drop in magnetization has been ob-\nserved due transition from IC to AFM phase. In inset\nof Figure 2(a), temperature dependence of speci\fc heat\n(Cp(T)) is plotted at zero \feld. A very sharp peak has\nbeen observed at 2.9 K in Cp(T) due to PM-IC tran-\nsition. Field dependence of magnetization along c-axis\nfor some selected temperatures both above and below\nT1and T 2are plotted in Figure 2(b). In inset of Fig-\nure 2(b), a closer view of M(H) at 1.8 K is given which\nexhibits multiple magnetization plateaux due to \feld in-\nduced magnetic phase transitions, similar to previously\nobserved data.13M is very small up to 200 Oe then it\nshows step like jump at \frst critical \feld Hc1and obtains\n1/3 of saturation magnetization value in a certain \feld\nrange. This step like increase in magnetization can be ex-\nplained from magnetic phase diagram by S. Kobayashi.14\nThe saturation magnetization value is consistent with a\nCo2+moment of about 3 \u0016B. At 1.8 K, the system re-\nmains at AFM phase below 200 Oe, then it enters to fer-\nrimagnetic phase via an incommensurate phase with in-\ncreasing \feld. So this \feld induced AFM to ferrimagnetic\nphase transition is re\rected in sharp step-like increase in\nM(H) curve around 200 Oe. Another increase in M(H)\naround 3.8 kOe (second critical \feld Hc2) is observed due\n12 3 4 5 0123450\n4 8 1 2012340\n1 2 3 4 5 6 7 010203040500\n120.00 .30 .60 .9c (emu/mol Oe)T\n (K) zfc \nfcH|| c100 Oea\n)C\np (JK-1mol-1)T\n (K)H\n|| cM (emu/g)H\n (T) 1.8K \n3K \n5K \n9K \n14Kb)H\nc1H (T) M (mB/f.u.) 1.8 KH c2FIG. 2. (a) Plots of \u001f(T) with zero \feld cool and \feld cool\ncondition for CoNb 2O6at 100 Oe \feld applied along caxis.\nInset:Speci\fc heat ( Cp) versus temperature plot for CoNb 2O6\nat zero \feld. (b) Isothermal magnetization at some selected\ntemperatures when \feld is applied along caxis. Inset shows\nthe closer view of M-H curve at 1.8 K.\nto \feld induced transition from ferrimagnetic state to sat-\nurated PM state. With increasing temperature step-like\nincrease in M(H) curve gradually disappears. Just above\n3 K, magnetization linearly increases with H and then\nsaturates. Further increase of temperature makes the M\nalmost linear with H.\nIV. DIELECTRIC CONSTANT MEASUREMENTS\nThe isothermal dielectric constant measurements per-\nformed with a frequency 1 kHz, as a function of external\nmagnetic-\feld at some selected temperatures are plot-3\n-0.9-0.6-0.30.00.30.60.9-0.020-0.015-0.010-0.0050.0000.0050\n.00 .20 .40 .60 .81 .0-0.020-0.015-0.010-0.0050.0000.005De'/e'0 %H\n (T) 1  \n2 \n3 \n4 \n51.5 KH || ca\n)b\n)H|| cDe'/e'0 %H\n (T) 1.5 K \n1.6 K \n2 K \n2.3 K \n4.3 K \n5.5 K\nFIG. 3. (a) Five segment curves for relative change of dielec-\ntric constant as a function of magnetic \feld are plotted at 1.5\nK when magnetic \feld applied along caxis. (b) Plot of rel-\native change of dielectric constant as a function of magnetic\n\feld for some selected temperatures when magnetic \feld is\napplied along caxis.\nted in Figure 3. Here external magnetic \feld is applied\nalongcaxis and electric \feld is applied along aaxis.\nActually, Figure 3 shows percentage of relative change\nin dielectric constant (\u0001 \"0/\"00=[\"0(H)-\"00]/\"00), where\n\"00is the dielectric constant of the sample in absence\nof magnetic \feld. Sharp anomalies are found in isother-\nmal relative change of dielectric constant at 1.5 K in \fve\nsegment curve, shown in Figure 3(a). With increasing\n\feld \u0001\"0/\"00% remains constant initially and then ex-\nhibit a sharp negative peak around Hc1, then it shows\nalmost a constant positive plateau region in a certain\n\feld range. With further application of magnetic \feld,\n\u0001\"0/\"00% displays a sharp step-like jump around Hc2andhysteresis has also been found here. Then, \u0001 \"0/\"00% de-\ncreases very slowly with increasing H. Depending on \feld\nstrength, \u0001 \"0/\"00% obtains positive value as well as neg-\native value. With increasing temperature, these anoma-\nlies gradually disappear, shown in Figure 3(b). At 4.3\nK, sharp peak-like feature around Hc1is totally disap-\npeared but \u0001 \"0/\"00% becomes positive in a certain \feld\nrange though the plateau-like behavior is disappeared.\nNo positive region has been found with further increase\nof temperature where dielectric constant monotonically\ndecreases with increasing \feld. At 5.5 K, where system\nis aboveT1, peak around Hc1and step-like jump around\nHc2in dielectric constant disappear and it shows almost\nlinear dependence with \feld. According to the magnetic\nphase diagram with \feld applied along c-axis, drawn from\nneutron di\u000braction data, CoNb 2O6undergoes multiple\n\feld induced magnetic transitions at 1.5 K.14In low \feld\nregion, below T 2, CoNb 2O6exhibits successive \feld in-\nduced antiferromagnetic (AF) to incommensurate (IC)\nthen IC to ferrimagnetic (FR) transitions around 225 Oe\nand 395 Oe respectively. Dielectric constant shows two\nsuccessive slope changes in low \feld region which can be\ninterpreted by these AF to FR transition via intermediate\nIC phase. Above a certain critical \feld, this system en-\nters to \feld induced FR state and remains in this state up\nto 3.2 T where corresponding dielectric constant exhibits\na plateau-like feature obtaining a certain positive value.\nObserved sharp step-like jump in \u0001 \"0/\"00% around 0.33\nT may be related to transition from \feld induced FR to\nsaturated PM state. Field induced magnetic transitions\nare re\rected in \feld dependent dielectric constant mea-\nsurements also. So it is clear that dielectric constant is\nmagnetically coupled in this system.\nV. THERMAL EXPANSION MEASUREMENTS\nWe have also performed magnetostriction measure-\nments at some selected temperatures. Figure 4(a) shows\nmagnetostriction \u0001 L(H)/L0=[L(H)-L0]/L0, whereL0is\nthe length of the sample in absence of magnetic \feld, at\n1.6 K for \feld increasing and decreasing conditions. Simi-\nlar to dielectric constant measurement, magnetostriction\nat 1.6 K also exhibits two anomalies around Hc1and\nHc2. Hysteresis has also been found here. \u0001 L(H)/L0\nshows a weak cusplike anomaly around 500 Oe where a\npeak type feature has been found in dielectric constant\nmeasurement. A very pronounced peak has been found\nin \u0001L(H)/L0around 3500 Oe where dielectric constant\nexhibits a very sharp step-like jump almost near about\nthis \feld. The very sharp peak around 3500 Oe is as-\nsociated with the transition from the ferrimagnetic state\nto a saturated PM high-\feld phase. Apart from this,\nmagnetostriction measurement shows an interesting be-\nhavior. \u0001L(H)/L0obtains positive value as well as neg-\native value. Similar kind of behavior is also observed\nin magnetostriction of EuTiO 3where it shows a sign\nchange with increasing magnetic \feld10. With increasing4\n0.00 .20 .40 .60 .81 .0-3-2-101230\n.00 .40 .81 .21 .62 .0-3-2-101234561.6 KDL/L0(10- 5)H\n (T)H|| ca)b\n)H\n|| cDL/L0(10- 5)H\n (T) 1.6 K \n2.1 K \n2.9 K \n4 K \n5.5 K\nFIG. 4. (a)Magnetostriction, \u0001 L(H)=L0at 1.6 K for both\n\feld increasing and decreasing condition when magnetic \feld\napplied along caxis. (b) Magnetostriction, \u0001 L(H)=L0, at\nseveral temperatures both above and below T1andT2for\nsome selected temperatures when magnetic \feld is applied\nalongcaxis.\n\feld \u0001L(H)/L0remains negative up to 4700 Oe then it\nbecomes positive and increases linearly with \feld above\n6000 Oe. Magnetostriction curves for some selective tem-\nperatures both below and above T1andT2are shown in\nFigure 4(b). At 2.1 K, weak anomaly around Hc1dis-\nappears but sharp peak around Hc2has been observed.\nWith increasing temperature, height of the peak around\n3500 Oe decreases gradually and disappears above 4 K.\nAt 5.5 K, \u0001 L(H)/L0exhibits very small value up to\n6000 Oe and then it increases monotonically with in-\ncreasing \feld but it remains always positive throughoutthis region. Particularly at low temperature, \feld in-\nduced metamagnetic transitions are re\rected in magne-\ntostriction measurements also which suggests that spins\nare strongly coupled with lattice in this system.\nVI. SUMMARY\nWe have carried out magnetostriction and magne-\ntodielectric measurements on single crystalline CoNb 2O6\nsample at low temperature. The samples are well char-\nacterized by magnetization and speci\fc heat measure-\nments. We have related \feld dependence of dielec-\ntric constant and thermal expansion measurements with\nmagnetic phase diagram by neutron di\u000braction and com-\npared with magnetization. Multiple phase transitions ob-\nserved by neutron di\u000braction data are clearly re\rected\nin \feld dependence of dielectric constant and magne-\ntostriction measurements. Field dependence of the di-\nelectric constant display several similarities with magne-\ntostriction measurements in CoNb 2O6. Both \feld depen-\ndence of dielectric constant and magnetostriction exhibit\ntwo anomalies around two critical \felds of metamagnetic\ntransitions and obtain positive as well as negative value.\nSo it can be concluded that spin-lattice coupling plays a\nkey role and spins are magnetically coupled with charges\nin this system.\n1S. Seki, Y. Onose, and Y. Tokura, Phys. Rev. Lett. 101, 067204\n(2008).\n2K. Singh, A. Maignan, D. Pelloquin, O. Perez, and Ch. Simon,\nJ. Mater. Chem. 22, 6436 (2012).\n3N. Bellido, Ch. Simon, and A. Maignan, Phys. Rev. B 77, 054430\n(2008).\n4N. Bellido, Ch. Simon, and A. Maignan, J. Magn. Magn. Mater.\n321, 1770 (2009).\n5T. Basu, K. K. Iyer, K. Singh, and E. V. Sampathkumaran, Sci.\nRep. 3, 3104 (2013).\n6R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D.\nPrabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer,\nScience 327 177 (2010).\n7S. Mitsuda, S. Kobayashi, K. Aga, H. Katagiri, H. Yoshizawa,\nM. Ishikawa, K. Miyatani, and K. Kohn, J. Phys. Soc. Jpn. 64,\n2325 (1995).\n8S. Kobayashi, S. Mitsuda, M. Ishikawa, K. Miyatani, and K.\nKohn, Phys. Rev. B 60, 3331 (1999).\n9C. Heid, H. Weitzel, P. Burlet, M. Bonnet, W. Gonschorek, T.\nVogt, J. Norwig, and H. Fuess, J. Magn. Magn. Mater. 151, 123\n(1995).\n10P. G. Reuvekamp, R. K. Kremer, J. K ohler, and A. Bussmann-\nHolder, Phys. Rev. B 90, 094420 (2014).\n11U. Adem, L. Wang, D. Fausti, W. Schottenhamel, P. H. M. van\nLoosdrecht, A. Vasiliev, L. N. Bezmaternykh, B. B uchner, C.\nHess, and R. Klingeler, Phys. Rev. B 82, 064406 (2010).\n12D. Prabhakaran, F. R. Wondre, A. T. Boothroyd, J. Cryst.\nGrowth 250, 72 (2003).\n13I. Maartense, I. Yaeger, B.M. Wanklyn, Solid State Commun. 21\n93 (1977).\n14S. Kobayashi, S. Mitsuda, and K. Prokes, Phys. Rev. B 63,\n024415 (2000)."    },    {        "title": "2104.14819v1.Current_Induced_Magnetization_Control_in_Insulating_Ferrimagnetic_Garnets.pdf",        "content": " 1 Current-Induced Magnetization Control in Insulating Ferrimagnetic Garnets Can Onur Avci Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus de la UAB, Bellaterra, 08193, Spain  The research into insulating ferrimagnetic garnets has gained enormous momentum in the past decade. This is partly due to the improvement in the techniques to grow high-quality ultrathin films with desirable properties and the advances in understanding the spin transport within the ferrimagnetic garnets and through their interfaces with conducting materials. In recent years, we have seen remarkable progress in controlling the magnetization state of ferrimagnetic garnets by electrical means in suitable heterostructures and device architectures. These advances have readily placed ferrimagnetic garnets in a favorable position for the future development of insulating spintronic concepts. The purpose of this article is to review recent experimental results of the current-induced magnetization control and associated phenomena in ferrimagnetic garnets, as well as to discuss future directions in this rapidly evolving area of spintronics.             2 1. Introduction The discovery of ferrimagnetic garnets (FMGs) dates back to 1950s1,2) followed by several decades of extensive investigations to understand their structural and magnetic properties.3) In the early 1970s, the development of liquid phase epitaxy has allowed the growth of micrometer-thick FMG films such as yttrium iron garnet (Y3Fe5O12, YIG) with ultrahigh quality and low production cost.4) Consequently, FMGs have become a model system for engineering magnetic and magneto-optical properties with high precision through their composition. FMGs with suitable domain properties and large magneto-optic response have led to technological developments such as magnetic bubble memory,5) magnetic bubble display,6) and magneto-optic printers.7) These advances marked the first integration of magnetic materials into modern electronic technologies in the late 1970s and early 1980s. The 1980s have marked a paradigm shift in magnetism research. The discovery of giant magnetoresistance has initiated the spintronics revolution and boosted the magnetic recording industry.8,9) For the next two decades, other prominent phenomena such as interfacial perpendicular magnetic anisotropy10) (PMA) and spin-transfer torques11-13) (STTs) have spearheaded spintronics research and related applications.14) All these effects exclusively required conducting ferromagnetic structures. During this period, FMGs have been set aside because of their insulating character, and metallic multilayers have become the preferential materials in this exciting field. Magnetic insulators, and in particular FMGs, possess extraordinary properties that could be beneficial in spintronics. They are often characterized by low damping and long magnon diffusion lengths, enabling efficient spin current generation and its nonlocal transport.15-19) They can now be grown in nanometer form, in a highly ordered single-crystal structure, desirable for low-pinning domain wall20-26) and skyrmion motion.27,28) They offer multiple degrees of freedom for tuning relevant magnetic properties such as saturation magnetization, magnetocrystalline anisotropy, etc., through composition and strain engineering.3,29,30) They are naturally protected and highly robust against heat, oxidation, aging, and degradation. Finally, the recently discovered transport phenomena such as spin pumping,31,32) spin Seebeck effect,33) and spin Hall magnetoresistance34 suggest FMGs as highly efficient spin current generating and detecting materials.   3 Recently, a better understanding of electrical transport in bulk and interfaces of certain material systems characterized by large spin-orbit coupling revealed that relativistic spin-dependent effects could generate pure spin currents as a result of a charge current.35) The resulting spin-orbit torques (SOTs) can lead to magnetization switching,36-40) domain wall (DW) motion,21,22,24,26) magnon generation/suppression,41) etc., in suitable magnetic heterostructures. In the SOTs scheme, in contrast to the STTs, charge and spin currents travel orthogonally, which provides a simple means to generate spin-torques, among others, on FMGs neighboring to a SOT-source, usually a nonmagnetic metal (NM). The intersection of FMGs and the SOT-driven magnetization control have created fertile grounds in spintronics. We have seen some remarkable results of SOT-induced magnetization control in FMGs in less than half a decade. In this Review, we will give an overview of some of the pioneering experiments on the SOTs-driven control of FMGs. We will first discuss the characteristics of FMGs, SOTs and other relevant physical phenomena. We will then review some key results and discuss them in a broader context. We will finally discuss the potential directions and future opportunities in this rapidly evolving and exciting area of spintronics.  2. General Properties of Ferrimagnetic Garnets 2.1 Composition and structure FMGs constitute a large family of magnetic oxides with the general formula of X3Fe5O12, where X is either a non-magnetic rare-earth element such as yttrium (Y) or a magnetic rare-earth element such as gadolinium (Gd), terbium (Tb), thulium (Tm), etc. The lattice structure consists of two octahedral Fe3+ ions, three tetrahedral Fe3+ ions, and three X3+ ions at the dodecahedral sites, per formula unit. The octahedral and tetrahedral Fe3+ ions are coupled antiparallel to each other while X3+, if magnetic, is aligned parallel to the octahedral Fe3+. All magnetic interactions are mediated via superexchange interaction through O2- anions. In the case of YIG, there is no magnetic compensation, and it preserves ferrimagnetic order up to 559 K. In iron garnets with magnetic rare-earth elements, the total magnetic moment depends sensitively on the rare-earth; hence the magnetism differs significantly depending on the rare-earth choice, composition, and temperature. Notably, there exists a temperature at which the opposing magnetic sublattices are equal to each other and the film shows zero net magnetization, thereby acts like a collinear antiferromagnet. Nonetheless, the ordering temperature is mainly determined by the magnetic moment of Fe, independent of the rare-earth choice, and hence is about the same (530-580 K)  4 in all FMGs,42) which makes the entire family of FMGs suitable for room temperature research and applications  2.2 Growth Ultrathin films of FMGs are typically grown on lattice-matched gadolinium gallium garnet (Gd3Ga5O12, GGG) or similar garnet substrates by pulsed laser or magnetron sputter deposition using stoichiometric targets.43-45) For the sputter deposition, an off-axis method (i.e., the target and substrate are held at 90° with respect to each other) is generally preferred due to the enhanced stoichiometry of the deposited films.46) Typically, the substrate is kept at a high temperature (650-850°C) during deposition and slowly cooled down to room temperature to obtain the desired high crystallinity, homogeneity and surface smoothness. Alternatively, room temperature deposition and post-growth annealing are also reported to yield similar results.47) By employing the mentioned methods, films of thicknesses ranging from a few to tens of nm can be obtained with sub-nm surface roughness.  2.3 Perpendicular magnetic anisotropy Of particular interest to this Review, some FMGs possess bulk PMA, which is an essential property in spintronic devices since it enhances the thermal stability of magnetic elements and ensures long-term data retention. The PMA in FMGs is mainly of magneto-elastic origin, mediated by lattice strain. When grown epitaxially, a small lattice mismatch between the substrate and the FMG film leads to compressive or tensile strain, giving rise to negative magneto-elastic energy and consequently to PMA.48) Thus far, PMA has been reported for thin films of TmIG,49-52) TbIG53, EuIG,52,53) GdIG,54) YIG,55) and Bi substituted YIG18) (Bi:YIG) grown on GGG or similar substrates, polycrystalline EuIG grown on a quartz substrate,30) and DyIG grown on a SiO2 substrate by pulsed laser deposition.56) There were also successful efforts to grow TmIG and YIG films by r.f. magnetron sputtering with similar magnetic and structural properties compared to those obtained by pulsed laser deposition.46,47,57) Recent studies suggest that there is an additional interface contribution to the PMA from the metal capping of FMGs.58) For the complementary metal-oxide-semiconductor (CMOS) technology integration, magnetron sputtering is the preferred method since films with high uniformity can be grown on large-scale wafers. Tunable PMA through material and interface engineering and the use of magnetron sputtering together make FMGs very attractive for spintronics.  5 3. Spin-Orbit Torques and Electrical Detection of Magnetization in Ferrimagnetic Garnets SOTs are presently the state-of-the-art current-induced magnetic manipulation method in spintronic devices, including the ones based on FMGs.35) Their experimental discovery dates back to the end of the 2000s and early 2010s, first in semiconducting heterostructure,59) and then in metallic multilayers.36,60) Initially, the SOTs research has mostly focused on heavy metal/ferromagnet bilayers such as Pt/Co,36,37,60,61) Ta/CofeB38,40,62) and W/CoFeB63) due to their historical significance, technological relevance, the convenience of deposition and fabrication, and large torque-to-current ratios reported. Later on, topological insulators,64,65) semimetals,66,67) oxidized metals68-70) and intermetallic alloys71,72) combined with appropriate magnetic materials, have been considered. Many SOTs generating mechanisms have been discovered thus far, including the spin Hall effect63) (SHE), interfacial Rashba-Edelstein effect,60) topological surface states,73) bulk crystal asymmetry,74) anomalous Hall effect,75) (AHE), planar Hall effect76) (PHE), interface spin filtering,77) etc. Among these, the SHE is primarily considered as a SOT source for current-induced magnetization manipulation in FMGs. The SHE emerges in the bulk of the materials characterized by strong spin-orbit coupling (Pt, W, Ta, etc.). It describes the spin-dependent scattering of conduction electrons which generates a pure spin current transverse to the charge current injection direction. A magnetic layer in interfacial contact with a SHE material can then absorb the resulting spin current, which acts as a spin torque on the magnetization and enables its reversal (Fig. 1). In the other listed SOT sources, while the spin current generation mechanism differs, the applied torque has comparable symmetry. 3.1 Symmetry and characterization of spin-orbit torques  SOTs consist of damping-like and field-like components with the symmetries 𝛕\"#∝𝐦×(𝐲×𝐦) and 𝛕)#∝𝐦×𝐲, respectively (see Fig. 2a). Here m is the magnetization unit vector and y is the in-plane axis perpendicular to the current flow direction. 𝛕\"# has the suitable geometry to induce 180° switching of the in-plane magnetization along the axis perpendicular to the current injection as well as the perpendicular magnetization; though in the latter, an in-plane magnetic field is required to break the rotational symmetry of 𝛕\"# effective field (Fig. 2a). Furthermore, 𝛕\"# can move homochiral Néel-type DWs and skyrmions in the PMA systems with high efficiency and large velocities.  6 A handful of electrical and optical techniques have been developed to characterize the SOTs.78) Perhaps the simplest and most accessible method is the harmonic Hall voltage detection of the SOT effective fields.61,79) It consists of applying an a.c. current through Hall bar devices and detecting and analyzing the harmonic Hall voltage response proportional to m (Fig. 2b). The Hall voltage in metallic magnetic systems predominantly consists of AHE and PHE where the former (latter) is proportional to the out-of-plane (in-plane) component of the magnetization vector: 𝑉+=𝑅.+/𝐼cos𝜃+𝑅6+/𝐼sin9𝜃sin2𝜑. Here 𝜃 and 𝜑 represent the polar and azimuthal angle of m and I is the injected current (see Fig. 3a). Upon an a.c. current injection, m oscillates about its equilibrium position due to oscillating SOTs and generates a harmonic response in the Hall voltage, twice the frequency of the input current, hence generating a second harmonic Hall voltage (𝑉9<). 𝑉9< contains contributions from SOTs but also thermoelectric effects (anomalous Nernst and spin Seebeck effects) driven by Joule heating-induced temperature gradients.80) Self-consistent analysis of the data taken in suitable experimental geometries, detailed in Refs.61,79,80), leads to vector quantification of SOTs in both in-plane and PMA systems, and the proper separation of the non-SOT originating signals.80,81) 3.2 Spin Hall magnetoresistance in magnetic insulators A charge current cannot pass through FMGs henceforth cannot generate AHE and PHE. However, it has been discovered that FMGs, and in general magnetic insulators, can influence the electrical resistance of the material that is in interfacial contact with it and give rise to spin Hall magnetoresistance34,82-84) (SMR). SMR, first discovered in YIG/Pt bilayers, describes the changes in the electrical resistivity of a nonmagnetic metal adjacent to a magnetic insulator depending on the magnetization orientation. The effect arises due to the asymmetry in the scattering of the SHE-induced spin current at the FMG/NM interface, which depends on the relative direction of the magnetization with respect to the spin polarization (collinear or orthogonal). The back-scattered spin current contributes to the charge current via the inverse SHE, which ultimately manifests itself as a modification in the longitudinal and transverse resistivity. In a Hall voltage measurement, like the AHE and PHE in conducting magnets, the SMR gives rise to signals proportional to the in-plane and out-of-plane components of the magnetization vector due to the real and imaginary part of the interface spin mixing conductance,85) respectively. The Hall voltage then reads: 𝑉+=𝑅6+/=>?@𝐼sin9𝜃sin2𝜑+𝑅.+/=>?@𝐼cos𝜃+𝑅A+/𝐼𝐻C (1).  7 Note, however, that the amplitude of the 𝑅6+/=>?@ is generally much larger than 𝑅.+/=>?@, contrary to the situation in the metallic systems. Moreover, there is a non-negligible ordinary Hall effect signal (𝑅A+/) linearly proportional to the out-of-plane component of the external field (𝐻C), but unrelated to the magnetization vector, which is generally neglected in all-metal systems due to much larger AHE and PHE contributions. Figure 3 shows a typical characterization of the SMR and ordinary Hall effect signals using a Hall bar device made of TmIG/Pt.86)  The harmonic Hall voltage method can be used also for quantifying the SOTs in insulating magnets, e.g., in FMG/Pt. Due to the dominant 𝑅6+/=>?@ contribution, the in-plane oscillations of m induced by SOTs produce the largest 𝑉9<, henceforth, in FMG/Pt the measurement geometry differs from all-metal systems. Furthermore, thermoelectric contributions to 𝑉9< should be treated with utmost care. In metallic systems, the main thermoelectric contribution is found to originate from the anomalous Nernst effect produced within the magnetic layer itself, whereas in magnetic insulators the spin Seebeck effect (SSE), the subject of the next section, gives the dominant thermoelectric contribution to 𝑉9<. The detailed description of how to extract the damping-like and field-like SOTs and thermoelectric signals in FMG/NM systems with PMA is given in Refs.87,88) 3.3 Spin Seebeck effect The SSE describes spin current generation by applying a temperature gradient (∇T) across a magnetically ordered material33. In a simplified picture, the temperature difference along the gradient direction creates an imbalance in the magnon population, generating a pure spin current flow along ∇T with polarization parallel to m. The SSE-originated spin current can diffuse into an adjacent metal with a large SHE and generate an inverse SHE voltage or alternatively called a spin Seebeck voltage (𝑉>>/). Additional to the SMR, the in-plane magnetization vector can be probed by measuring 𝑉>>/ in FMGs by simply using current-induced Joule heating as a means to generate ∇T perpendicular to the layers.89,90) In contrast to the SMR, the SSE can be used to sense 180° rotation of m since 𝑉>>/ is expected to change sign upon the reversal of m. With an a.c. current injection, the SMR and SSE together can be measured by 𝑉< and 𝑉9<, respectively, for effectively characterizing the magnetization vector in the 3D space and other relevant properties such as coercivity (𝐻F), magnetic anisotropy (𝐻G), etc. (see Fig. 4). We note that, in addition to the SMR and SSE, it is also possible to detect the magnetization vector orientation in FMGs through spin-torque ferromagnetic resonance  8 measurements91-93. However, such measurements rely on a different and more complex experimental setup and signal analysis, hence not depicted in Fig. 4. 4. Spin-Orbit Torque Switching in Ferrimagnetic Garnets Over the past decade, the characterization of various spin-dependent transport phenomena such as SSE, spin pumping, and SMR has shown a significant spin current transmission through FMG/NM interfaces. However, the first demonstration of robust SOT-induced magnetization switching in an FMG came only in 2017. Avci et al. have shown that the magnetization of an 8 nm-thick TmIG can be switched by SOTs generated by a 5 nm-thick Pt overlayer.86) Through Hall effect measurements, the magnetization was found to switch between the up and down states after consecutive millisecond-long pulses of alternating polarity in the presence of an in-plane field (Fig. 5a-b). The switching polarity was found to change upon reversing the in-plane field, expectedly of the SOT-induced switching.  Furthermore, the critical current density was characterized with a variety of in-plane fields and found to be inversely proportional to the external field strength (Fig. 5c-d) similar to previous reports.38) The switching current densities through Pt were comparable to those used in metallic Pt/Co(~1nm) bilayers despite the large magnetic volume (8 nm) of TmIG. However, after considering the difference in the saturation magnetization -which is about one order of magnitude lower in FMGs with respect to Co- the switching efficiency in TmIG/Pt was found to be comparable to Pt/Co and other Pt-based metallic heterostructures.  Later on, the SOT-induced switching in TmIG/Pt was also demonstrated with a few nanosecond-long pulses and with a low in-plane field requirement of only 2 Oe in 2 µm-wide Hall bars94) (Fig. 5e-f). Following these initial demonstrations, several other works have reported SOT-induced switching in TmIG/Pt, TbIG/Pt, YIG/Pt and Bi:YIG/Pt bilayers with similar or higher efficiencies.57,95-98) In particular, Vélez et al. have reported local switching of magnetization in a continuous TmIG film by sending current through patterned Pt structures.96) Local control of the magnetization in extended films is unique to magnetic insulators, and may find interest in the implementation and in-situ reconfiguration of synthetic magnetic structures for magnonic applications.  The SOT switching of TmIG is not limited to the use of Pt. Shao et al. have shown efficient switching in TmIG/W bilayers with TmIG thickness varying between 3.2 and 15 nm.99 The switching polarity was found the opposite to that obtained in TmIG/Pt for a fixed current  9 polarity, verifying the predominant SHE origin of the switching torques from Pt and W. Note that the SHE in Pt and W has an opposite sign. Switching in thick, bulk-like TmIG has demonstrated the significant potential of SOTs for controlling the magnetization in a wide thickness range. This study has also revealed that the SOT switching efficiency strongly depends on the TmIG thickness and that above 10 nm of TmIG, the SOT reaches the maximum efficiency (Fig. 6). It was argued that the spin current absorption is less efficient in thinner films due to the reduced magnetic moment density, which effectively reduces the exchange interaction between the current-induced accumulated spins and the local magnetic moments of TmIG near the interface, producing smaller torques. When the Ms is close to the bulk limit, though, the SOT efficiency was maximized. Similar conclusions have been reached in another study by using different SHE metals grown on TmIG.100) Interestingly, while switching of FMGs with PMA has been demonstrated in several different materials, SOT-induced switching of in-plane FMGs has remained little explored.101) Limitations in suitable device fabrication with strong in-plane anisotropy and magnetization detection may be some of the reasons for the lack of such experiments. Overall, the various reports on FMG-based heterostructures have demonstrated that the SOTs act similarly upon magnetization in insulating magnets and hence revealing a significant potential for consideration in electrically-controlled magnetic devices. 5. Chiral Domain Walls and Their Current-Driven Motion in Ferrimagnetic Garnets Magnetic DWs and their current-driven dynamics have been a prominent subject in magnetism and spintronics due to their intriguing physics and potential use in data storage and logic devices.102,103) Earlier experiments of current-driven DW motion relied on less efficient STTs. Recently though, SOTs emerged as more efficient alternative to displace DWs, which, moreover, does not require the magnetic layer to be conducting.  SOT-induced DW motion is most effective in PMA layers possessing Néel type DWs due to the orthogonal alignment of DW internal spins and 𝛕\"#, which maximizes the effective field on the DW. Earlier, it was believed that in the standard PMA systems like Pt/Co, the DWs are of Bloch type governed by magnetostatics. Lately, though, it was understood that the interfacial Dzyaloshinskii-Moriya interaction (DMI) in Pt/Co and other similar NM/ferromagnet layers, gives rise to homochiral Néel DWs in structurally asymmetric multilayers.104) Therefore, SOTs have been shown to move DWs in metallic heterostructures effectively.21,22)  10 The efficient SOT-induced switching in FMGs, as well as the increased interest in the interfacial DMI, have triggered research into DWs and their current-induced dynamics in perpendicular FMGs. In 2019, Avci et al. have shown, through measurements of SOT-induced DW motion, that TmIG and TbIG possess DMI-stabilized left-handed Néel DWs.105) A series of measurements with Pt and Cu/Pt capping layers on TmIG and TbIG indicated that the DMI predominantly originates from the FMG interface with the GGG substrate and not from the interface with the metal on top. This was the first direct demonstration of chiral magnetism occurring at a magnetic oxide interface. Shortly after, Vélez et al. and Ding et al. have reached the same conclusion that the GGG/TmIG substrate interface gives rise to the interfacial DMI and left-handed DWs in TmIG.95,96) Vélez et al., moreover, have found, through nitrogen-vacancy center magnetometry measurements, that the Pt capping contributes to the DMI with an opposite sign; thereby concluded that the DMI contributions from the two interfaces compete with each other (see Fig. 7). Nevertheless, the reported DMI effective fields were only in the order of a few mT but strong enough to turn the equilibrium DW configuration from the Bloch-type to the partially or fully Néel-type.  Later on, Ding et al. and Caretta et al. have confirmed the interfacial origin of the DMI through TmIG thickness-dependent experiments.100,106) Furthermore, Caretta et al. have reported that the DMI originates from the rare-earth orbital magnetism since the effect was absent in perpendicular Bi:YIG films but present in TmIG and TbIG grown on GGG substrates (Fig.8). Given the wide tunability of FMGs through composition and substrate choice, understanding the origins of DMI is an important step for designing materials with chiral spin textures. Despite experimental progress, though, the theoretical foundations of the interfacial DMI at oxide-oxide garnet interfaces remains yet to be established.  In the study mentioned above by Avci et al.,105) it was also shown that the DMI-stabilized chiral DWs in TmIG can be moved in a DW track (Fig. 9a) by current injection through the Pt overlayers as fast as 800 m/s with current densities of the order of 1.2x108 A/cm2 (Fig. 9b). The fast domain wall dynamics were a consequence of ferrimagnetic ordering and the absence of the precessional dynamics. In ferromagnetic systems such as Pt/Co, the DW velocity in the flow regime linearly increases as a function of current and saturates when the DL-SOT effective field becomes comparable to the DMI effective field.104) This is due to the rotation of the DMI-stabilized DW spins towards the transverse direction (parallel to 𝛕\"#) at large excitations. However, in ferrimagnets, the antiferromagnetic coupling between the two  11 sublattices prevents the net moment from precessing; hence the limiting factor is not anymore the DMI effective field but rather the much larger exchange field.25) This intriguing physics makes the ferrimagnetic materials appealing for DW-based ultrafast spintronic devices.  In compensated FMGs, additional to the magnetic compensation temperature, there exists a second temperature at which the angular momentum is compensated and the magnetization dynamics resembles to that of antiferromagnets, allowing fast domain wall motion.25,107) A recent study has shown that out-of-plane field assisted current-induced domain wall motion in GdIG with PMA can be as fast as 6000 m/s near the angular momentum compensation.54) This result marks the highest domain wall velocity reported to date and demonstrates yet other degrees of freedom offered by FMGs for ultrafast spintronics. Very recently, Caretta et al. have reported a fascinating feature of current-induced DW motion in FMGs98. They have shown that in Bi:YIG, DWs can be propelled at velocities in excess of 4300 m/s (Fig.9 c), the highest reported in any material so far without the need of an out-of-plane assisting field. The velocity was found to saturate to a universal limit above a critical current density and applied in-plane fields. Supported by analytical and atomistic modeling, this saturation anomaly was attributed to Lorentz contraction, a consequence of special relativity on the extremely fast-moving magnetic solitons. The reported velocities were calculated to be within 10% of the relativistic limit. These experiments open doors to realizing relativistic phenomena such as spin-wave Cherenkov radiation, higher-dimensional relativistic dynamics, and a host of related phenomena that previously seemed out of reach. And to achieve these behaviors in room-temperature, table-top experiments will hopefully lead to significant advances. Importantly, understanding the fundamental limits to the speeds of magnetic DWs and solitons, and the relation to the underlying material parameters can be used to maximize their speeds in future devices.  6. Signatures of Skyrmions in Ferrimagnetic Garnets The exciting discovery of chiral magnetism at FMGs’ interfaces and ultrafast DW motion have also accelerated the research into magnetic skyrmions in these materials. A skyrmion is a topologically protected magnetic texture stabilized by the DMI.108) Skyrmions can be as small as a few nm in diameter up to hundreds of nm determined by the material parameters, and can be moved by SOTs,27) Henceforth, they offer desirable prospects for efficient data storage, processing, and logic applications.108)  12 So far, the direct observation of skyrmions in FMGs has been elusive. Though, strong evidence of skyrmions in TmIG and YIG has been found through the measurements of the topological Hall effect109,110) (THE). The THE emerges due to the transverse deflection of electrons passing through the electromagnetic field generated by a skyrmion111) (Fig. 10a). It gives a Hall signal contribution proportional to the skyrmion density in the (Hall cross area of the) film and is generally characterized by additional bumps overlapping the standard Hall resistance curves within a characteristic out-of-plane applied field range (Fig. 10b). Characterization of the THE as a function of FMG thickness, temperature and applied magnetic fields has revealed that there is a phase pocket in which the THE emerges (Fig. 10c). The common observation in such studies is that the THE occurs in ultrathin FMG films (<6 nm), particularly at higher temperatures, when the PMA becomes weaker. The collective understanding of these findings pinpoints the existence of skyrmions in ultrathin FMGs at the right combination of different material and experimental parameters.  Independently of the above studies, bubble domains of only a few hundred nm of diameter (similar to typical skyrmion size) have been observed in relatively thick (25.6 nm) TmIG by scanning transmission x-ray microscopy.112) Photoemission electron microscopy analysis further showed that the DWs at the boundaries of the observed bubbles are of Bloch-type with arbitrary chirality. Such bubbles have been known since 1970s in micrometer-thick films,113) but the current study marked their first nucleation, observation and detailed characterization in nanometer-thick FMGs. The established sample fabrication procedure and experimental scheme for observing such small magnetic textures, combined with the efforts of electrically characterizing skyrmions in ultrathin FMGs holds promise for the direct observation and engineering of insulating skyrmions in the near future.  7. Conclusions The results summarized in this Review were obtained using only several FMGs and with a few years of experimental efforts. Yet, some impressive results such as highly competitive SOT-induced switching, interfacial chiral magnetism, ultrafast current-driven DW motion and relativistic current-induced dynamics have been achieved. The vast family of FMGs with many parameter tuning knobs combined with alternative SOT source materials creates a fertile ground for future studies and application prospects. Thus far, the SOT source has been limited to the SHE in Pt and W. There exist many other SOT materials and mechanisms including Weyl semimetals, topological insulators, two-dimensional electron gas interfaces, the interfacial  13 Rashba-Edelstein effect and more recently orbital Hall currents114) are to be considered for more effective magnetic manipulation of FMGs in suitable devices.  The advances in the deposition of FMGs shall inevitably open the doors to investigations of magnetic coupling phenomena in multilayers. Thus far, only single FMG layers have been considered in the studies summarized here. Sequential deposition of FMGs with varying magnetic and structural properties may lead to novel properties and functionalities that may not be achievable by individual layers. For instance, a recent report showed antiferromagnetic coupling between YIG and GdIG.115) Similarly coupled FMGs with PMA could reveal novel device concepts and architectures in electrically-controlled insulating spintronic devices.   The lack of in-depth understanding of the SOT-induced switching characteristics in FMGs opens new arenas for exploration. In all-metal systems, a significant effort has been spent to understand the SOT switching dynamics, device size and shape dependence, speed limits, etc. in the past couple of years, revealing outstanding results.116-118) For instance, the state-of-the-art switching speed in all-metal systems are in the order of a few picoseconds,119) some three orders of magnitude lower than that obtained in FMGs. Due to smaller Gilbert damping in FMGs in comparison to metallic ferromagnets, the switching dynamics in short timescales will be considerably different. FMGs are best suited to explore and understand the effect of the Gilbert damping on the switching, domain wall and skyrmion dynamics since it can be effectively tuned by the rare-earth choice. Efforts for understanding fast switching dynamics, device size and shape dependencies and the effect of the Gilbert damping in FMG-based devices are lagging behind and could reveal unpredictable twists. 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A charge current through the normal metal (e.g., Pt) can generate a spin accumulation at the interface with the FMG, which then generates a damping-like SOT on the magnetization. The resulting effective field (HDL) can switch the magnetization between the up and down state in the presence of an in-plane field applied along the current injection direction that breaks the rotational symmetry of HDL. (Adapted from Ref. 87)          \n 23   Figure 2 – (Color online) (a) Damping-like and field-like SOTs (𝝉IJ, 𝝉KJ), and their associated effective fields (𝐻IJ, 𝐻KJ) for the given magnetization and current injection geometry. Note that the torque/field signs and amplitudes are arbitrary as they are material-dependent. (b) A visualization of the harmonic measurements. The magnetization vector oscillates with an a.c. current injection due to the oscillating SOTs and produce harmonic Hall voltage signals. (Adapted from Ref. 38)       \n 24  Figure 3 – (Color online) (a) Schematic of a setup for the Hall effect measurement on a device made of TmIG/Pt. (b) Typical Hall resistance signal for TmIG/Pt layers with PMA, recorded during an out-of-plane field sweep, consisting of the anomalous and ordinary Hall effect contributions as shown. (c) Hall resistance measured with an in-plane field (HIP) sweep applied at various angles. The curves at 45° and 135° represent the typical SMR behavior where a large positive and negative signal appears, and m is fully rotated in-plane. Orange curves are macrospin simulations based on experimental parameters and estimated PMA of about 2700 Oe. (d) Hall resistance signal amplitude measured at various in-plane angles reveals the expected sin2j behavior (see Eq. 1). (Adapted from Ref. 86) \n 25  Figure 4 – An electrical detection toolbox for magnetic insulators having out-plane (top row) or in-plane (bottom row) equilibrium magnetization orientation.        \n 26  Figure 5 – (Color online) Current-induced magnetization switching experiments in TmIG/Pt bilayers on Hall bar devices similar to the one in Fig. 3a. (a) Reference Hall resistance signal after subtraction of the OHE slope. (b) Magnetization switching experiments by consecutive application of 5 ms-long pulses through the device in the presence of an in-plane field of +/-500 Oe. The switching polarity reverses upon inverting the field polarity, as expected of the SOT-induced switching behavior. (c) A current sweep in the presence of a constant in-plane field, showing reversible switching at critical current density of about 1.8x1011 A/m2. (d) Switching phase diagram constructed based on measurements reported in (c) with different in-plane fields. (e) Partial switching of TmIG by electrical pulses of 1-5 ns length and 25-35 V \n 27 amplitude. 100% corresponds to the full switching of the Hall cross area and 0% corresponds to no switching. (f) In-plane field dependence of the switching characterized by stepping the field and applying positive and negative pulses and measuring RH consecutively. These measurements have revealed an in-plane field requirement as low as 2 Oe for switching TmIG.  (Adapted from Refs. 86 and 94) \n Figure 6 – (Color online) TmIG saturation magnetization (Ms) dependence of the damping-like SOT efficiency in TmIG/W bilayers. The enhanced torque efficiency at larger Ms (which is obtained at larger TmIG thicknesses) is attributed to the stronger exchange coupling between the SHE-induced interface spin accumulation and the local magnetization in TmIG. (Adapted from Ref. 99)  \n 28  Figure 7 – (Color online) Nitrogen-vacancy center microscopy characterization of domain wall internal magnetization in TmIG and TmIG/Pt bilayers. (a) Schematics of the experimental setup and the DW types. (b) Color map of the magnetization orientation measured by magnetic stray fields (BNV) showing a straight vertical domain wall crossing the Pt edge. (c,d) BNV measured along the line scans on the TmIG/Pt and bare TmIG regions. Fits showing that the domain wall in TmIG/Pt is a Bloch-type while in TmIG only it is a left-handed Néel type.  (Adapted from Ref. 96) \n 29  Figure 8 – (Color online) The interfacial DMI energy measured in various FMG-based systems. (Adapted from Ref. 106)          \n 30  Figure 9 – (a) (Color online) A typical DW device with integrated Hall arms. The magnetization can be probed either by a laser spot, as shown, or magnetic imaging by using magneto-optic Kerr effect.  DW velocity measurements as a function of the current density in TmIG/Pt (b) and in Bi:YIG/Pt (c). (Adapted from Refs. 98 and 105) \n 31  Figure 10 – (Color online) (a) Schematics of a magnetic sykrmion and the emergence of the topological Hall effect due to polarization-dependent electron deflection. (b) The transverse Hall resistivity in TmIG(1.85 nm)/Pt at room temperature as a function of an out-of-plane field sweep and the extracted topological Hall effect signal (bottom panel). (c) Colored phase diagram showing the temperature and field dependence of the emergent topological Hall signal in layers with different TmIG thickness. (Adapted from Refs. 110 and 111)  \n"    },    {        "title": "1504.01951v1.Magnetic_and_nonmagnetic_phases_in_doped_AB2_t_J_Hubbard_chains.pdf",        "content": "arXiv:1504.01951v1  [cond-mat.str-el]  8 Apr 2015Magnetic and Nonmagnetic Phases in Doped AB2t-JChains\nR. R. Montenegro-Filho and M. D. Coutinho-Filho\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, 50670-901, Recife-PE , Brazil\nWe discuss the rich phase diagram of doped AB2t-Jchains using data from DMRG and exact\ndiagonalization techniques. The Jvsδ(hole doping) phase diagram exhibits regions of itinerant\nferrimagnetism, Incommensurate, RVB, and Nagaoka States, Phase Separation, and Luttinger Liq-\nuid (LL) Physics. Several features are highlighted, such as the modulated ferrimagnetic structure,\nthe occurrence of Nagaoka spin polarons in the underdoped re gime and small values of J= 4t2/U,\nwheretis the first-neighbor hopping amplitude and Uis the on-site repulsive Coulomb interaction,\nincommensurate structures with nonzero magnetization, an d the strong-coupling LL physics in the\nhigh-doped regime. We also verify that relevant findings are in agreement with the corresponding\nones in the square and n-leg ladder lattices. In particular, we mention the instabi lity of Nagaoka\nferromagnetism against Jandδ.\nI. INTRODUCTION\nThet-Jversion of the Hubbard Hamiltonian [1] is a\nkey model for the understanding of strongly correlated\nelectron systems. The model is defined through only\ntwocompeting parameters: the hopping integral t, which\nmeasures the electron delocalization through the lattice,\nand the exchange coupling J= 4t2/U, whereU >> tis\ntheon-siteCoulombrepulsion. In fact, severalversionsof\nthe simplest Hubbard Hamiltonian, with a single orbital\nat each lattice and the on-site Coulomb repulsion, have\nbeen extensively used to model a variety of phenomena,\nsuch as: metal-insulator transition [2], quantum mag-\nnetism [3] and High- Tcsuperconductivity [4]. Moreover,\nexact solutions [1] and rigorous results [5, 6] have played\na central role in this endeavor.\nWe emphasize Lieb’s theorem [7], a generalization of\nthe one by Lieb and Mattis [8] for Heisenberg systems,\nwhich asserts that the ground state (GS) total spin of\na bipartite lattice at half filling and U >0 is given\nbySGS=|NA−NB|/2, where NA(NB) is the num-\nber of sites on sublattice A(B); indeed, Lieb’s theorem\nhas greatly enhanced the investigation of new aspects of\nquantum magnetism [6]. In particular, we mention the\noccurrence of ferrimagnetic GS, in which case we select\nstudiesusingHubbardor t-Jmodels[9–15], includingthe\nHeisenberg strong-coupling limit [16–18], on chains with\nAB2orABCtopological structures with SGS= 1/2 per\nunit cell [9–13, 16, 17], which implies ferromagnetic and\nantiferromagnetic long-range orders [10]. Further, the\ninclusion of competing interactions or geometrical and\nkinetic frustration [19–21], enlarge the classes of models,\nthereby allowing ground-states not obeying Lieb or Lieb\nandMattis theorems. Thesestudieshaveprovedeffective\nin describing magnetic and other physical properties of a\nvariety of organic, organometallic, and inorganic quasi-\none-dimensional compounds [19, 22].\nOf particular physical interest are doped systems, al-\nthough in this case rigorous results are much rare [6].\nOne exception is Nagaoka’s theorem [23], which asserts\nthat for J= 0 (U→ ∞) thet-Jmodel with one holeadded to the undoped system (half-filled band) is a fully\npolarized ferromagnet , favored by the hole kinematics, if\nthe lattice satisfies the so-called connectivity condition\n[24]. A long-standing problem about this issue is the sta-\nbility of the ferromagnetic state for finite hole densities\nand finite values of J. Numerical results have indicated\n[25, 26] that two-dimensional lattices display a fully po-\nlarized GS for J= 0 and δ/lessorsimilar0.2, where δ=Nh/N, with\nNh(N) the total number of holes (sites); while, analyti-\ncal studies [27, 28] havesuggested that this state is stable\nup toJt∼δ2.\nFurther, an ubiquitous phenomenon in doped strongly\ncorrelated materials is the occurrence of inhomogeneous\nstates, particularly spatial phase separation in nano- and\nmesoscopic scales [29] and incommensurate states [29,\n30]. In underdoped High- Tcmaterials, dynamical and\nstatical stripes in copper oxide planes has been the focus\nof intensive research [31]. Concerning two-dimensional t-\nJorHubbardmodels, phaseseparationintohole-richand\nno-hole regions was discussed in the large −and small −J\nlimits [32]. Howeverthe precisechargedistributionin the\nground state remains controversial. The use of distinct\nand refined numerical methods have pointed to striped\n[33] or uniform phases [34]; recently, it was claimed that\nthe origin of this issue relies on the strong competition\nbetween these phases [35]. For the linear t-JHubbard\nchain the physics is more clear [36], and phase separation\ntakes place for J= 2.5−3.0, depending on the doping\nvalue, but it is absent in the small −Jregime.\nIn this work, we use Density Matrix Renormalization\nGroup (DMRG) [37] technique and Lanczos exact diago-\nnalization(ED)toobtainthegroundstatephasediagram\nand the low-energy excitations properties of the doped t-\nJmodel on AB2chains [9] for J= 0.0−0.4. We verify\nthe occurrence of an itinerant modulated ferrimagnetic\n(FERRI) phase in the underdoped regime, regions of in-\ncommensurate (IC) states and Nagaoka ferromagnetism\n(F), and two regions of phase separation (PS), in which\nIC and F states coexist with the resonating valence bond\nstate (RVB), respectively. In addition, we find that the\nRVB state is the stable phase at δ= 1/3, and identify\na crossover region that ends at the onset of a Luttinger2\n0 0.23 1/30.1δ00.10.20.30.4 J\nFERRI PS\nFIC\nPS(IC - RVB)\n(F - RVB)δPS, J\nδFERRI, J\nJF, δ\n2/31CrossoverLL\n(RVB)~~(a)\n(b)\nB\nB\nδ = 1/3RVB at F\nδ = 2/3A\n21FERRI\nOnset of LL at IC\n00.050.10.15\nδ00.51\nSGS / SLJ = 0.1\nJ = 0.3(c)\nFIG. 1. (Color online). (a) GS phase diagram for the AB2\nt-Jmodel (error bars account for the discrete values assumed\nbyδin a finite-size system). The phases are illustrated in\n(b): modulated ferrimagnetism (FERRI), incommensurate\n(IC), Nagaoka ferromagnetism (F), short-range resonating va-\nlence bond (RVB) states, phase separation (PS), and Lut-\ntinger liquid (LL). The estimated transition lines δFERRI,J,\nδPS,J, andJF,δare also pointed out. (c) Ground state total\nspin,SGS, normalized by its value in the undoped regime:\nSL≡(Nc/2)−0.5, as function of δfor the indicated values\nofJandN= 3Nc+1 = 100.\nliquid (LL) phase at δ= 2/3, above which the LL physics\n[38] sets in.\nII. PHASE DIAGRAM\nThet-Jmodel reads:\nHt−J=−t/summationdisplay\n<i,j>,σPG(c†\niσcjσ+H.c.)PG(1)\n+J/summationdisplay\n<i,j>(Si·Sj−1\n4ninj),\nwhereciσannihilateselectronsofspin σat sitei,niisthe\nnumber operator at site iandPG=/producttext\ni(1−ni↑ni↓) is the\nGutzwiller projector operator that excludes states with\ndoubly occupied sites. In our simulations, we set t= 1\nand have consideredchains with Nc(N) unit cells (sites).\nIn ED calculations closed boundary conditions are used\nwithNc= 8 (N= 3Nc), while in the DMRG simulations\nopen boundary conditions are used and the system sizesB+B2 1B+B2 1B+B2 1B+B2 1A(a)\nA A A A4321l = a\n00.5<Sz\nl> (b)δ = 0.04 J = 0.1 <nh,l>\n-0.500.5(c)δ = 0.18 J = 0.1\nFIG. 2. (Color online). (a) Effective linear chain (spacing\na≡1) associated with N= 3Nc+ 1 = 100 sites for J= 0.1\nused to illustrate the hole, /angbracketleftnh,l/angbracketright, and spin, /angbracketleftSz\nl/angbracketright, profiles: (b)\nδ= 4/100 (FERRI phase) and (c) δ= 18/100 (IC phase).\nranged from Nc= 33 (N= 3Nc+1 = 100) to Nc= 121\n(N= 364). We retain from 243 to 364 states in the\nDMRG calculations, and the typical discarded weight is\n1×10−7.\nThe ground state (GS) phase diagram, shown in Fig. 1\n(a), displays the regions of the above-mentioned phases,\nillustrated in Fig. 1(b), including the estimated transi-\ntion lines and the crossover region. A special feature of\ntheAB2chain is its symmetry [12, 13, 17] under the ex-\nchange of the labels of the Bsites in a given unit cell l\n[identified in the FERRI state, Fig. 1(b)]. This symme-\ntry implies in a conserved parity pl=±1 in each cell of\nthe lattice. The phase diagram of a chain with Ncunit\ncells is calculated by obtaining the lowest energy for all\nsubspaces with xcontiguous cells of parity −1 and the\nothersNc−xcells with parity +1, with x= 0...Nc,\nfor fixed δandJ. In the phase diagram shown in Fig.\n1(a),p≡/summationtextNc\ni=1pl= +1 for δ≥1/3,p/negationslash=±1 in the PS\nregion, and p=−1 forδ < δPS,J. The magnetic configu-\nration of a phase is identified by the total spin SGS, local\nmagnetization, magnetic structure factor, and spin cor-\nrelation functions. In what follows, we shall characterize\nthe phases shown in Fig. 1(a).\nIII. FERRIMAGNETISM AND TRANSITION\nTO IC STATES\nAtδ= 0 and J/negationslash= 0, the insulating Lieb ferrimag-\nnetic state with total spin quantum number SGS=SL≡\nNc/2−0.5≡SLis found for a chain with open bound-3\n0 0.5q/π01S(q)0\n0.02\n0.04\n0.06\n0.08\n0.10\n0.12(a)δJ = 0.3\n0.5 1q/π0123S(q) 00.05 0.1δ00.10.20.30.4\n∆q/π(b)\nFIG. 3. (Color online). Chain with N= 3Nc+1 = 100 and\nJ= 0.3. (a) and (b): Magnetic structure factor S(q) for the\nindicated values of δ. Inset of (b): ∆ q≡qmax−π, whereqmax\nis the value of qat which the local maximum of S(q), near\nq=π, is observed.\nary conditions, N= 3Nc+ 1 = 100, with an Asite on\neach side. In order to evaluate the stability of this state\nagainst doping, we calculate SGSas a function of δfrom\nthe energy degeneracy in Sz. As shown in Fig. 1(c),\nas hole doping increases from δ= 0 to a critical value\nδ=δFERRI,J, the value of SGSdecreases linearly from\nSLto 0 or a residual value, signaling a smooth transi-\ntion to the IC phase. However, for low enough J,SGSof\nthe IC phase increases linearly with δup toδ=δPS,J,\nthe line at which PS occurs [see Fig. 1(a)], or up to\nthe boundary, JF,δ, of the Nagaoka F phase. This unex-\npected behavior claims for an explanation.\nIn order to understand the behavior of SGSfor low\nJwe have calculated the profiles of the magnetization,\n/angbracketleftSz\nl/angbracketright, in the spin sector Sz=SGS, and of the hole den-\nsity,/angbracketleftnh,l/angbracketright, forJ= 0.1 (see Fig. 2). To help in the data\nvisualization, we use a linearized version of the lattice,\nas illustrated in Fig. 2(a). As shown in Fig. 2(b), for\nδ= 0.04 the holes distort the ferrimagnetic structure,\nwhich display a modulation with wavelength λ≈17, in15 30 45 60l\n-0.4-0.200.20.40.6 0.53IC(p = −1)\nRVB\n(p = +1)\n J = 0.30.280.44\n0.20\n<nh,l>\n<SB1SB2>\nFIG. 4. (Color online). Phase separation (IC-RVB) for a\nchain with N= 3Nc+ 1 = 100 sites, J= 0.3, andNh= 18\nholes: spin correlation function between Bspins at the same\ncell,/angbracketleftSB1,l·SB2,l/angbracketright, and hole density profile, /angbracketleftnh,l/angbracketright.\nanti-phase with that exhibited by the hole (charge) den-\nsity wave. We have thus identified a modulated itinerant\nferrimagnetic phase in this underdoped regime. On the\nother hand, as shown in Fig. 2(c), for δ= 0.18 the mag-\nnetization has local maxima in coincidence with those of\nthe holedensity profile. In this case, the IC phaseis char-\nacterized by the presence of ferromagnetic Nagaoka spin\npolarons [28, 39] due to hole density wave with λ≈4.\nOur results point to a value of J(∼0.2) below which\nferromagnetic “bubbles” appear as precursors of the F\nphase found for J < JF,δ[see Fig. 1(a)].\nForJ= 0.3,SGS= 0 in the IC phase, as shown in\nFig. 1(c). In Figs. 4 (a) and (b) we present the magnetic\nstructure factor\nS(q) =1\nSL(SL+1)2Nc+1/summationdisplay\nl,meiq(l−m)/angbracketleftSl·Sm/angbracketright,(2)\nwherel,mandSrefertothe latticerepresentationshown\nin Fig. 2(a), for this value of Jand doping ranging from\nδ= 0 up to δ= 0.12. In a long-range ordered ferrimag-\nnetic state, sharp maxima at q= 0 (ferromagnetism) and\nq=π(antiferromagnetism) are observed in the curve\nS(q) forδ= 0. Adding two holes to the undoped state,\nsharp maxima at q= 0 and πare also observed, while\nbroad maxima occur for δ= 0.04, indicating short-range\nferrimagnetic order which evolves to the IC phase by in-\ncreasingdoping, beforephaseseparation(IC-RVB)atthe\nlineδ=δPS,J[see Fig. 1(a)]. In the inset of Fig. 4(b) we\nshow the departure of the maximum of S(q) fromq=π.\nIV. PHASE SEPARATION, RVB STATES AND\nLUTTINGER LIQUID\nIn Fig. 1(a) the dashed line inside the PS regionfix the\nboundary between two types of phase separation: in one\ncase, the separation occurs between Nagaoka ferromag-\nnetismandshort-rangeRVBstates(F-RVB); whileinthe\nother, it occurs between IC and short-range RVB states4\n2/3 1\nδ0.81R0.00\n0.05\n0.10\n0.15\n0.20\n0.25\n0.30\n0.35\n0.40\n2/3 1\nδ0.51\nKρ(a) (b)J\nFIG. 5. (Color online). Luttinger liquid behavior for a chai n\nwithN= 3Nc= 24 (ED results). (a) Ratio R=uρ//radicalbig\nDχ/π\nas a function of δfor the indicated values of J. (b) Exponent\nKρas a function of δ.\n(IC-RVB). Indeed, for 0 ≤J/lessorsimilar0.063 and δF−RVB≤\nδ <1/3, the GS phase separates with F and short range\nRVB states under coexistence, where δF−RVBdenotes\nhole density values along the phase separation line F-\nRVB, thereby extending our previous result [13] valid\nonly for J= 0. However, for 0 .063/lessorsimilarJ≤0.4 the sys-\ntem behaves differently. The new PS (IC-RVB) region is\nhere illustrated for J= 0.3,N= 3Nc+ 1 = 100 sites,\nandNh= 18 holes: we thus find that there are 26 cells\nwith odd parity ( pl=−1), associated with the IC phase,\nand the remaining 7 cells with even parity ( pl= +1),\nassociated with the RVB phase. In this case, as shown\nin Fig. 4(c), the hole-poor IC phase presents a local\nspin correlation function /angbracketleftSB1,l·SB2,l/angbracketright ≈0.2, average\nhole density per site ≈0.16, estimated from the sites\nindicated by arrows [one Asite and two Bsites in the\ncontext of the effective linear chain shown in Fig. 2(a)],\nand hole-density wave with λ≈4; while the hole-rich\nRVB phase presents /angbracketleftSB1,l·SB2,l/angbracketright ≈ −0.4 and average\nhole density per site ≈1/3, estimated from a cell with A\nandBsites indicated by arrows. Therefore, apart from\nboundary effects, the above results thus indicate that the\nphase separation for a given Jvalue is defined by the co-\nexistence ofthe two phaseswith the hole densities δIC-PS\n(≈0.16 forJ= 0.3) andδPS-RVB(≈1/3 forJ= 0.3)\nfixed at the IC-PS and PS-RVB boundaries, respectively,\nwhile the size of the phases are fixed by the chemical\ndopingδ=Nh/N(= 0.18 forN= 100 and Nh= 18).\nWe also remark that the stable RVB phase observed at\nδ= 1/3 and 0 ≤J≤0.4, which has finite charge and\nspin gaps, is in agreement with predictions for J= 0.35\n[12] and J= 0 [13].\nFor 0≤J≤0.4 and 1/3< δ <2/3, a crossover\nregion with the presence of long-range RVB states after\nhole addition away from δ= 1/3 is observed [see Fig.\n1(a)]. At the commensurate filling δ= 2/3, the system\npresentsachargegap,whilethespinexcitationisgapless,\nalso extending our previous result for J= 0 [13].\nWith the aim of investigating the LL behavior as a\nfunction of Jandδ≥2/3, we have calculated, through0 0.2 0.4 0.6 0.8 11.2\nJ / JF,δ(Nc)-0.002-0.0010(EGS− EF) / Nc\nNc = 33, δ = 0.100\nNc = 67, δ = 0.109\nNc = 121, δ = 0.104(a)\n00.05 0.1\nδ00.010.020.03\nJF,δNc=33\nNc=67\nNc=121\n0.6 δ2 + 3.3 δ3(b)\n0 10 20 30 40 5060l\n-0.500.51<Sz\nl>\nSz = 45, J = 0.0000\nSz= 39, J = 0.0025\nSz = 39, J = 0.0050\nSz = 39, J = 0.0075(c)\nδ = 0.1\n0 10 20 30 40 5060\nl00.050.10.150.2\n<nh, l>\nSz= 39, J = 0.0025\nSz = 39, J = 0.0050\nSz = 39, J = 0.0075\nNon-interacting \nspinless fermions(d)\nδ = 0.1\nFIG.6. (Color online). (a)Shift EGS−EFperunitcell, where\nEFis the energy of the fully polarized ferromagnetic state, as\na function of J/JF,δfor the indicated values of Ncandδ. (b)\nInstability line of the Nagaoka ferromagnetic phase. (c) Sp in\nand (d) hole profiles, /angbracketleftnh,l/angbracketrightand/angbracketleftSz\nl/angbracketright, respectively, for a chain\nwithN= 3Nc+1 = 100, δ= 0.1, and the indicated values of\nSzandJ.\nED techniques, the ratio R=uρ//radicalbig\nDχ/π,where\nχ=Nc\n4[E(Nh+2)+E(Nh−2)−2EGS(Nh)] (3)\nis the charge susceptibility, and E(Nh±2) is the total\nenergy for Nh±2 holes;\nD=Nc\n4π/bracketleftbigg∂2E(Φ)\n∂Φ2/bracketrightbigg\nΦmin(4)\nis the Drude weight, where E(Φ) is the lowest energy for\na magnetic flux Φ through a closed chain, and Φ minits\nvalue at EGS;\nuρ=E(kGS+∆k,S= 0)−EGS(kGS,SGS= 0)\n∆k(5)\nis the charge excitation velocity, where ∆ k= 2π/Nc, and\nE(kGS+∆k,S= 0) is the lowest energy with wavenum-\nberk=kGS+ ∆kand total spin S=SGS= 0. If\nthe low-energy physics of the system is that of a LL, we\nshould find R= 1 [40]; moreover, the exponent govern-\ning the asymptotic behavior of the correlation functions,\nKρ, satisfies the relation Kρ=πuρ/2χ. As shown in\nFig. 5(a), Ris indeed very close to 1 for δ >2/3; in\naddition, as shown in Fig. 5(b), we find 0 .7/greaterorsimilarKρ/greaterorsimilar0.5\nforδ >2/3. Remarkably, as shown in Figs. 5(a) and (b),\nthe data for RandKρexhibit data collapse as a function\nofδfor 0≤J≤0.4. In short, the results above clearly\nindicate that for δ >2/3 and 0 ≤J≤0.4 the system\nbehaves as a LL in the strong coupling regime.5\nV. STABILITY OF NAGAOKA\nFERROMAGNETISM\nIn this Section, we shall provide strong evidence that\nfor0≤J≤JF,δand 0< δ≤δF−RV B, the kinetic energy\nofholesisloweredinafullypolarizedferromagneticstate,\nan extension of Nagaoka ferromagnetism [23, 24], with\nthe GS energy equal to that of non-interacting spinless\nfermions: EGS=EF.\nThe estimate of JF,δis based on the data for the shift\n(EGS−EF)/Ncas a function of J, as illustrated in Fig.\n6(a) forδclose or equal to 0.1. We stress that the shift\ndecreases as Ncincreases for 0 <(J/JF,δ)<1, and goes\nto zero in the thermodynamic limit. In addition, one\nshould notice that, by examining the data above and be-\nlowJ=JF,δ, particularly for N= 3Nc+1 = 364 sites,\n∂EGS/∂Jappears to be discontinuous at J=JF,δin the\nthermodynamic limit, thus suggesting a first-order tran-\nsition to the IC phase at ( J/JF,δ) = 1. In Fig. 6(b) we\nshow that our estimated transition line, JF,δ, [see also\nFig. 1(a)] is almost not affected by finite size effects and\nimpliesδF,J∼√\nJasδ→0, as found from analytical\nresults [27, 28] for the t-Jmodel in a square lattice. In\nparticular, for J= 0 the instability of the Nagaoka state\noccurs at δ≈0.23, which is very close to the values of\nhole doping estimated for n-leg ladder systems [25] and\nthe square lattice [25, 26].\nThe spin profile for a chain with N= 3Nc+1 = 100,\nJ/greaterorsimilar0andδ= 0.1isalsoin verygoodagreementwith the\nNagaoka state, as shown in Fig. 6(c), although boundaryeffects are visible for J/greaterorsimilar0; in fact, Szchanges from 45\nto 39 (on average, three spins at each boundary are not\nfully polarized), but one should notice that the change\nsaturates as Jslightly increases above zero. This fact\nis corroborated by the hole density shown in Fig. 6(d),\nwhosedataforthe referredstateswith Sz= 39atδ= 0.1\nare very well described by the Nagaoka profile.\nVI. DISCUSSION AND CONCLUDING\nREMARKS\nThe presented phase diagram of doped AB2t-Jchains\nis remarkably rich. Indeed, several magnetic and non-\nmagnetic phases manifest themselves in a succession of\nsurprising relevant features, some of which are similar to\nthose observed in the square and n-leg ladder lattices:\nall in a simple doped chain. In particular, we empha-\nsize the modulated ferrimagnetic structure, the occur-\nrenceof Nagaokaspin polaronsin the underdoped regime\nand small values of J, incommensurate structures with\nnonzero magnetization, the strong-coupling LL physics\nin the high-doped regime, and the instability of Nagaoka\nferromagnetism against Jand doping. 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Rev B\n53, 11721 (1996)"    },    {        "title": "1511.01985v1.Coupled_Cluster_Treatment_of_the_Alternating_Bond_Diamond_Chain.pdf",        "content": " \n 1Coupled Cluster Treatment of the Alternating Bond D iamond Chain \nJian-Jun Jiang 1, Yong-Jun Liu 2, Fei Tang 3, and Cui-Hong Yang 4 \n 1Department of Physics, Sanjiang College, Nanjing 21 0012, China  \n2School of Physics Science and Technology, Yangzhou University, Yangzhou 225002, China  \n3Department of Electronic and Information Engineerin g, Yangzhou Polytechnic Institute, Yangzhou 225127,  China \n4Faculty of Mathematics and Physics, Nanjing Univers ity of Information Science and Technology, Nanjing 210044, China \nAbstract: By the analytical coupled cluster method (CCM), we study both the ground state and \nlowest-lying excited-state properties of the altern ating bond diamond chain. The numerical exact \ndiagonalization (ED) method is also applied to the chain to verify the accuracy of CCM results. \nThe ED results show that the ground-state phase dia gram contains two exact spin cluster solid \nground states, namely, the tetramer-dimer (TD) stat e and dimer state, and the ferrimagnetic \nlong-range-ordered state. We prove that the two exa ct spin cluster solid ground states can both be \nformed by CCM. Moreover, the exact spin gap in the TD state can be obtained by CCM. In the \nferrimagnetic region, we find that the CCM results for some physical quantities, such as the \nground-state energy, the sublattice magnetizations,  and the antiferromagnetic gap, are comparable \nto the results obtained by numerical methods. The c ritical line dividing the TD state from the \nferrimagnetic state is also given by CCM and is in perfect agreement with that determined by the \nED method. \n1. Introduction  \nLow-dimensional frustrated quantum spin systems hav e attracted continuous \nattention in recent years 1-10) . Owing to the interplay of strong quantum fluctuat ion in \nlow dimensionality and frustration, low-dimensional  frustrated quantum spin systems \nmay possess not only the state with quasiclassical magnetic order, but also exotic \nquantum states with no classical counterpart, such as the spin-liquid state and \nvalence-bond state 3, 8-10) . The impetus to the study of the properties of fru strated \nquantum spin systems was considerably enhanced afte r it was pointed out that there is \na close connection between spin liquids and the hig h-Tc superconductivity of strongly \ncorrelated systems 11) . Although most low-dimensional frustrated quantum spin \nsystems  cannot exactly solved, there are still some exactly  solvable ones, which are \nhelpful for people to understand the nature of the exotic quantum phase 12-14) . The \nMajumdar-Ghosh model, which has been intensively st udied, is a well-known \nexample 12) . The properties of some quasi-one dimensional mate rials, such as CuGeO 3, \ncan be described using that model 15) . The ground state (GS) of the Majumdar-Ghosh \nmodel is a spontaneous dimer state, which is the si mplest type of spin cluster solid  \n 2state  that is a tensor product of the exact local eigenst ates of cluster spins 16, 17) . \nAnother prototypical spin chain that possesses the exact spin cluster solid state, which \nis different from that of the Majumdar-Ghosh model,  is the pure spin-1/2 diamond \nchain 14) . The GS phase diagram of that chain is composed of  the ferrimagnetic phase, \ndimer phase, and TD phase. The latter two phases bo th belong to the exact spin cluster \nsolid state. Modifications of the diamond chain hav e attracted considerable attention \nin recent years, because of their rich spin cluster  solid state 16-22) . In addition to the \nvalue of theoretical research, it is found that a v ariant of the diamond chain called the \ndistorted diamond chain can be used to describe the  magnetic lattice of the mineral \nazurite, Cu 3(CO 3)2(OH) 2 18) . \nDespite the fact that the diamond chain and its var ious modifications have the exact \nspin cluster solid GS  in a certain parameter region, people often mainly used \nnumerical methods, such as the ED method and densit y-matrix renormalization group \n(DMRG) method, to obtain the accurate GS phase diag ram and lower-lying \nexcited-state properties of those chains in previou s research  14, 16-22) . Thus, it is an \nopen question whether an analytical method can give  the precise properties of the \nquantum diamond chain or its modifications. To answ er the above question, we \ninvestigated in this work the properties of an alte rnating bond diamond chain (ABDC), \na variant of the diamond chain, by an analytical me thod called CCM.  \nCCM is one of the most powerful methods of the quan tum many-body theory 23) . \nOver the last twenty or so years, it has been appli ed with much success to various \nfrustrated quantum spin systems in any dimension 8, 24-49) . Previous research shows \nthat the results obtained by CCM are fully competit ive with those obtained by other \nmethods, such as series expansions and  quantum Monte Carlo (QMC) 26, 29) . In this \nwork, we aim to use CCM to investigate the properti es of the ABDC. Our results \nindicate that CCM can yield accurate results for th e GS and lower-lying excited state \nregardless of whether the GS of that chain is in th e exotic quantum spin cluster state \nor ferrimagnetic state. \nAs shown in Fig. 1, the model Hamiltonian of the AB DC is  \n 3( ) ( ) ∑∑ ∑\n=− +\n=−\n=− − +⋅ +⋅ ++⋅ =3 /\n13 1 3 1 3 33 /\n13 1 3 23 /\n13 1 3 2 3 1N\nii i iN\nii iN\nii i i s s s J s s J s s s J Hrrr rr rrr, \nwhere is3r, 1 3−isr, and 2 3−israre spin-1/2 operators, and 1J, 2J, and 3J are \nantiferromagnetic interactions. As the unit cell is  made up of three sites, the total \nnumber of sites is N. For convenience, in what follows, we set αα==1 2 J J  and \nββ==1 3 J J . At 0=β , model (1) decouples into 3 /N  isolated triangles. The state \nof each spin triangle can be easily obtained. Thus,  we will discuss the property of the \nHamiltonian (1) at 0 >β  in the following discussion.  \nThe rest of the paper is arranged as follows. In Se ct. 2, we determine the quantum \nphase diagram of the model by the ED method. In Sec t. 3, CCM is used to discuss the \nproperty of the model. In Sect. 4, the CCM results are shown. A summary is given in \nSect. 5. \n2. Quantum Phase Diagram of the Alternating Bond Di amond Chain \nTo check the results of the CCM in the next section , we first give the quantum \nphase diagram of the ABDC by ED. Firstly, we consid er some special cases of the \nHamiltonian (1). The first one is the case of 0 =α . In this case, the Lieb-Mattis \ntheorem implies that the  magnitude of the total spin of the GS of the Hamilt onian (1), \ndefined by ∑\n==N\nll tol s s\n1rr, is 6 / N  50) . The ABDC possesses ferrimagnetic long-range \norder, and it is equivalent to an alternating bond antiferromagnetic mixed spin (1, 1/2) \nHeisenberg chain 51, 52) . It is reasonable to expect that the ferrimagnetic  long-range \norder will extend to a small- α-parameter region. In contrast, in the limiting cas e of \n∞→α , the spins 1 3−isr and is3r form a singlet dimer and the GS is the disordered \ndimer state. As all spins 2 3−isr are decoupled from each other in the dimer state, there \nis a 3 /2N-fold degeneracy in that state. The GS energy per u nit cell for the dimer state \nis given by  \nα75 . 0D− = e . (1) \n(2)  \n 4The case of 1=α  and 1=β  is the third one that we will discuss. The system reduces \nto a uniform diamond chain (UDC) in that case 14, 53) . The spin cluster state called the \nTD state is the exact GS of UDC when 1 =α  14, 18) . As displayed in Fig. 2, the \nquadruplets 2 3−isr, 1 3−isr, is3r, and 1 3+isr (1 3+isr, 2 3+isr, 3 3+isr, and 4 3+isr) of spins form \nsinglet tetramers, and the pairs 2 3+isr and 3 3+isr (1 3−isr and is3r) of spins construct \nsinglet dimers in the TD state. Let us define the f ollowing composite spin operators: 16, \n21, 53)  \n3 2 3 1 3 3 1 i i i i i q s s s s − − + = + + + r r r r r , \n3 1 3 i i i t s s −= + rr r . \nThen, the TD state can be represented as \n) (\n21 2\nTD1\nTD TD ψψ ψ ± =±, \nwhere 1\nTDψ and 2\nTDψ  are degenerate and they have the forms \n∏∏\n=−=− −\n=== ==== =\n6 /\n12 2 1 22\nTD6 /\n12 1 2 1 21\nTD\n1 , 0 , 00 , 1 , 0\nN\nii i iN\nii i i\nt q tt t q\nψψ\n \nwhere  \n) 2 2(\n121) (\n211, 0 , 0) (\n21) 2 2(\n1210 , 1 , 0\n4 3 3 3 2 3 1 3 4 3 3 3 2 3 1 34 3 3 3 2 3 1 3 4 3 3 3 2 3 1 3 4 3 3 3 2 3 1 34 3 3 3 2 3 1 3 3 1 3 3 1 3 2 2 1 23 3 2 3 3 3 2 31 3 3 1 3 2 3 1 3 3 1 3 2 3 1 3 3 1 3 2 3 1 3 3 1 3 2 31 3 3 1 3 2 3 1 3 3 1 3 2 3 2 1 2 1 2\n++++ ++++++++ ++++ ++++++++ − − −++ +++ −− + −− + −− + −−+ −− + −− − −\n↑↓↓↑+↓↑↑↓+↑↓↑↓−↓↑↓↑−↑↑↓↓−↓↓↑↑ −⊗↑↓−↓↑====↑↓−↓↑⊗↑↓↓↑+↓↑↑↓+↑↓↑↓−↓↑↓↑−↑↑↓↓−↓↓↑↑ −====\ni i i i i i i ii i i i i i i i i i i ii i i i i i i i i i ii i i ii i i i i i i i i i i i i i i ii i i i i i i i i i i\nt q tt t q\n, \nwhere ↑ and ↓ are the zs eigenstates. If the TD state is the exact GS of th e \nsystem, it is easy to obtain that the GS energy per  unit cell is  (3) \n, (5) \n(6) (4)  \n 5βα 5 . 0 25 . 0 5 . 0TD −−− = e .        \nCompare Eq. (2) with Eq. (7), and you will find tha t the dimer state may be the GS of \nthe Hamiltonian (1) only when βα+>1 . The case of ∞→β  is the last special one \nthat needs to be discussed. In that case, the three  spins 1 3−isr, is3r, and 1 3+isr form a \ntrimer. The GS wave functions of the ith trimer are  \n( ) ) 2 / 1 ( 2\n61\n1 3 3 1 3 1 3 3 1 3 1 3 3 1 3 = ↑↑↓+↓↑↑−↑↓↑=+ − + − + −+ z\ntol i i i i i i i i i i s φ ,       \n( ) 3 1 3 3 1 3 1 3 3 1 3 1 3 3 1 12 ( 1/ 2) \n6z\ni i i i i i i i i i tol s φ−\n− + − + − + = ↓ ↑ ↓ − ↓ ↓ ↑ + ↑ ↓ ↓ =− , \nwhere z\ntol s is the z-component of the total spin of the ith trimer. Using the \npseudo-operator iTr\n with the magnitude 1/2, one can express Eqs. (8) a nd (9) as 54, 55)  \n−+\n=⇓=⇑\ni ii i\nφφ\n, \nwhere i⇑ and i⇓ denote the eigenstate of z\niT with the eigenvalue 1/2 and the \neigenstate of z\niT with the eigenvalue -1/2, respectively. In the fou rth special case, the \n1J terms of the Hamiltonian (1) can be treated as per turbations. By using the \nfirst-order perturbation theory with respect to 1J, one can obtain the effective \nHamiltonian \n13 /\n1194\n+\n=∑⋅ − =iN\nii eff TT J Hrr\n. \nThis result means that the GS of the Hamiltonian (1 ) is also in the ferrimagnetic state \nin the case of ∞→β , just as in the first case discussed above.  \n  Next, we determine the phase diagram by ED. As 2\nitr, defined by ) 1 (2+=i i i t t tr\n, \ncommutes with the Hamiltonian H, we have a sequence of  good quantum numbers \n} ,, , {3 / 2 1 N itt t tLL . Thus, the GS of the ABDC belongs to one of the su bspaces that are (7) \n(11) (8) \n(10)  (9)  \n 6specified by } ,, , {3 / 2 1 N itt t tLL  22). As the magnitude of the composite spin itr is 0 or \n1, the correlation function between the spin pairs 1 3−isr and is3r takes a value of -0.75 \nor 0.25. One can then calculate the short-range cor relation function >⋅<− i is s3 1 3rr to \ndetermine the phase diagram of the ABDC. Our ED res ults show that the value of \n>⋅<− i is s3 1 3rr is equal to 0.25, -0.25, or -0.75 in the entire pa rameter region. Thus, as \nshown in Fig. 3, the GS phase diagram of the ABDC i s composed of the ferrimagnetic \nstate, TD state, and dimer state. Finite-size effec ts on the position of the phase \nboundary are very minimal, as can be seen from the comparison of the results for \nN=12 and  30. At 1=β , the ED results show that two critical points sepa rate the TD \nstate from the (i) ferrimagnetic state and (ii) dim er state. For a system with N=24, our \nresults show that the two critical points are 909. 0=α  and 2 respectively, which are \nconsistent with those given in Ref. (14). Fig. 3 sh ows that, as expected above, the \nABDC possesses the ferrimagnetic long-range order i n the small- α or large- β \n-parameter region. Moreover, the ferrimagnetic stat e is always the GS of the chain if \nα is less than a certain critical value TDα. TDα for a system with 30=N  is \nshown in Fig. 3. When the parameter α exceeds that critical point, the TD phase \nappears in the phase diagram and it exists in a fin ite-parameter region. Besides TDα, \nthe other critical value is Dα ( 1D=α ), beyond which the dimer phase is also \nincluded in the phase diagram. The straight line 1−=αβ  in Fig. 3 represents the \nexact boundary between the TD state and the dimer s tate.  \n3. Coupled Cluster Method Applied to the Alternatin g Bond Diamond Chain \nIn this section, we discuss the properties of the q uantum TD state, dimer state, and \nferrimagnetic state of the system determined by CCM . Since details of the CCM \napplied to quantum spin systems have been given els ewhere 23, 26, 27) , we present \nonly a brief description of the method that we used  to treat the ABDC. \nWe first describe how we analyze the properties of the TD state by CCM. The  \n 7starting and key point for a CCM calculation is to choose a suitable normalized \nreference state φ. In the past, people often chose the classical sta te or the quantum \nstate (such as the dimer state) of the spin systems  as the reference state of CCM to \ninvestigate the properties of the spin cluster stat e 8, 24) . Since the singlet tetramer and  \nsinglet dimer appear along the chain alternately in  the TD state, we use the collinear \nstate as shown in Fig. 1(a), but not the two types of state mentioned above, as the \nCCM reference state. As neighboring spins in the A and B sublattices are aligned \nparallel, whereas those in the C sublattice are aligned antiparallel, that referenc e state \nis also called the ferromagnetic-ferromagnetic-anti ferromagnetic (FFA) state in the \nfollowing discussion for convenience. After carryin g out a mathematical rotation of \nthe local axes of all the “up” spins: x xs s−→ , y ys s→ , and, z zs s−→ , all the \nspins in the reference state align along the negati ve z-axis. The reference state is then \ngiven by  \nL L ⊗↓⊗↓⊗↓⊗↓⊗↓⊗↓=+ + + − − 3 3 2 3 1 3 3 1 3 2 3 i i i i i i φ , \nand the CCM parameterizations of the ket and bra GS s of model (1) are expressed as \n26, 27)  \n1 2 1 2 \n1 2 , , \n1 , , ,\nl l \nlN\nS\ni i i i i i \nl i i i e S S s s s ψ φ + + + \n== = ∑ ∑ L\nLL , \n1 2 1 2 \n1 2 , , \n1 , , , 1 \nl l \nlN\nS\ni i i i i i \nl i i i Se S S s s s ψ φ − − − − \n== = + ∑ ∑ L\nL% % % % L . \nBecause it is impossible to consider all the spin c onfigurations in the S and S~ \ncorrelation operators in practice, we use the well- established LSUB n approximation \nscheme to truncate the expansions of S and S~ 26, 27) . Within the LSUB n \napproximation, only the configurations, including n or fewer correlated spins that \nspan a range of no more than n contiguous lattice sites, are taken into account. In this \npaper, we assume that the two sites are contiguous if they are connected by 1J, 2J, \nor 3J bonds. Although the number of fundamental configur ations contained in the (12)  \n(13)   \n 8LSUB n approximation grows rapidly with respect to the tr uncation index n, it can be \nreduced if we use the lattice symmetries and conser vation laws that pertain to the \nHamiltonian. Obviously, the LSUB n approximation becomes exact in the limit \n∞→n . \nNow, one can prove that the exact TD state of the H amiltonian (1) can be produced \nby CCM with the FFA reference state. If all correla tion coefficients contained in the \nket-state correlation operator S except those displayed in Fig. 4(a) \nare set equal to zero, S is reduced to \n∑ ∑ ∑ ∑\n=+\n++\n+\n=+\n+++\n++\n−\n=++\n−+\n−+\n−\n=+\n+++\n−+\n− ++ ++ + =6 /\n13 3 2 3 26 /\n11 3 3 1 3 1 3 26 /\n13 2 3 1 3 2 3 26 /\n11 3 3 1 3 2 3 4 ) ( ) (N\nii icN\nii i i ibN\nii i i iaN\nii i i i ss S s s ss S ss ss S s s ss SS . \nThe ket GS of model (1) is then given by \nLL\n⊗↑↑+↓↓⊗↑↓↑↓+↑↑↓↓+↓↑↓↑+↓↓↑↑+↑↑↑↑+↓↓↓↓⊗==\n++ +++ −− + −− + −−+ −− + −− + −−\n] [][\n3 3 2 3 2 3 3 2 31 3 3 1 3 2 3 2 1 3 3 1 3 2 3 2 1 3 3 1 3 2 3 21 3 3 1 3 2 3 2 1 3 3 1 3 2 3 4 1 3 3 1 3 2 3\ni ic\ni ii i i ib\ni i i ib\ni i i iai i i ia\ni i i i i i i iS\nSS S SS S eφψ\n.  \nBy “re-rotating” the local axes of spins that point  upward in the FFA reference state, \none can obtain the following state: 8)  \nLL\n⊗↓↑−↑↓⊗↓↓↑↑−↓↑↓↑−↑↑↓↓−↑↓↑↓−↓↑↑↓+↑↓↓↑⊗=\n++ +++ −− + −− + −−+ −− + −− + −−\n] [][\n3 3 2 3 2 3 3 2 31 3 3 1 3 2 3 2 1 3 3 1 3 2 3 2 1 3 3 1 3 2 3 21 3 3 1 3 2 3 2 1 3 3 1 3 2 3 4 1 3 3 1 3 2 3\ni ic\ni ii i i ib\ni i i ib\ni i i iai i i ia\ni i i i i i i i\nSS S SS S ψ\n. \nIt is obvious that an exact TD state is given by 14=S , 5 . 02 2==b aS S , and 12=cS .  \nMoreover, the GS energy per unit cell of the Hamilt onian (1) is written as \nβα α β φφ 5 . 0 25 . 05 . 0 25 . 0)5 . 0 25 . 0 ( )5 . 0 25 . 0 (3\n2 2 2 TD −−− = −−−+−−= =− c b a S SS S S HeeNe . \nIt is in agreement with Eq. (7).  \n  Next, we prove that the exact dimer state can als o be constructed within the CCM. \nTo achieve this goal, the state shown in Fig. 1(b) is chosen to be the CCM reference \nstate and we call it the ferromagnetic-ferromagneti c-ferromagnetic-I (FFFI) state. If \nonly the ket-state correlation coefficient shown in  Fig. 4(b) is not equal to zero, one \ncan obtain the following ket-state within the CCM: (16)  \n(17)  (15) (14)  \n 9L L ⊗↓↑−↑↓⊗↓⊗=− − − ] [3 1 3 2 3 1 3 2 3 i i i i i S ψ .     \nApparently, the above state is only the dimer state  if 12=S . The GS energy per unit \ncell of the Hamiltonian (1) obtained by CCM based o n the FFFI reference state is \nα α 75 . 0 )5 . 0 25 . 0(2 D − = −−= S e . \nIt is obvious that Eq. (19) is the same as Eq. (2).  \nAlthough the above two short-range correlated spin cluster states are the exact GS \nof the Hamiltonian (1), the exact solution to the f errimagnetic state cannot be obtained \nowing to quantum fluctuation. In the ferrimagnetic state, a pair of 1 3−isr and is3r \nforms a triplet dimer, and the  magnitude of the total spin of that state is 6 /N  as \nmentioned above. Thus, we choose the state displaye d in Fig. 1(c) as the CCM \nreference state to analyze the properties of the fe rrimagnetic state. For convenience, \nthe above reference state is also called the \nferromagnetic-ferromagnetic-ferromagnetic-II (FFFII ) state in the following \ndiscussion. We also calculate, aside from the GS en ergy, the typical physical quantity \nof the ferrimagnetic state, that is, the sublattice  magnetizations AM and C BM+ \nusing  \n/3 \n3 2 \n11\n/ 3 N\nz\nA i \niM s Nψ ψ −\n==− ∑% , \n/3 /3 \n3 1 3 \n1 1 1 1 \n/ 3 / 3 N N \nz z \nB C i i \ni i M s s N N ψ ψ ψ ψ + − \n= = =− − ∑ ∑ % % . \nCCM can be well applied to investigating the proper ties of the lowest-lying excited \nstate as well as the GS. The excited state wave fun ction eψ is determined after \napplying an excitation operator eX linearly to the ket-state wave function. It is giv en \nby 26)  \n∑ ∑\n=++ += =N\nl i i ii i i i i ie s e\ne\nll ls s s X eX\n1 , ,, ,\n2 12 1 2 1,\nLLL χ φ ψ .               \nAnalogous to the GS, the LSUB n approximation scheme is also used to truncate the (21)  (20)  (18)  \n(19)   \n 10 expansion of the operator eX. One can then use CCM to calculate the spin gap Δ of \nthe spin systems. It is given by the lowest eigenva lue of the following LSUB n \neigenvalue equation: 26)  \nφ φ χS e S\ni i ie\ni i i eXHe s s s\nl l],[\n2 1 2 1, ,− − − −= Δ LL  . \n  We calculate two types of spin gap by CCM in the present paper. One is the \nsingle-triplet energy gap STΔ, which is a representative physical quantity of th e \nHamiltonian (1) if its GS is the TD state. The spin  gap STΔ is defined as \ng tol ST E sE −==Δ ) 1 (1  ,                          \nwhere 1E and gE respectively denote the energy of the lowest-lying  state with \n1=tols  and the GS energy. In the ferrimagnetic phase, the  ABDC possesses an \nantiferromagnetic character as well as a ferromagne tic character 51, 52) . Thus, the other \nspin gap determined by CCM is the antiferromagnetic  gap, which is given by \ng tol N AF E N s E −+= =Δ+ ) 16 / (1 6 / ,             \nwhere 1 6 /+ NE  and gE are the energy of the lowest-lying state with 1 6 /+=N stol  \nand the energy of the GS, respectively. \n4. Results of the Coupled Cluster Method  \nFirstly, we present our CCM results for the GS. As TD and the dimer state are the \nexact GSs of the Hamiltonian (1), we focus on the p roperties of the ferrimagnetic state. \nWhen 0=α  and 1=β , the property of the Hamiltonian (1) is exactly eq uivalent to \nthat of the one-dimensional Heisenberg ferrimagneti c spin chain which has been \ninvestigated by various analytical and numerical me thods 51, 56-58) . Table I shows the \nresults of CCM in that case. One can find that CCM results for the GS physical \nquantities, such as the GS energy per unit cell and  the sublattice magnetization, \nconverge very rapidly with an increase in the level  of approximation. This \nphenomenon would be related to the short correlatio n length of the one-dimensional \nHeisenberg ferrimagnetic spin chain  58) . As a result, the energy per unit cell e and (22)  \n(24)  (23)   \n 11 the sublattice magnetization AM given by CCM at the LSUB12 level of \napproximation are in agreement with four decimal pl aces with the best results of the \nnumerical method, namely, those of the DMRG method  56) . For clarity, we only show \nthe results of the above physical quantities at the  LSUB12 level of approximation in \nthe following discussion 59) . To check the results of CCM, we also calculated t hose \nphysical quantities by ED and found that the result s obtained by ED also converge \nextremely fast. The physical quantities for a syste m with 30=N  are shown in Table I. \nIt can be seen that they are very close to those of  DMRG. Therefore, in the following \npart, the results of ED are also given for 30=N  sites for comparison with those of \nCCM. \nFig. 5 shows the GS energy e as a function of β given by CCM on the basis of \nthe FFFII reference state for three distinct values  of the parameter α: 0=α , 0.85 , \nand 1. It can be found that the CCM results are in good agreement with those of ED \nin all cases. e decreases monotonically with increasing in β in the first case, \nwhereas in the second (third) case, the energy dete rmined by CCM on the basis of the \nFFFII reference state and that given by Eq. (7) int ersect at two critical points (one \npoint). This finding proves that the transition bet ween the TD state and the \nferrimagnetic state belongs to the first-order tran sition. By CCM, we have obtained \nthe critical points at which the GS of the Hamilton ian (1) evolves from the \nferrimagnetic state to the TD state for any other p arameter α greater than TDα. \nThey are presented in Fig. 6. One can see that the boundary line between the TD state \nand the ferrimagnetic state determined by CCM and t hat obtained by ED for a system \nwith 30=N  almost overlap. \nThe results for the sublattice magnetizations AM and C BM+ when 0=α  are \npresented graphically in Fig. 7. As seen in that fi gure, the sublattice magnetization \ngiven by CCM coincides fairly well with that obtain ed by ED across the entire \nparameter range. AM and C BM+ both experience growth with the increase in β in  \n 12 the region 1 0<<β . At 1=β , they reach their maximum at the same time. \nAfterwards, they decrease with further increase in β. The reason for the evolution of \nsublattice magnetizations with the parameter β is that the increase in 1 −β  (β−1 ) \nhelps every three spins 1 3−isr, is3r, and 1 3+isr (2 3−isr, 1 3−isr, and is3r) form a trimer. As a \nresult, the magnetic long-range order of the Hamilt onian (1) is strongest when 1 =β . \nNext, we present CCM results for the single-triplet  energy gap STΔ and the \nantiferrimagnetic gap AFΔ, using 1=α  as an example. In that case, whether the GS \nof the ABDC is in the TD state or ferrimagnetic sta te depends on whether the \nparameter β is located in the region 18 . 1 0<<β  or 18 . 1>β . To check the \nresults of CCM, STΔ and AFΔ were also obtained by ED. \nIn Fig. 8, the ED results for STΔ are displayed when 18 . 1 0<<β . One can find \nthat the single-triplet gap of a finite system with  12≥N  reaches its value in the \nthermodynamic limit in the entire parameter region.  Fig. 9 shows the single-triplet gap \nSTΔ given by CCM. Apparently, CCM LSUB n results for STΔ converge rapidly \nwith an increase in n, and the spin gap in the limit ∞→n  is determined by CCM if \n10≥n . Results of the spin gap STΔ in some cases are shown in Table II. As seen in \nFig. 9 and Table II, the spin gap STΔ given by CCM with 10 ≥n  equals that \nobtained by ED in the entire parameter region. The results of CCM and ED both show \nthat the single-triplet gap obviously appears when 0>β , and increases with β in \nthe region 1 0<<β . When 1 =β , it reaches its maximum. Although it then \ndecreases with an increase in β, it does not vanish when 18 . 1 <β . Hence, our \ncurrent findings as well as the results of previous  research indicate that the TD state is \ngapful 6, 60) .  \nFinally, we turn to our CCM results for the antifer romagnetic gap AFΔ. The AFΔ  \n 13 values for 0=α  and 1=β  obtained from CCM are listed in Table I. One can s ee \nthat our CCM results for AFΔ are highly converged. The antiferromagnetic gap AFΔ \ngiven by CCM at the LSUB12 level of approximation i s in agreement up to three \ndecimal places with that obtained by QMC 51) . The antiferromagnetic gap is plotted as \na function of β in Fig. 10 when 1 =α  and 18 . 1>β . AFΔ values obtained by ED \nfor a system with 30 =N  are also displayed in that figure for comparison w ith the \ncorresponding CCM data. One can find that AFΔ increases with increasing in β, \nalthough the rate of increase gradually decreases. The size of the antiferromagnetic \ngap obtained by CCM is in good agreement with that given by ED in the entire \nparameter region. Thus, CCM can also be used to acc urately analyze the lowest-lying \nexcited-state properties of the ABDC. \n5. Conclusions \nIn this paper, the CCM method, a powerful analytica l tool for treating the frustrated \nHeisenberg chain in any dimension, was applied to t he ABDC. To verify the accuracy \nof CCM results, we have also investigated the prope rties of the ABDC by the ED \nmethod. The ED results show that the GS phase diagr am is composed of the TD state, \ndimer state, and ferrimagnetic state. We have shown  that the former two exact spin \ncluster solid GSs can both be formed by CCM. Some p hysical quantities of the \nferrimagnetic state, such as the GS energy and subl attice magnetizations, have been \ndetermined by CCM up to high orders of approximatio n. The results of the above \nquantities obtained by CCM are compared with those given by numerical methods. \nThe case of 0 =α  and 1=β  is a typical example, in which the results of CCM are \nsufficiently accurate to be comparable to those of the numerical Monte Carlo or \nDMRG method. For any other parameter, the CCM resul ts are also in perfect \nagreement with the results of the numerical method,  namely, those of ED. Thus, it is \nnatural to observe that CCM as well as ED can be us ed to precisely determine the \nphase boundary between the TD state and the ferrima gnetic state. \nWe have also calculated, aside from the GS physical  quantities, the single-triplet  \n 14 energy gap and antiferromagnetic gap  of the ABDC by CCM and compared them with \nthose given by ED. Our results show that the single -triplet energy gap in the \nthermodynamic limit can be obtained by CCM. It is a lso found that the \nantiferromagnetic gap obtained by CCM is comparable  to that determined by ED.  \nTherefore, the properties of the ABDC can be precis ely analyzed by analytical \nCCM. Moreover, our findings provide a typical examp le of a powerful CCM \napplication to frustrated quantum spin systems, eve n though its GS is in the quantum \nstate with no classical analogy. \nAcknowledgments \nWe thank Dr. Damain Farnell for his help with the a pplication of CCM to spin \nsystems. 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Jpn. 69 , 2710 (2000).   \n 18  \n \n \n \n \n \n \n \n  \n   \n    \n    \n    \n    \n      \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  e AM AFΔ \nLSUB8 -1.454172 0.292129  1.760226 \nLSUB10 -1.454109 0.292403  1.759433 \nLSUB12 -1.454096 0.292472  1.759224 \nED ( N=30) -1.454095 0.292478  1.759174 \nLinear spin wave theory (SWT) 56)  -1.436 0.195 1    \nSecond-order SWT 57)  -1.454322 0.293884  - \nQMC 51, 58)  −1.455 ± 0.001  0.29 1.75914 \nDMRG 56)  −1.45408 0.29248  - Table I. Results obtained for the ABDC using the CC M in the case of 0=α  and 1=β . The \nGS energy per unit cell e, the sublattice magnetization AM, and the antiferrimagnetic gap \nAFΔ obtained by CCM are shown. These results are compa red with those obtained by other \nmethods. \n  \n 19  \n \n \n \n \n \n \n \n  \n   \n    \n    \n    \n    \n    \n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n β LSUB8 LSUB10  LSUB12  ED \n0.10  0.090396  0.080051  0.080051  0.080051  \n0.20  0.166259  0.146264  0.146264  0.146264  \n0.30  0.228444  0.198044  0.198044  0.198044  \n0.40  0.276621  0.235127  0.235127  0.235127  \n0.50  0.310449  0.257644  0.257644  0.257644  \n0.60  0.329849  0.266132  0.266132  0.266132  \n0.70  0.335139  0.261484  0.261484  0.261484  \n0.80  0.327065  0.244852  0.244852  0.244852  \n0.90  0.306736  0.217528  0.217528  0.217528  \n1.00  0.275494  0.180828  0.180828  0.180828  \n1.10  0.234759  0.136015  0.136015  0.136015  \n1.15  0.211265  0.110931  0.110931  0.110931  Table II. Results of the single-triplet energy gap using CCM-LSUB n approximation with \nn={ 8, 10, 12} when 1=α . These results are compared with those obtained by  ED for N=30 \nsites.   \n 20  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure captions \nFig. 1. Sketches of the FFA reference  state (a), FFFI reference state (b), and FFFII ref erence state \n(c) of the ABDC.  \nFig. 3. GS phase diagram obtained by ED. The solid line 1−=αβ  represents the exact \nboundary between the dimer phase and the TD phase.  \nFig. 4. Illustration of fundamental configurations retained in the ket-state correlation operator \nS for CCM based on FFA reference state (a) or FFFI r eference state (b). The centers of the \nshaded circles mark the flipped spins with respect t o the reference state. \nFig. 8. Spin gap STΔ of the ABDC versus β using ED when 1=α .  \nFig. 9. Spin gap STΔ of the ABDC versus β using CCM based on FFA reference state and ED \nwhen 1=α .  Fig. 5. GS energy per site e versus β using CCM based on FFFII reference state, ED, and  \nEq. (7) for different α values.  \n \nFig. 10. Spin gap AFΔ  versus β using CCM based on FFFII reference state and ED wh en 1=α .  \n Fig. 7. Sublattice magnetizations AM and C BM+ versus β using CCM based on FFFII  \nreference state and ED when 0=α .  Fig. 2. Schematic picture of the TD state. The rect angles and ellipses represent the tetramers and \nsinglet dimers, respectively. \nFig. 6. Boundary line between the TD state and the ferrimagnetic state determined by CCM and \nED. \n  \n 21  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 1J\n2J\n2 3−iS\n1 3−iSiS3\nA\nBC\n3J\n1J\n2J\nA\nBC\n3J\n2 3−iS\n1 3−iSiS32 3−iS\n1 3−iSiS3\n1J\n2J\nA\nBC\n3J(a) \n(b) \n(c) Figure 1  \n 22  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 3−iS\n1 3−iSiS3\n23+iS33+iS\n1 3+iS\n3 4 iS+\n1 3−iSiS3\n1 3+iS\n23+iS33+iSFigure 2  \n 23  \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 3  \n 24  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\naS22 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\nbS2\n2 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\ncS22 3−ii3\n1 3+i\n1 3−i 2 3+i3 3+i\n4S\n2 3−ii3\n1 3−i\n2S(a) \n(b) Figure 4  \n 25  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 5  \n 26  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 6  \n 27  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 7  \n 28  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 8  \n 29  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 9  \n 30  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 10 "    },    {        "title": "2201.03781v1.Nonreciprocal_dynamics_of_ferrimagnetic_bimerons.pdf",        "content": "Nonreciprocal dynamics of ferrimagnetic bimerons\nLaichuan Shen,1Jing Xia,2Zehan Chen,3, 4Xiaoguang Li,5Xichao Zhang,6\nOleg A. Tretiakov,7Qiming Shao,3, 4, 8,\u0003Guoping Zhao,2Xiaoxi Liu,6Motohiko Ezawa,9,yand Yan Zhou1,z\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China\n2College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n3Department of Electronic and Computer Engineering, The Hong Kong University\nof Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China\n4Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China\n5Center for Advanced Material Diagnostic Technology, College of Engineering\nPhysics, Shenzhen Technology University, Shenzhen 518118, China\n6Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan\n7School of Physics, The University of New South Wales, Sydney 2052, Australia\n8Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic\nTechnology, The Hong Kong University of Science and Technology, Hong Kong, China\n9Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan\n(Dated: January 12, 2022)\nMagnetic bimerons are topologically nontrivial spin textures in in-plane easy-axis magnets, which can be used\nas particle-like information carriers. Here, we report a theoretical study on the nonreciprocal dynamics of\nasymmetrical ferrimagnetic (FiM) bimerons induced by spin currents. The FiM bimerons have the ability to\nmove at a speed of kilometers per second and do not show the skyrmion Hall effect at the angular momentum\ncompensation point. Our micromagnetic simulations and analytical results demonstrate that spin currents are\nable to induce the nonreciprocal transport and a drift motion of the FiM bimeron even if the system is at the\nangular momentum compensation point. By analyzing the current-induced effective fields, we find that the\nnonreciprocal transport is attributed to the asymmetry of the bimeron structure. Our results are useful for\nunderstanding the physics of bimerons in ferrimagnets and may provide guidelines for building bimeron-based\nspintronic devices.\nIntroduction. Reciprocity is a fundamental principle in\nmany fields, such as in mechanics and thermodynamics [1].\nHowever, when a certain symmetry of the system is bro-\nken, the reciprocal relation may be violated and nonrecipro-\ncal phenomena appear [1–3]. Nonreciprocal transport which\nplays an important role in various application devices, such\nas in the diode [4–7] and shift register [8], has been reported\nfor (quasi-)particles, for instance, electrons [9], phonons [4],\nphotons [10] and magnons [11]. For topological solitons,\ne.g. skyrmions [12], they also show such a transport in the\nasymmetrical racetrack which requires sophisticated repro-\ncessing [13–16]. However, nonreciprocal transport attributed\nto the intrinsic characteristics of topological solitons still re-\nmains to be discovered.\nDifferent types of topological spin textures have been in-\nvestigated for a few decades, such as domain walls [17, 18],\nskyrmions [12, 19, 20] and bimerons [21–31], which emerge\nin ferromagnetic (FM) [17, 19, 25], ferrimagnetic (FiM) [32–\n35] and antiferromagnetic (AFM) [18, 36, 37] materials. In\nparticular, FM skyrmions are promising as nonvolatile infor-\nmation carriers to serve the future memory and logic com-\nputing devices [38–44]. However, the FM skyrmion shows\nthe transverse drift during its motion due to the existence of\na nonzero Magnus force, which may lead to the annihila-\ntion of the fast-moving skyrmion at the sample edge. This\n\u0003Email: eeqshao@ust.hk\nyEmail: ezawa@ap.t.u-tokyo.ac.jp\nzEmail: zhouyan@cuhk.edu.cnphenomenon is referred to as the skyrmion Hall effect [45–\n47]. Compared to the FM skyrmion, the AFM skyrmion is\nfree from the skyrmion Hall effect, as the compensated lat-\ntice structures of antiferromagnets lead to a perfect cancel-\nlation of the Magnus force [48, 49]. However, the compen-\nsated magnetic moments in antiferromagnets give rise to the\ndifficulties in detecting AFM spin textures [50]. Recently,\nFiM materials have received great attention, since the AFM\nspin dynamics is realized in ferrimagnets at the angular mo-\nmentum compensation point [33, 51] and unlike the antiferro-\nmagnet, even for compensated ferrimagnet, we can detect the\nmagnetization of one sublattice using magnetotransport mea-\nsurements, such as anomalous Hall effect or tunnel magne-\ntoresistance. On the other hand, a magnetic bimeron consist-\ning of two merons is considered as the topological counter-\npart of a magnetic skyrmion in in-plane magnets and is stabi-\nlized in various magnetic materials [22, 24, 26–31]. Recent\nreports show that two-dimensional CrCl 3[52, 53] and van der\nWaals LaCl/In 2Se3heterostructures [54] are promising can-\ndidates for hosting bimerons. Additionally, the bimeron is\na stable solution in ferromagnets [25, 26, 55], antiferromag-\nnets [28, 30, 37] and frustrated magnets [29, 56]. Although a\nbimeron is topologically equivalent to a skyrmion, the former\nhas richer dynamics [30, 55].\nIn this work, based on the Landau-Lifshitz-Gilbert (LLG)\nequation [57], we theoretically study the current-induced dy-\nnamics of FiM bimerons with intrinsic asymmetrical shape.\nNumerical and analytical results demonstrate that the FiM\nbimeron driven by opposite currents could exhibit different\nspeeds, that is, it shows the nonreciprocal dynamics. By an-arXiv:2201.03781v1  [cond-mat.mes-hall]  11 Jan 20222\nalyzing the current-induced effective fields, it is found that\nsuch a nonreciprocal behavior is attributed to the asymmetry\nof the bimeron structure. In addition to the nonreciprocal dy-\nnamics, a drift motion of the FiM bimeron may be induced\nby the current even if the system is at the angular momentum\ncompensation point.\nProposal of nonreciprocal transport of FiM bimerons. We\nconsider a FiM film with two sublattice magnetization M1and\nM2[Fig. 1(a)], and the interfacial Dzyaloshinskii-Moriya in-\nteraction (DMI) [38, 40] is introduced, which can be induced\nat the magnetic layer/heavy metal interface. To form FiM\nbimerons, the ferrimagnets with in-plane magnetic anisotropy\n(such as DyCo 5[58]) are promising materials. Here we focus\non the study of a FiM film with in-plane easy-axis anisotropy,\nin which the asymmetrical bimeron is a stable solution, sim-\nilar to the cases of FM [55] and AFM [30] bimerons. For\nthe interfacial DMI shown in Fig. 1(a), it is usually responsi-\nble for stabilizing the skyrmion with rotational symmetry (the\naxis of rotation is parallel to the polar axis that is perpendicu-\nlar to thex-yplane) [59]. For the FiM system we consider, the\nin-plane magnetic anisotropy forces the magnetization to tilt\naway from the polar axis, which violates the rotational sym-\nmetry dominated by DMI. As reported in Refs. [27, 59], in\nthe tilted magnetic phases (the magnetization of a homoge-\nneously magnetized state is tilted away from the polar axis),\nthe rotationally symmetrical spin texture is an incompatible\nform and the asymmetrical spin textures appear. Figure 1(b)\nshows the spin structure of a FiM bimeron, and the compo-\nnents of its reduced magnetization si=Mi=jMijand Néel\nvector n= (si\u0000sj)=2are presented in the Fig. 1(c). The\nNéel vector nin real space for a FiM bimeron is plotted in\nFig. 1(d), showing that although the size of the left meron\nis different from that of the right meron, the bimeron’s spin\nstructure still has mirror symmetry about the x-zplane.\nAdditionally, we derive a closed equation for the Néel vec-\ntorn[60, 61] (see Supplemental Material [62] for details),\n\u00160\u001a2(1+\u000b2)\n2\u0015n\u0002n+\u001b_n=\u0000n\u0002f\u0003\nn+\u000b\u001an\u0002_n+2\u000fn\u0002p\u0002n.\n\u001a=MS1\n\r1+MS2\n\r2and\u001b=MS1\n\r1\u0000MS2\n\r2are the staggered and\nnet spin densities, respectively [61]. MSi,\ri,\u00160,\u0015and\u000b\nare the saturation magnetization, gyromagnetic ratio, vacuum\npermeability constant, homogeneous exchange constant and\ndamping constant, respectively. f\u0003\nnand\u000frelate to the ef-\nfective field and current density, respectively (Supplemental\nMaterial [62]). In the above equation, only the damping-like\nspin torque is considered, while the field-like spin torque is\nnot included. The effect of field-like spin torque on the FiM\nbimeron has been discussed in Supplemental Material [62].\nThe above equation indicates that the current-induced effec-\ntive field relates to the cross product of the Néel vector nand\npolarization vector p. Such current-induced effective fields\nare of interest to us in the following symmetry analysis. As\nmentioned earlier, the bimeron’s spin structure is symmetric\nabout thex-zplane, so that the Néel vector nis canceled in\ntheydirection and the ycomponent of nwill not contribute\nto the nonreciprocal dynamics. Thus, we only need to pay\nattention to the components of nin thex-zplane and their\ncorresponding current-induced effective fields ( n\u0002p). As\nshown in Figs. 1(e)-1(j), we sketch two vectors nLandnRto represent the x-z-plane components of the Néel vector n\nfor two merons, respectively. Based on the symmetry con-\nsideration, nLandnRare symmetric about the xaxis for a\nbimeron with a symmetrical shape (see Fig. S1 of Supple-\nmental Material [62]), while for the bimeron studied here it\nhas an asymmetrical shape, resulting in the breaking of this\nsymmetry. Figure 1(e) shows the results of n\u0002pforp=ex,\nwhere the cross product operation causes the current-induced\neffective fields to be perpendicular to the x-zplane. When we\nchange the sign of the current, which is equivalent to chang-\ning the direction of p,i.e.,p=ex!\u0000ex, the corresponding\ncurrent-induced effective fields are still perpendicular to the\nx-zplane [Fig. 1(f)]. By comparing Fig. 1(e) with Fig. 1(f),\nwe see that for p=exand\u0000ex, the current-induced ef-\nfective fields are symmetric about the x-zplane, so that the\nbimeron does not exhibit the nonreciprocal motion behavior.\nSimilar mirror symmetry is observed for the case of p=\u0006ez\n[Figs. 1(i) and 1(j)], so there is no nonreciprocal phenomenon.\nHowever, for p=ey[Fig. 1(g)] and\u0000ey[Fig. 1(h)], n\u0002p\nis in thex-zplane, and obviously nL\u0002eyis not mirror-\nsymmetrical to nL\u0002(\u0000ey), so that the opposite currents have\ndifferent effects on the meron on the left in Fig. 1(d) [this re-\nsult also applies to the meron on the right in Fig. 1(d)]. For\na bimeron with a symmetrical shape, as mentioned above,\nnLandnRare symmetric about the xaxis, indicating that\nthe effect of the positive current on the left meron (the right\nmeron) is equivalent to that of the negative current on the right\nmeron (the left meron). Therefore, although opposite currents\nhave different effects on each meron, the structural symmetry\ncauses the opposite currents to have the same effects on the\nwhole, so that the bimeron with a symmetrical shape will not\nshow nonreciprocal transport, which has been confirmed in\nSupplemental Material [62]. For the FiM bimeron studied in\nthis work, it has an asymmetrical shape [Fig. 1(c)], resulting in\nthe presence of nonreciprocal phenomena. Namely, nonrecip-\nrocal transport is attributed to the asymmetry of the bimeron\nstructure.\nCurrent-induced nonreciprocal transport and drift motion\nof FiM bimerons. To verify the above analysis, we have sim-\nulated the magnetization dynamics of FiM bimerons and ob-\ntained the bimeron speeds, as shown in Figs. 2(a)-2(c). We\nindeed observe that only when the polarization vector pis\nalong theydirection (perpendicular to the symmetry plane\nof the bimeron’s spin structure), the FiM bimeron driven by\nopposite currents has different speeds, that is, it exhibits the\nnonreciprocal transport [Fig. 2(b)]. As expected by the above\nsymmetry analysis, for the cases where pis along the xorz\ndirections [Figs. 2(a) and 2(c)], such a nonreciprocal trans-\nport does not appear. Here we employ the LLG equation\nwith the damping-like spin torque [63, 64] to simulate the\ndynamics of FiM bimerons, and the simulation details are\ngiven in Supplemental Material [62]. We also simulate the\ncreation of FiM bimerons (see Figs. S4 and S5 of Supplemen-\ntal Material [62]) and the creation process is given in Supple-\nmental Movie 1-3. Moreover, based on the definition of the\nguiding center ri=\u00001=(4\u0019Q)R\n[in\u0001(@xn\u0002@yn)]dS[65],\nwe obtain the time evolution of riand the bimeron veloc-\nityvi= _ri(see Fig. S8 of Supplemental Material [62]).3\nFIG. 1. (a) Schematic of the studied model, where M1andM2denote two sublattice magnetization of ferrimagnet. The interface-induced\nDMI and in-plane easy-axis magnetic anisotropy Kare considered. (b) The real-space spin structure of a FiM bimeron. (c) The components\nof reduced magnetization sand the Néel vector nfor a FiM bimeron with a positive topological charge Q. (d) The Néel vector nin real space.\nThe color represents the zcomponents of n, andnzof two merons has opposite signs. (e)-(j) The sketch of two vectors nLandnR, and the\ncross product of the vector nL,Rand polarization vector p.\nFIG. 2. (a)-(c) The FiM bimeron speeds as functions of the current density jforp=ex,\u0000ey, and\u0000ez. (d)-(f) The angle between the actual\nand desired motion directions for p=ex,\u0000ey, and\u0000ez. Here, the FiM bimeron has a positive Q, and the numerical and analytical results\nare obtained by solving LLG equation and Eq. (1), respectively. (g) The alternating current pulse adopted in our simulation. (h)-(j) The time\nevolution of the guiding center ( rx,ry) for different polarization vectors p.\u000b= 0:05,MS1= 1:1MS2and\r1= 1:1\r2.\nQ=\u00001=(4\u0019)R\ndS[n\u0001(@xn\u0002@yn)]is the topological\ncharge [22, 49].\nWe now discuss the current-induced drift motion of FiM\nbimerons. Figures 2(a)-2(c) present the bimeron speeds in\nthe desired motion direction (it is in y,xandydirections for\np=ex,\u0000eyand\u0000ez, respectively). The FiM bimeron at\nthe angular momentum compensation point has an ability to\nmove with a speed of about km s\u00001, similar to the AFM spin\ntextures [28]. However, spin currents may induce a drift speed\nwhich is perpendicular to the desired motion direction, even\nif the FiM system is at the angular momentum compensation\npoint ( i.e.,\u001b=MS1\n\r1\u0000MS2\n\r2= 0). As shown in Figs. 2(d) and2(f) where p=exand\u0000ezrespectively, the angle between\nthe actual and desired motion directions is not zero, that is,\nthe FiM bimeron shows a drift motion, and such an angle in-\ncreases with the applied currents j. For the case of p=\u0000ey,\nthe drift motion can be safely disregarded [Fig. 2(e)].\nTo explain the simulation results, we derived the Thiele\nequation (see Supplemental Material [62] for details) [61, 66–\n68], from which we obtain the steady motion speeds,\n\u0012\nvx\nvy\u0013\n=1\n\u0011\u0012\n\u000bLyy\u0000G\u0000\u000bLxy\nG\u0000\u000bLxy\u000bLxx\u0013\u0012\nFx\nFy\u0013\n;(1)\nwhere\u0011=\u000b2(LxxLyy\u0000L2\nxy)+G2.Lij=\u00160\u001atzR\ndS(@in\u00014\nFIG. 3. The numerical (symbols) and analytical (curves) velocities\nfor a FiM bimeron in systems with different values of MS1=M S2,\nwhere three polarization vectors (a) p=ex, (b)p=\u0000eyand (c)\np=\u0000ezare considered. Here, j= 5 MA cm\u00002,MS2= 376 kA\nm\u00001,\r1= 1:1\r2, and other parameters are the same as those used\nin Fig. 2. The numerical and analytical results are obtained from the\nLLG equation and Eq. (1), respectively.\n@jn)andG= 4\u0019Q\u0016 0tz\u001bwith the layer thickness tz.Fi\ndenotes the driving force induced by the damping-like spin\ntorque (its expression is given in Supplemental Material [62]).\nIfG= 0, Eq. (1) indicates that the bimeron speed is inversely\nproportional to the damping (see Fig. S9 of Supplemental Ma-\nterial [62]).\nTo verify the above analytical formula, we simulate the\nmotion of FiM bimerons and calculate the bimeron veloci-\nties for different values of MS1=M S2. Figure 3 shows the\ncomparison of the numerical and analytical velocities, where\nthe analytical velocities for all polarization vectors pare in\ngood agreement with the numerical results. Moreover, Fig. 3\nshows that one of the velocity components ( vx,vy) is sym-\nmetric about MS1=M S2= 1:1, while the other component is\nantisymmetric. From Eq. (1), we see that one of the veloc-\nity components is proportional to 1=(C+G2)with a con-\nstantC,i.e.,vi/1=(C+G2), while the other component\nvj/G=(C+G2), whereG/(MS1=M S2\u0000\r1=\r2)be-\ncause we fixed the values of MS2and\r1in the simulation.\nForvi/1=(C+G2)it presents a symmetric curve, while for\nvj/G=(C+G2), an antisymmetric curve is obtained.\nAssuming that the main driving force is in the ydirection\n(Fy6= 0) and the system is at the angular momentum com-\npensation point ( G= 0), from Eq. (1), the drift speed is ob-\ntained,vx=\u000b(LyyFx\u0000LxyFy)=\u0011. Note thatLyy(orLxx)\nis always nonzero for spin textures. According to the aboveformula of the drift speed, we find that there are two factors\nwhich cause the drift motion even if G= 0. The first factor\nis the presence of a nonzero Lxy[27] and the second factor\nis that an additional force perpendicular to the desired motion\ndirection is induced by the applied currents. In order to ver-\nify the above analysis, we calculated the numerical values of\nLandF(Figs. S10 and S12 of Supplemental Material [62]),\nand then substituting them into Eq. (1) gives the analytical\ndrift speeds which are consistent with the numerical results, as\nshown in Figs. 2(d)-2(f). Moreover, the numerical values of L\nandFconfirm that the drift motion presented in Figs. 2(d) and\n2(f) is due to the presence of a nonzero Lxyand an additional\nforce [they originate from the deformation of the bimeron’s\nspin structure after the spin currents are applied (Fig. S13\nof Supplemental Material [62])]. The drift speed due to the\nnonzeroLxyis greater than that due to the presence of an ad-\nditional force for the case of Fig. 2(d), while the drift speed in\nFig. 2(f) is dominated by the additional force.\nSince FiM bimerons exhibit the nonreciprocal transport, an\nalternating current pulse presented in Fig. 2(g) induces the\nbimeron to show a ratchet motion [69–73] if we take p=\u0000ey\n[Fig. 2(i)]. Thus, FiM bimerons are ideal information carriers\nin AC racetrack storage devices [71]. For p=exand\u0000ez,\nthe bimeron does not show the nonreciprocal motion in the y\ndirection and the final value of ryis zero [Figs. 2(h) and 2(j)],\nwhile due to the presence of the drift motion [Figs. 2(d)\nand 2(f)], the final values of rxare not equal to zero.\nTo quantify the nonreciprocal transport of FiM bimerons,\nthe speed difference \u0001vi=jvi(+j)j\u0000jvi(\u0000j)jis defined.\nIn Figs. 4(b)-4(d), \u0001viis calculated as a function of the cur-\nrent density j, the damping \u000band the ratio of MS1andMS2.\nAccording to the fitting results shown in Fig. 4(b), the rela-\ntionship between \u0001vxandjis well described by this func-\ntion\u0001vx=k2j2+k4j4, where we take p=\u0000eyand\nG= 0. To understand the results shown in Fig. 4(b), let\nus return to Eq. (1). For G= 0, Eq. (1) is simplified as\nvx=Fx=(\u000bLxx), whereLxxandFxare related to the\nbimeron’s spin structure so their values are affected by the ap-\nplied currents. As mentioned earlier, opposite currents have\ndifferent effects on the FiM bimeron with an asymmetrical\nshape, so that the value of jFx=Lxxjfor a positive current is\ndifferent from that for a negative current. Thus, the relation\nbetweenFx=Lxxandjmust include even terms in addition\nto odd terms, so the speed vxis written as a general polyno-\nmial form,vx=P\nl=1kljl=2with coefficient kl. Consider-\ning the first two terms of such a polynomial, the fitting results\nalmost match the numerical simulations [Fig. 4(a)] (if more\nhigh-order terms are considered, the gap between the fitting\nresults and numerical simulations will be narrowed). From\nthe above speed vx, we obtain the speed difference that only\ncontains even terms, \u0001vx=k2j2+k4j4+\u0001\u0001\u0001(the magnitude\nofk2,k4and\u0001\u0001\u0001is directly related to the strength of nonrecip-\nrocal transport). In addition, taking different damping \u000b,\u0001vx\nis calculated and summarized in Fig. 4(c), showing that \u0001vx\nis inversely proportional to \u000b, as indicated by this equation\nvx=Fx=(\u000bLxx). Moreover, by changing the value of MS1,\n\u0001vxand\u0001vyfor different MS1=M S2are obtained, as shown\nin Fig. 4(d), where \u0001vxreaches its maximum value at the an-5\nFIG. 4. (a) The FiM bimeron speeds vxas functions of the current\ndensityj. The symbols are obtained from the numerical simulations\nand the curves are the fitting results. For the case where the fitting\nfunction isvx=P2\nl=1kljl=2,k1= 50:158andk2=\u00000:375[the\nunit ofklis m s\u00001(MA cm\u00002)\u0000l]. Forvx=P4\nl=1kljl=2,k1=\n41:726,k2=\u00000:274,k3= 3:04\u000210\u00003andk4=\u00003:085\u000210\u00005.\nThe adopted parameters are the same as those used in Fig. 2(b). (b)-\n(d) The speed differences \u0001vias functions of the current density\nj, the damping constant \u000band the ratio of MS1andMS2. The nu-\nmerical and analytical results are obtained by solving LLG equation\nand Eq. (1), respectively. In panel (b), the green dashed and red solid\ncurves are the fitting results of \u0001vx=k2j2(withk2=\u00000:375) and\n\u0001vx=k2j2+k4j4(withk2=\u00000:274andk4=\u00003:085\u000210\u00005),\nrespectively. The default parameters are p=\u0000ey,j=\u000625MA\ncm\u00002,\u000b= 0:05,MS1= 1:1MS2and\r1= 1:1\r2.\ngular momentum compensation point ( MS1=M S2= 1:1), and\n\u0001vxand\u0001vyalmost are symmetric and antisymmetric about\nMS1=M S2= 1:1, respectively.\nGeneralization of nonreciprocal transport to magnetic\nskyrmions. In the above sections, we discussed the nonrecip-\nrocal transport of the FiM bimeron with a positive topological\nchargeQ. Such a nonreciprocal transport is also observed for\nthe FiM bimeron with a negative Q(see Fig. S17 of Sup-\nplemental Material [62]). The results of nonreciprocal trans-\nport of FiM bimerons can be extended to other types of spin\ntextures with broken symmetry. For general topological soli-\ntons, e.g. skyrmions, they have a symmetrical structure so\nnonreciprocal transport does not appear, while the bimerons\nunder investigation have intrinsic asymmetrical shape, result-\ning in the presence of nonreciprocal dynamics. Therefore, in\norder to attain the nonreciprocal transport, a break in struc-\ntural symmetry of spin textures is required. As shown in Fig.\nS18 of Supplemental Material [62], when an in-plane mag-\nnetic field is utilized to break the rotational symmetry of a FM\nskyrmion, the skyrmion driven by opposite currents exhibits\nnonreciprocal transport. Compared to the intrinsic asymme-\ntry of bimerons ( k2\u0019\u00000:3, Fig. 4), this externally inducedasymmetry of skyrmions gives rise to a weak nonreciprocity\n(k2\u0019\u00000:01, Fig. S18 of Supplemental Material [62]).\nConclusions. We have analytically and numerically studied\nthe drift and nonreciprocal motions of FiM bimerons driven\nby spin currents. Our results demonstrate that due to the defor-\nmation of the bimeron’s spin structure, spin currents may in-\nduce a drift speed which is perpendicular to the desired motion\ndirection, even if the FiM system is at the angular momentum\ncompensation point. Moreover, the symmetry analysis shows\nthat since the FiM bimeron studied here has an asymmetrical\nshape, the bimeron driven by opposite currents exhibits non-\nreciprocal transport. Our analysis of nonreciprocal transport\nof FiM bimerons is applicable to other types of spin textures\nwith broken symmetry and our results are useful for building\nbimeron-based spintronic devices, such as bimeron diode and\nAC racetrack memory.\nThis study is supported by Guangdong Spe-\ncial Support Project (Grant No. 2019BT02X030),\nShenzhen Fundamental Research Fund (Grant No.\nJCYJ20210324120213037), Shenzhen Peacock Group\nPlan (Grant No. KQTD20180413181702403), Pearl River\nRecruitment Program of Talents (Grant No. 2017GC010293)\nand National Natural Science Foundation of China (Grant\nNos. 11974298 and 61961136006). J.X. acknowledges the\nsupport by the National Natural Science Foundation of China\n(Grant No. 12104327). X.Li acknowledges the support by the\nGuangdong Basic and Applied Basic Research Foundation\n(Grant No. 2019A1515111110). X.Z. was an International\nResearch Fellow of Japan Society for the Promotion of Sci-\nence (JSPS). X.Z. was supported by JSPS KAKENHI (Grant\nNo. JP20F20363). O.A.T. acknowledges the support by the\nAustralian Research Council (Grant No. DP200101027),\nthe Cooperative Research Project Program at the Research\nInstitute of Electrical Communication, Tohoku University\n(Japan), and by the NCMAS 2021 grant. Q.S. acknowledges\nfunding support from the Shenzhen-Hong Kong-Macau\nScience and Technology Program (Category C, Grant No.\nSGDX2020110309460000), Research Grant Council-Early\nCareer Scheme (Grant No. 26200520), and the Research\nFund of Guangdong-Hong Kong-Macao Joint Laboratory for\nIntelligent Micro-Nano Optoelectronic Technology (Grant\nNo. 2020B1212030010). G.Z. acknowledges the support by\nthe National Natural Science Foundation of China (Grant\nNos. 51771127, 51571126, and 51772004), the Scientific\nResearch Fund of Sichuan Provincial Education Department\n(Grant Nos. 18TD0010 and 16CZ0006). 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Raabe,6 Nyun Jong Lee,7 Sang-Il Kim,7 Seung-Young Park,7 Younghak Kim,8 Jae-Young Kim,8 Dongjoon Lee,1,9 OukJae Lee,1 Jun Woo Choi,1,10 Byoung-Chul Min,1,10 Hyun Cheol Koo,1,9 and Joonyeon Chang1,10  Affiliations: 1Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea 2Department of Physics, Sookmyung Women’s University, Seoul 04130, Korea 3School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China 4Department of Applied Physics, University of Tokyo, Hongo 7-3-1, Tokyo 113-8656, Japan 5Department of Electrical and Computer Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan 6Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland 7Spin Engineering Physics Team, Division of Scientific Instrumentation, Korea Basic Science Institute, Daejeon, 305-806, Korea 8Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 37673, Korea 9KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02792, Korea 10Department of Nanomaterials Science and Engineering, Korea University of Science and Technology, Daejeon 34113, Korea  † These authors contributed equally to this work. * Authors to whom correspondence should be addressed: shwoo_@kist.re.kr    Page 2 of 24 Magnetic skyrmions are swirling magnetic textures with novel characteristics suitable for future spintronic and topological applications. Recent studies confirmed the room-temperature stabilization of skyrmions in ultrathin ferromagnets. However, such ferromagnetic skyrmions show undesirable topological effect, the skyrmion Hall effect, which leads to their current-driven motion towards device edges, where skyrmions could easily be annihilated by topographic defects. Recent theoretical studies have predicted enhanced current-driven behaviour for antiferromagnetically exchange-coupled skyrmions. Here we present the stabilization of these skyrmions and their current-driven dynamics in ferrimagnetic GdFeCo films. By utilizing element-specific X-ray imaging, we find that the skyrmions in the Gd and FeCo sublayers are antiferromagnetically exchange-coupled. We further confirm that ferrimagnetic skyrmions can move at a velocity of ~50 m s-1 with reduced skyrmion Hall angle, θSkHE ~20°. Our findings open the door to ferrimagnetic and antiferromagnetic skyrmionics while providing key experimental evidences of recent theoretical studies.   Page 3 of 24 Introduction Magnetic skyrmions are non-trivial topological objects1,2 that have been greatly highlighted recently, mainly due to their unique and fascinating topological characteristics suitable for future spintronic applications such as skyrmion-based racetrack memory.3–5 Magnetic skyrmions can be stabilized in the presence of a strong Dzyaloshinskii-Moriya interaction (DMI),6,7 which prefers non-collinear spin orientation between neighbouring magnetic moments. Recent investigations have revealed that, in structures where ultrathin ferromagnets are interfaced with large spin-orbit coupling materials, such as Ta/CoFeB/MgO,8 Pt/Co/Ir,9 Pt/Co/MgO,10 Pt/Co/Ta and Pt/CoFeB/MgO,11,12 the interfacial DMI can be strong enough to stabilize the chiral skyrmion structure even at room temperature. It has also been shown that skyrmions can move along magnetic tracks upon the injection of an electrical current,8,11,13,14 which indicates that the skyrmions can be adopted in practical devices.5 However, contrary to theoretical predictions, ferromagnetic skyrmions have shown relatively slow and pinning-dominated current-driven dynamic behaviours at room temperature.8,11,13–15 More seriously, ferromagnetic skyrmions exhibit an inevitable topology-dependent effect, namely, the skyrmion Hall effect,16–19 where the magnetic skyrmions do not move collinear to the current flow direction but acquire a transverse motion due to the appearance of a topological Magnus force acting upon the non-zero topological charge. The Magnus force is opposite for a ferromagnetic skyrmion whose topological charge (Q) is either Q = 1 or -1. To avoid this issue, theoretical studies have suggested that the skyrmion Hall effect can be suppressed by utilizing antiferromagnetically exchange-coupled skyrmions with Q = 1 and -1 in antiferromagnetic materials.20–23 Along with the recent interest in antiferromagnets resulting from their intrinsic ultrafast dynamics and insensitivity to disturbing magnetic fields,24 these advantages in skyrmion dynamics have generated an intense interest in antiferromagnetic skyrmion textures. However, since such an excitation behaves as if it were a simple bubble with Q = 0, it is quite difficult to experimentally verify if it is really a topological object. In this work we investigate a similar object in ferrimagnet multilayers. Namely, we consider skyrmion excitations in one magnetic layer and in the magnetic other layer. Since they are antiferromagnetically coupled, their topological charges are opposite. Let us refer to such a pair of skyrmions as a ferrimagnetic skyrmion. The distinctive nature is the topological Magnus forces are not balanced exactly because the magnetization is different between two layers. We predict that ferrimagnetic skyrmions show a small but nonzero skyrmion Hall effect, which we may use as a verification of the topological excitation. We Page 4 of 24 note a recent study reporting the stabilization of ferrimagnetic skyrmions in Fe/Gd multilayers,25 in which Bloch-type skyrmions are stabilized by long-range dipolar interactions. However, current-driven dynamics of these skyrmions cannot be used for actual applications, mainly due to the random chirality of each individual skyrmion that results in a complex and random current-driven motion.26,27 Thus, the actual observation of deterministic and efficient current-driven dynamics of chiral ferrimagnetic skyrmions remains elusive so far and needs to be pursued for the realization of skyrmion-based memory and logic computing devices.  Results X-ray microscopy observation of magnetic domains In our multilayer stack, the Pt heavy metal underlayer is used to induce a strong DMI that stabilizes chiral magnetic textures. Note that the [Pt(3 nm)/Gd25Fe65.6Co9.4(5 nm)/MgO(1 nm)]n structure with a large repetition number, n = 20, is used in this study, due to: i) the sizable internal demagnetization field that effectively drives as-grown the magnetization state into the multi-domain state (see Supplementary Fig. 1 and Supplementary Note 1 for its hysteresis behaviours and the top left panel in Fig. 1a for its as-grown multi-domain state), ii) a large amount of magnetic material that resulted in the enhanced magnetic contrast in our X-ray transmission measurement, and iii) promising skyrmion characteristics observed in the structure, which will be presented throughout this Article. Using vibrating sample magnetometry (VSM) measurements, we estimate the magnetization compensation point, TM, of our GdFeCo film to be > 450 K (see Supplementary Fig. 2 and Supplementary Note 2 for details). Hence, our magnetic devices remain at an uncompensated ferrimagnetic state throughout all the room-temperature X-ray measurements. Moreover, to correctly extract the magnetic parameters of our ferrimagnetic films, we have performed several experiments including ferromagnetic resonance (FMR), spin-torque FMR (ST-FMR), and X-ray magnetic circular dichroism (XMCD) spectroscopy measurements on a companion GdFeCo film grown on SiOx/Si substrate (see Supplementary Figs. 3-6, Supplementary Note 3 and Methods for detailed measurement descriptions and acquired parameter values). To reveal the nature of ferrimagnetic skyrmions in our system, we first performed element-specific scanning transmission X-ray microscopy (STXM) in the presence of an external perpendicular magnetic field, Bz. Figure 1a shows the STXM images of the domain structure in a patterned 2.5-µm-wide and 5-µm-long Pt/GdFeCo/MgO film with decreasing external magnetic field from Bz = 0 mT to Bz = -130 mT. The upper and lower panels show Page 5 of 24 corresponding STXM images taken at the absorption edges of Fe (L3-edge) and Gd (M5-edge), respectively. In these STXM images, dark and bright contrasts correspond to upward (+Mz) and downward (-Mz) magnetization directions, respectively. At Bz = 0 mT, a labyrinth stripe domain state with the average domain width of ~220 nm is achieved. Moreover, it is immediately obvious that STXM images at Fe- and Gd-absorption edges show opposite contrast, revealing their expected antiferromagnetic spin ordering within the GdFeCo alloy. Note that, since the measurements are conducted at room temperature, which is lower than the compensation point TM, the magnetic moment of Gd aligns parallel with the external magnetic field while the moment of Fe aligns in an anti-parallel fashion.28 As the magnetic field increases, fewer domains remain and magnetic configurations become less complex. Eventually, by reaching a magnetic field of Bz = -130 mT, we observe multiple isolated skyrmions, and it is evident that the Gd and Fe magnetic moments are still antiferromagnetically exchange-coupled within these skyrmions, therefore confirming that we observed ferrimagnetic skyrmions. The high spatial resolution (~25 nm) of STXM allows us to measure the diameter of the observed skyrmions as discussed in Supplementary Figs. 7-8 and Supplementary Note 4, and we find that the average skyrmion diameter is roughly ~180 nm, which is as small as the skyrmions found in ferromagnetic multilayers with a large DMI value, 1.5 ~ 2 mJ m-2, studied in Refs. [9,11,12]. We later confirm that our skyrmions exhibit a left-handed Néel-type chirality by observing their current-driven behaviours. Figure 1b schematically illustrates the orientation of the antiferromagnetically exchange-coupled internal magnetic moments within the observed ferrimagnetic skyrmion structure.  Current-driven behaviours of ferrimagnetic skyrmions Having established that ferrimagnetic skyrmions can form at a finite external field in this material, we next study their current-induced dynamics in the magnetic track. Figure 2a shows a schematic drawing of our ferrimagnetic track and electric contacts patterned on a 100-nm-thick Si3N4 membrane for transmission X-ray measurements. The actual scanning electron microscopy (SEM) micrograph of our device is also included, and two indicated areas within the image, (i) and (ii), are used to analyse current-driven skyrmion behaviours shown in Fig. 2b and 2c, respectively. In Fig. 2b and 2c, each STXM image, taken at Fe-edge, was acquired after injecting a single current pulse, with the various pulse amplitudes of between 4.90×1010 A m-2 ≤ |ja| ≤ 3.55×1011 A m-2 and the pulse duration of 5 ns (see Supplementary Fig. 9 and Supplementary Note 5 for details on the electronic connections and the actual pulse shape). Note that each skyrmion is colour-circled, and a single colour is used Page 6 of 24 for the same skyrmion throughout each sequence. The pulse polarity, defined by the electric current flow direction, is indicated in the figure as red and blue pulse-shaped arrows. It is also noteworthy that, while skyrmion core points +z and –z directions in STXM images in Fig. 2b and Fig. 2c, respectively, the effective core magnetization points -z and +z directions in Fig. 2b and Fig. 2c, respectively, because Gd moments are dominant in our material at room temperature as discussed above. Fig. 2b first shows a sequence of STXM images of skyrmions stabilized by a magnetic field of Bz = 145 mT near the left Au electrode on a magnetic track. In our field of view, there are initially two skyrmions, and as we inject leftward current pulses, additional two skyrmions appear while all of the skyrmions show homogeneous propagations. Moreover, it is noticeable that, when a train of the skyrmions propagates along the track, their alignment/trajectory shows a finite angle with respect to the current flow direction, which is the hallmark of the skyrmion Hall effect. We then reversed the magnetic field to Bz = -145 mT and investigated another region near the right Au electrode on the same track, as shown in Fig. 2c. While no skyrmion is observed in the field of view of the first image, up to four skyrmions appear with pulse injections, and all of them show pinning-free homogeneous displacements, as was observed for the other polarity of skyrmions in Fig. 2b. Note that a train of these bright skyrmions also shows transverse velocity component, and surprisingly, the sign of slope is opposite to the case of dark skyrmions. This is due to the opposite topological polarity of skyrmions for two cases, which experience the opposite sign of the topological Magnus force that consequently provides opposite transverse propagation directions. This observation of finite skyrmion Hall effect and the symmetry of skyrmion Hall angle in our ferrimagnetic material agrees well with the cases of ferromagnetic skyrmions in Ta/CoFeB/TaOx17 and Pt/CoFeB/MgO.18 Moreover, in the last image of each sequence, we observe the repulsion between skyrmions and the edges of the sample, which locates right underneath the field of views as shown in Fig. 2a. The repulsion occurs due to the DMI boundary condition,29 which results in the skyrmion motion i) back toward the sample center (blue-circled skyrmion in Fig. 2b) or ii) straight along the sample edge (yellow-circled skyrmion in Fig. 2b and blue-circled skyrmion in Fig. 2c), which also agrees with previously observed ferromagnetic skyrmion motion along the edge in Ref. [17]. With these observations shown in Fig. 2b and 2c, three important qualitative conclusions on the skyrmion physics within ferrimagnetic material can be drawn. First and most importantly, our investigation reveals that ferrimagnetic skyrmions can also be displaced by electric currents at room temperature just as the ferromagnetic skyrmions.8,11–Page 7 of 24 14,17,18 Moreover, we show that the skyrmion propagation direction is along the current flow direction (against the electron flow) for both +Mz-core and -Mz-core skyrmions, and this same directionality agrees well with the spin Hall current-driven motion of homochiral left-handed Néel-type hedgehog skyrmions stabilized by interfacial DMI in Pt/ferromagnet thin films.11–14,18 This implies that the interfacial DMI at the Pt/GdFeCo interface plays a crucial role in stabilizing skyrmions and also driving them on the track in our ferrimagnetic structure. Furthermore, the skyrmion pinning, which was often observed in many of ferromagnetic systems,11,13,14,17 is significantly reduced in our ferrimagnetic material. This low pinning may originate from the amorphous nature of GdFeCo because the absence of grain boundaries leads to lower skyrmion pinning, as was observed in amorphous CoFeB ferromagnetic films.11 Overall, it is noteworthy that our observation serves as the first experimental observation of current-driven excitation of nanoscale magnetic skyrmions in ferrimagnets. The pulse amplitude-dependent skyrmion velocity and its skyrmion Hall angle are plotted in Fig. 2d and 2e, respectively. To correctly calculate the distance and angle between two images, we have performed image-displacement correction using the edge between our magnetic track and Au electrode. It is first noticeable that skyrmion velocity increases linearly with pulse amplitudes, and the maximum velocity approaches ~50 m s-1 at |ja| = 3.55×1011 A m-2, which is comparable to the current state-of-the-art skyrmions observed in a few ferromagnetic heterostructures.11,14,18 Moreover, as shown in Fig. 2e, we observe a very small skyrmion Hall angle, |θSkHE| up to ~20°, which is far lower than the skyrmion Hall angles, |θSkHE| > 30°, observed for ferromagnetic skyrmions in Ta/CoFeB/MgO and Pt/CoFeB/MgO structures.17,18 Antiferromagnetic coupling between two sublayers and the corresponding largely-reduced net magnetization within GdFeCo films have led to the effective inhibition of the skyrmion Hall effect. It is noteworthy that relatively large skyrmion Hall effect in conventional ferromagnets may lead skyrmions toward device edges, where they could easily be annihilated by topographic defects.20,30 Moreover, the skyrmion Hall effect-driven strong transverse motion may pose a limitation for the maximum skyrmion speed due to the finite edge repulse. Therefore, we believe that our ferrimagnetic multilayers can serve as an important magnetic material for such future skyrmionic devices that could replace conventional ferromagnets with enhanced reliability and mobility. Note that skyrmion Hall angle increases monotonically at low current densities, |ja| < ~2×1011 A m-2, and saturates at high current densities, because the skyrmion dynamics is dominated by pinning sites at low driving forces, which is similar to the creep motion of ferromagnetic skyrmions in low-current-density regime caused by the pinning potential.15,17 The remnant finite Page 8 of 24 skyrmion Hall angle, θSkHE ~ 20°, results from the uncompensated magnetic moments between Gd and FeCo sublayers at 300 K < TM, where MS_Gd ≠ MS_FeCo. (see Methods). Therefore, by adjusting material compositions, it will be possible to further reduce the effective skyrmion Hall angle in ferrimagnetic GdFeCo films.  Micromagnetic simulation on ferrimagnetic skyrmion dynamics For more comparison, we simulated the current-driven dynamics of a ferrimagnetic skyrmion (see Methods for more modelling details) in a checkerboard-like two-sublayer spin system based on the G-type antiferromagnetic structure with simple square lattices,20,30,31 where the two sublayers, corresponding to Gd and FeCo, are coupled in a ferrimagnetic manner with a net saturation magnetization, while each sublayer is ferromagnetically ordered. We have also examined simulations using the two-sublattice model with classic J1-J2-J2’ Heisenberg exchange interactions as shown in Supplementary Figs. 10-12 and Supplementary Note 6. While we find that the intra-sublattice exchange interactions indeed affect on the ferrimagnetic skyrmion size and dynamics, however, the influence of these effects on the overall dynamics, especially on the skyrmion Hall effect, turns out not to be significant in both qualitative and quantitative results. Simulations were performed with both models: with and without pinning defects, using experimentally measured materials parameters given in Methods. Moreover, in simulations, we considered the error range of damping coefficient measurement shown in Supplementary Fig. 3 and Supplementary Note 3, because the ferrimagnetic skyrmion dynamics can be strongly influenced by small changes in damping coefficient (see Supplementary Figs. 13-14 and Supplementary Note 7 for details). Note that the error ranges of simulations are shown as shaded areas in Fig. 2d and 2e. The simulated skyrmion velocity as a function of the current density is first shown in Fig. 2d. Simulation results show qualitative and quantitative agreement with experimental observations, revealing that the ferrimagnetic skyrmion velocity is linearly proportional to the driving current density. We also calculated the skyrmion Hall angle as a function of the current density as shown in Fig. 2e. It is first noticeable that the skyrmion Hall angle in the model without pinning defects is independent of the current density, while the skyrmion Hall angle in the model with certain pinning defects increases with increasing current density and approaches a constant value calculated with the pinning-absent model. This linear increase is qualitatively consistent with our experimental observations and also with previous report17, indicating the existence of certain pinning effects due to impurities or defects in the real material. Moreover, the Page 9 of 24 calculated skyrmion Hall angle considering the damping errors shows a fair quantitative agreement with experimental observation. Although the averaged skyrmion Hall angle for simulation is still slightly larger than experiments, we speculate that the small difference may originate from the larger effective damping associated with skyrmion dynamics. Weindler et al. recently reported that local FMR (α = 0.0072) and domain wall dynamics measurements (α = 0.023) yield very different damping parameters for the same material, Permalloy, and magnetic texture-induced nonlocal damping may be responsible for the increase in effective damping.32 Gerrits et al. also reported that, unlike small-angle magnetization dynamics such as conventional FMR, large-angle magnetization dynamic could induce an increase in the apparent damping,33 and we believe this scenario could also be used to explain our case, where skyrmion motion involves the large-angle magnetization dynamics. Moreover, because a patterned 2.5-µm-wide and 5-µm-long nanowire structure was used for skyrmion study while continuous films were employed in material parameter analysis, roughness-induced extrinsic damping enhancement could be another source of damping increase.34 Nevertheless, by considering above possible scenarios, our quantitative results on the suppression of the skyrmion Hall effect can be reasonably understood.  Discussion We have so far observed that the current-driven behaviours of ferrimagnetic skyrmions are indeed attractive compared with their ferromagnetic counterparts. However, it may also be possible that the small value of skyrmion Hall angle in a ferrimagnetic material is only from the small saturation magnetization in a ferrimagnet and not from its antiferromagnetic characteristic. Thus, for a fair comparison between ferrimagnetic and ferromagnetic skyrmions, we have performed simulations on the current-driven dynamics of skyrmions with the same low (net) saturation magnetization, as shown in Fig. 3. It should first be pointed out that, by using the same material parameters, the sizes of the ferrimagnetic and ferromagnetic skyrmions are measured to be identical at certain out-of-plane magnetic fields (Fig. 3a, inset). Figure 3a shows the velocities of the current-driven ferrimagnetic and ferromagnetic skyrmions as a function of the driving current density. It can be seen that both the ferrimagnetic and ferromagnetic skyrmion velocities increase with increasing driving current density, where the mobility of ferrimagnetic skyrmion is slightly larger than that of ferromagnetic skyrmion. More significantly, as shown in Fig. 3b, the skyrmion Hall angle of ferrimagnetic skyrmion is significantly smaller than that of the ferromagnetic skyrmion by a Page 10 of 24 factor of 2, even the net saturation magnetizations are the same for both skyrmions. Note that skyrmion Hall angles are independent of the current densities for both cases, as the pinning effect is not considered in this comparison. To compare the effect of pinning on ferrimagnetic and ferromagnetic skyrmions more carefully, we have simulated the current-driven dynamics of both ferrimagnetic and ferromagnetic skyrmions in the same disorder model, which is shown in Supplementary Fig. 15 and Supplementary Note 8. It is found that the given pinning effect on the current-driven skyrmion dynamics is not significant for both ferrimagnetic and ferromagnetic cases, especially at a large driving current density (e.g. 5×1011 A m-2), however, at a small driving current density (e.g. 1×1011 A m-2), the ferrimagnetic skyrmion motion is more influenced by the pinning effect. The reason could be that the ferromagnetic skyrmion experiences a stronger Magnus force, which helps it in overcoming obstacles.35,36 Nevertheless, larger skyrmion velocity and smaller skyrmion Hall angle for ferrimagnetic case were maintained over the whole range of examined current densities. Overall, simulation results indicate that, even when the ferrimagnetic and ferromagnetic skyrmions have the same low (net) saturation magnetization, the current-driven dynamics of the ferrimagnetic skyrmion is much more reliable for the transport in narrow channels as information carriers. In conclusion, we have observed and studied the stabilization and current-driven dynamics of antiferromagnetically exchange-coupled skyrmions in ferrimagnetic GdFeCo films. By utilizing the element-specific X-ray imaging, we have identified that the ferrimagnetic skyrmion in the GdFeCo films consists of two antiferromagnetically exchange-coupled skyrmions in the Gd and FeCo sublayers. We further confirm that current-driven ferrimagnetic skyrmions can move at a velocity of ~50 m s-1 with reduced skyrmion Hall angle, |θSkHE| ~20°. With micromagnetic simulations, we reveal that ferrimagnetic skyrmions are much more attractive than their ferromagnetic counterparts in many technological-relevant aspects, such as larger skyrmion mobility and strongly suppressed skyrmion Hall effect, mainly due to their antiferromagnetic nature. Our findings reveal the promising dynamic properties of ferrimagnetic skyrmions, and highlight the possibility to build more reliable skyrmionic devices using ferrimagnetic and antiferromagnetic materials.   Page 11 of 24 Methods Sample preparation and experimental method The [Pt(3 nm)/Gd25Fe65.6Co9.4(5 nm)/MgO(1 nm)]20 films were grown by DC magnetron sputtering at room temperature under 1 mTorr Ar for Pt and GdFeCo and 4 mTorr Ar for MgO at a base pressure of roughly ~2×10-8 Torr. Samples were grown on a 100-nm-thick SiN substrate and then patterned using electron beam lithography and lift-off technique. Nominally sample films were grown on SiOx/Si substrate for vibrating-sample magnetometry (VSM), ferromagnetic resonance (FMR), spin-torque FMR (ST-FMR), asymmetric bubble expansion measurements. The series of measurements yielded material constants: anisotropic field µ0Hk = 0.15 T, net saturation magnetization MS = 2×105 A m-1, uniaxial anisotropy, Ku = 4.01×104 J m-3, damping coefficient, α = 0.205±0.035, spin Hall angle, θSH = 0.055, magnetization ccompensation ratio, 𝑛=𝑀!!\"#$𝑀!!\"=0.78 and DMI magnitude, D = -0.98 mJ m-2 (see Supplementary Note 3 for details).  All microscopy images were acquired using the STXM installed at the PolLux (X07DA) beamline of the Swiss Light Source (SLS) at the Paul Scherrer Institute in Villigen, Switzerland. The device used for the experiments was 2.5-µm-wide and 5-µm-long, which yielded an electrical resistance of ~57 Ohms measured in 2-point. This resistance reduced the impedance mismatch of the device, and allowed an almost complete transmission of 5-ns-long short electrical pulses across the device. This was verified by simultaneously measuring the injected (through a -20 dB pickoff T) and transmitted samples with a real-time oscilloscope. Pulse current densities above ~4×1011 A m-2 led to a damage of the Au contact, which eventually limited the maximum current applied in Fig. 2. Skyrmion velocities were determined using the total displacements, measured by acquiring XMCD-STXM images before and after the injection of the pulses, and the integrated pulse time. Three to ten displacements were recorded for each pulse amplitude, and the average value and standard deviations of the individual velocity measurements are plotted in Fig. 2d. Current densities were calculated by dividing the injected current with stripe width and effective total thickness of Pt and GdFeCo.  Simulation method The spin dynamics simulation is carried out by using the Object Oriented MicroMagnetic Framework (OOMMF) with the home-made extension modules for the periodic boundary condition.37 The model is treated as a checkerboard-like two-sublattice Page 12 of 24 spin system based on the G-type antiferromagnetic structure with simple square lattices, where the two sublattices are coupled in a ferrimagnetic manner with a net spontaneous magnetization, while each sublattice is ferromagnetically ordered. The Hamiltonian is based on the classical Heisenberg model, given as ℋ=−𝐽!\"𝐒!∙𝐒!!!,!!−𝐽!\"!𝐒!∙𝐒!≪!,!≫!−𝐽!\"!𝐒!∙𝐒!≪!,!≫!+𝐷𝐮!\"×𝑧∙𝐒!×𝐒!!!,!! −𝐾𝐒!!!!−𝜇!𝐒!∙𝑯!+𝐻!!\"                                           (1) where 𝐒! represents the local spin vector reduced as 𝐒!=𝑴!𝑀!! at the site i, and 𝐒! represents the local spin vector reduced as 𝐒!=𝑴!𝑀!! at the site j. 𝑴! and 𝑴! are the magnetization at the site i and j, respectively. 𝑀!! denotes the saturation magnetization of the sublattice i, while the saturation magnetization of sublattice j is defined as 𝑀!!=𝑛𝑀!! with the compensation ratio n. <i, j> runs over all the nearest-neighbor sites in the two-sublattice spin system. <<i, j>>A and <<i, j>>B run over all the nearest-neighbor sites in the sublattice A and sublattice B, respectively (see Supplementary Fig. 10). 𝐽!\" is the exchange coupling energy constant between the two adjacent spin vector 𝐒! and 𝐒!, which has a negative value (𝐽!\"<0) representing the antiferromagnetic spin ordering of the two sublattices. 𝐽!\"! and 𝐽!\"! are the exchange coupling energy constants for the sublattice A and sublattice B, respectively, which are positive numbers (𝐽!\"!>0, 𝐽!\"!>0) representing the ferromagnetic intra-sublattice coupling. 𝐷 is the interface-induced DMI constant, 𝐮!\" is the unit vector between spins 𝐒! and 𝐒!, and 𝑧 is the interface normal, oriented from the heavy-metal layer to the ferrimagnetic layer. 𝐾 is the perpendicular magnetic anisotropy (PMA) constant, H is the applied magnetic field, and 𝐻!!\" stands for the dipole-dipole interaction, i.e., the demagnetization effect. The time-dependent dynamics of the spin system is controlled by the Landau-Lifshitz-Gilbert (LLG) equation augmented with the damping-like spin Hall torque, which is expressed as !𝐒!!\"=−𝛾!𝐒!×𝐇!\"\"+𝛼𝐒!×!𝐒!!\"+𝜏𝐒!×𝑝×𝐒!                (2) where 𝐇!\"\"=−1𝜇!𝑀!!∙𝛿ℋ𝛿𝐒! is the effective field on a lattice site, 𝛾! is the Gilbert gyromagnetic ratio, and α is the phenomenological damping coefficient. The coefficient for the spin Hall torque is given as 𝜏=𝛾!ℏ𝑗𝜃!\"2𝜇!𝑒𝑀!!𝑏, where 𝑗 is the applied charge current density, 𝜃!\" is the spin Hall angle, and 𝑏 is the thickness of the ferrimagnetic layer. 𝑝=𝒋×𝒛 denotes the spin polarization direction. Page 13 of 24 For the simulation on the multilayer structure, we employed an effective medium approach11 with the lattice constant of 5 Å, which improves the computational speed by converting the multilayer into a two-dimensional effective model with reduced parameters. The intrinsic magnetic parameters used in the simulation are measured from our experimental samples as well as adopted from Refs. [11,18,38]: the damping coefficient α = 0.205 ± 0.035, the inter-sublattice exchange stiffness AGd-Fe = -10 pJ m-1, AGd-Gd = 5 pJ m-1, AFe-Fe = 5 pJ m-1, the spin Hall angle θSH = 0.055, the DMI constant D = -0.96 mJ m-2, the PMA constant Ku = 4.01×104 J m-3, and the net saturation magnetization 𝑀!=𝑀!!\"−𝑀!!\"=200 kA m!!. The compensation ratio is measured as 𝑛=𝑀!!\"𝑀!!\"=0.78. In the experimental multilayer system, the thickness of one ferrimagnetic layer is tm = 5 nm, the thickness of one repetition is tr = 9 nm, and the number of repetitions is nrep = 20. For the simulation on the model with pinning defects, the defects with the size of 10 Å × 10 Å and a higher PMA (Kp = 5Ku) are randomly distributed in the whole ferrimagnetic layer. The density of the defects in the whole model equals 5 %.  Data availability Data supporting the findings of this study are available within the article and its Supplementary Information files and from the corresponding author upon request. The micromagnetic simulator OOMMF used in this work is publicly accessible at http://math.nist.gov/oommf.   Page 14 of 24 References 1. Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006). 2. Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915–919 (2009). 3. Jonietz, F. et al. Spin Transfer Torques in MnSi at Ultralow Current Densities. Science 330, 1648–1651 (2010). 4. Yu, X. Z. et al. Skyrmion flow near room temperature in an ultralow current density. Nat. 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Phys. Rev. Lett. 109, 96602 (2012).    Page 18 of 24 Acknowledgements This work was primarily supported by Samsung Research Funding Center of Samsung Electronics under Project Number SRFC-MA1602-01. Part of this work was performed at the PolLux (X07DA) beamline of the Swiss Light Source. S.W. and J.W.C. acknowledge the support from KIST Institutional Program. K.M.S acknowledges the support from the Sookmyung Women's University BK21 Plus Scholarship. X.Z. was supported by JSPS RONPAKU (Dissertation Ph.D.) Program. Y.Z. acknowledges the support by the President's Fund of CUHKSZ, the National Natural Science Foundation of China (Grant No. 11574137), and Shenzhen Fundamental Research Fund (Grant Nos. JCYJ20160331164412545 and JCYJ20170410171958839). M.E. acknowledges the support by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP17K05490, 25400317 and JP15H05854), and also the support by CREST, JST (Grant No. JPMJCR16F1). S.F acknowledges the support by the EU Horizon 2020 MAGicSky project (Grant No. 665095). K.M.S. and J.W.C. acknowledge the travel fund supported by the National Research Foundation of Korea (NRF) funded by the MSIP (2016K1A3A7A09005418). S.-Y.P. and B.-C.M. acknowledge the support from the National Research Council of Science & Technology (NST) grant (No. CAP-16-01-KIST) by the Korea government (MSIP). S.-Y.P. also acknowledges the support from KBSI Grant (D37614). S.W. also acknowledges S. Emori for his helpful comments on the manuscript.  Author Contributions S.W. designed and initiated the study. K.M.S. optimized structure, fabricated devices and performed the film characterization. S.W., K.M.S., S.F. and J.R. performed X-ray imaging experiments using STXM at Swiss Light Source in Villigen, Switzerland. X.Z. and Y.Z. performed the numerical simulations. M.E. carried out the theoretical analysis. During the revision of this article, N.J.L., S.-I.K. and S.-Y.P. performed ferromagnetic resonance (FMR), D.L. and O.L. performed spin-torque FMR, and S.W., K.M.S., Y.H.K., J.-Y.K. and J.W.C. performed XMCD spectroscopy at 2A beamline at Pohang Accelerator Laboratory in Pohang, Korea. S.W., X.Z. and M.E. drafted the manuscript and revised it with assistance from X.L., D.L., O.L., S.-Y.P., J.W.C., B.-C.M., H.C.K. and J.C.. All authors commented on the manuscript.  Competing Financial Interest The authors declare no competing financial interests. Page 19 of 24  Author Information S.W. and K.M.S. contributed equally to this work. Correspondence and requests for materials should be addressed to S.W. (shwoo_@kist.re.kr). Page 20 of 24 Figure Legends  Figure 1. Scanning transmission X-ray microscopy (STXM) imaging of domain structure upon magnetic field application. a, STXM images acquired by sweeping the external perpendicular magnetic field from Bz = 0 mT to Bz = -130 mT. Dark and bright contrasts correspond to magnetization oriented up (along +z) and down (along -z), respectively. Upper panel and lower panel show corresponding images acquired at the L3 and M5 absorption edges of Fe and Gd, respectively. Note that, due to longer penetration depth associated with the higher energy used for Gd, ~1189 eV, compared with that of Fe, ~709 eV, magnetic contrast under Au electrodes is visible for Gd magnetic moment imaging. b, Schematic of antiferromagnetically exchange-coupled ferrimagnetic skyrmion on a magnetic track as observed in our GdFeCo films as indicated in the red dashed-square boxes in the last image of a. Scale bar, 1 µm.  Figure 2. Current-driven behaviour of ferrimagnetic skyrmions and their velocity and skyrmion Hall effect. a, Schematic of scanning transmission X-ray microscopy (STXM) geometry, and a scanning electron microscopy (SEM) image of the actual device used for experiments. Scale bar, 2 µm. Sequential STXM images taken at Fe-edge showing the responses of multiple skyrmions after injecting unipolar current pulses along the track at b, Bz = 145 mT and c, Bz = -145 mT, respectively. With a fixed pulse-length of single pulse, 5 ns, the pulse amplitude is changed between 4.90×1010 A m-2 ≤ |ja| ≤ 3.55×1011 A m-2. Pulse polarities are indicated as red- and blue-coloured arrows inside each image. Within STXM images, the same skyrmion is indicated with the same colour. Scale bars, 500 nm. d, Experimental and simulated average skyrmion velocity of Pt/GdFeCo/MgO versus current density. e, Experimental and simulated average skyrmion Hall angle of Pt/GdFeCo/MgO versus current density. In d-e, The shaded areas in plots represent the simulation results considering the damping coefficient error ranges of α = 0.205 ± 0.035, which was measured experimentally as described in Supplementary Fig. 3 and Supplementary Note 3. Note that pulse current densities above ~4×1011 A m-2 led the damage of the Au contact, which eventually limited the maximum applicable current to our sample. Error bars denote the standard deviation of multiple measurements.    Page 21 of 24 Figure 3. Comparison between the current-driven dynamics of ferrimagnetic and ferromagnetic skyrmions of the same net saturation magnetization. (a) The ferrimagnetic and ferromagnetic skyrmion velocities as a function of the driving current density. Inset shows the close-up top view of the ferrimagnetic skyrmion in a square film with periodic boundary conditions in both x and y directions. The lattice constant is set as 5 Å. (b) The ferrimagnetic and ferromagnetic skyrmion Hall angle as a function of the driving current density. The net saturation magnetization was set to be MS = 2×105 A m-1 for both cases. Scale bar, 5 nm.    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